Mathematical glossary

Abel summation evaluates a power series just inside its radius of convergence and then takes the limit as the variable approaches the endpoint.

Instead of adding terms a_n directly, form the convergent series of a_n x^n for x a little below 1, then ask what happens as x rises toward 1. For 1 − 2 + 3 − 4 + ..., the closed form x/(1 + x)^2 tends to 1/4. Substituting x = 1 into the closed form is not part of the definition, even when it happens to give the right number. Abel summation can succeed where ordinary and Cesàro summation fail, and it can still fail for series such as 1 + 2 + 3 + ....

Example The Abel sum of 1 − 2 + 3 − 4 + ... is 1/4.

Connected ideas Geometric series, Divergent series, Regularisation, Analytic continuation, Cesàro summation

Explore it The Sum That Is Not a Sum

The absolute value of a real number is its distance from zero, so it is never negative.

On a number line, −5 and 5 are both five units from zero. Vertical bars record this distance: |−5| = |5| = 5. The absolute difference |a−b| measures how far two real numbers are apart without caring which comes first. That makes it useful when a process compares neighbouring values. For complex numbers, the same distance idea extends to the plane and is often called modulus.

Example |3−8| = 5 and |8−3| = 5.

Connected ideas Modulus, Squared error

Explore it Differences Around a Circle, Gilbreath’s Difference Field

An absorbing barrier is a place where a random walk stops for good once it arrives, so the walk can never leave it.

Most random walks can wander forever, but a walk with an absorbing barrier is over the first time it touches one. With two barriers every game ends in one of exactly two ways, and the question becomes how the probability divides between them and how long the walk takes. That the walk does end has to be shown rather than assumed: here it follows because a long enough run of steps in one direction always has a positive chance.

Example A bankroll of 0 is absorbing: with no money left, no further round can be staked.

Connected ideas Random walk, Markov chain, Probability, Gambler’s ruin

Explore it The Gambler’s Ruin

An aliquot sequence repeatedly replaces a number with the sum of its proper divisors.

The map links divisor structure to iteration. Perfect numbers are fixed points, amicable numbers form two-cycles and other starts may grow, fall or enter a longer loop. Many questions about termination and boundedness remain open, so computed terms need an explicit starting value and limit. Proper divisors here are positive divisors smaller than the number itself. The number 1 has none, so its divisor sum is zero and this laboratory ends the chain.

Example The proper-divisor sum of 12 is 1 + 2 + 3 + 4 + 6 = 16.

Connected ideas Proper divisor

Explore it Sum the Divisors and Go Again

Two distinct positive integers are amicable when each is the sum of the other’s positive proper divisors.

The proper divisors of 220 add to 284, and the proper divisors of 284 add to 220. The aliquot procedure therefore alternates between the two instead of stopping at a single fixed value. Requiring distinct numbers separates an amicable pair from a perfect number repeating itself. Other starting values can enter the same pair after several steps, but that does not make each of those earlier values an amicable number.

Example 220 → 284 → 220 forms a two-number cycle.

Connected ideas Proper divisor, Cycle, Perfect number

Explore it Sum the Divisors and Go Again

Analytic continuation extends a function beyond its original formula while preserving agreement where both are defined.

The extension is forced locally by the function’s analytic behaviour, rather than chosen independently at each point. A continuation can exist at values where the original series diverges, but evaluating it there is not the same as summing that series term by term. This distinction is central to careful discussions of regularisation. Analytic means locally expressible by a convergent power series. Requiring this structure is essential: without it, merely extending a function can be done in many arbitrary ways.

Example The zeta function continues beyond its initial series domain.

Connected ideas Regularisation, Divergent series, Cesàro summation, Abel summation, Riemann zeta function

Explore it The Sum That Is Not a Sum

An angle measures the amount of turn between two directions meeting at a point.

Opening a pair of scissors changes the angle between its blades without changing their lengths. A full turn is 360 degrees, and a right angle is 90 degrees. Radians measure the same turn using the arc length on a unit circle. When a drawing places n points evenly around a circle, neighbouring points are separated by 360/n degrees. This angular spacing is different from the length of the straight chord between them.

Example Ten equally spaced rim points have a 36-degree angle between neighbours at the centre.

Connected ideas Radian, Circular arc, Unit circle, Rotational symmetry

Explore it Times Tables on a Circle, Tiles That Never Repeat

A pattern is aperiodic when it has no non-zero translation that leaves the entire pattern unchanged.

Aperiodic does not mean random or without order. Penrose tilings, for example, use local matching rules that force long-range structure while forbidding a repeating lattice. Any finite window can look compatible with repetition, so the claim concerns every possible translation of the infinite object. A tile set is called aperiodic in a stronger sense when it admits tilings but forces every allowed tiling to be nonperiodic. Matching rules matter to this distinction.

Example A Penrose tiling is aperiodic.

Connected ideas Tiling, Golden ratio, Periodic sequence

Explore it Tiles That Never Repeat

The arithmetic mean of a non-empty list is its sum divided by the number of values.

To average 2, 4 and 9, add them to get fifteen, then divide by three to get five. The mean describes a balance point and need not be one of the observed values. Large outliers can move it substantially. A sample mean is an arithmetic mean computed from observed data; an expected value is an average under a probability model, which may weight possible outcomes unequally.

Example The arithmetic mean of 2, 4 and 9 is 5.

Connected ideas Expected value, Variance, Estimator, Probability, Standard deviation

Explore it Three Guesses Beat One, The Sum That Is Not a Sum

An arithmetic progression is a list of numbers in which each term exceeds the one before it by a fixed amount.

The progression 3, 7, 11, 15, 19 starts at three and adds four each time, so its terms are exactly the whole numbers that leave a remainder of three on division by four. Describing a progression that way, as a starting remainder together with a step, turns a question about spacing into a question about remainders. Dirichlet proved that whenever the starting value and the step share no factor above one, the progression contains infinitely many primes. A progression whose start and step do share a factor contains at most one prime.

Example 3, 7, 11, 15 starts at 3 and steps by 4.

Connected ideas Prime number

Explore it The Prime Race That Fixes Its Own Result

An attractor is a state or collection of states that nearby or associated paths approach over time.

In the Kaprekar map, many different starts eventually arrive at 6174 and remain there. Other rules lead into repeating cycles, and continuous systems can approach more complicated sets without reaching them exactly. An attractor describes how surrounding paths behave, not just where one path ends. A fixed point that repels nearby starts is therefore not attracting. The collection of starts drawn to an attractor is its basin.

Example 6174 is an attractor for most four-digit Kaprekar starts.

Connected ideas Cycle, Fixed point, Basin, State space, Iterated function system

Explore it The 6174 Attractor, Happy Numbers, The Collatz Orbit Atlas

The average waiting time for a pattern is the mean number of flips, over many runs, before the pattern first appears.

Flip a fair coin until HHT first appears, count the flips, and repeat many times. The average of those counts is the waiting time, and for HHT it is 8. HTH takes 10 and HHH takes 14, because a pattern that can overlap itself tends to wait longer. The average describes runs where the pattern is alone. It does not say which of two patterns comes first when they race, and two patterns with the same average wait, such as HHT and THH, can race three to one.

Example A fair coin takes 8 flips on average to show HHT and 14 to show HHH.

Connected ideas Expected value, Probability

Explore it The Coin Bet You Can Always Beat

An axiom is a starting statement accepted within a mathematical system, from which other statements can be proved.

A proof does not begin from nothing: it uses stated assumptions and rules for drawing conclusions. Axioms specify some of those starting assumptions. Different axiom choices can describe different structures, so calling something an axiom does not make it an unquestionable claim about the physical world. In logic, it matters whether the axioms can be generated by an algorithm and whether they can lead to a contradiction. These are properties of the chosen system.

Example A geometry can take the existence of a unique line through two distinct points as an axiom.

Connected ideas Proof, Theorem, First incompleteness theorem

Explore it A Sentence About Its Own Number

A basin of attraction contains the starting states whose paths approach a particular attractor.

Imagine marking every start that heads to the same destination. Those starts form its basin, even when their paths take different numbers of steps or merge along the way. In a finite deterministic system, approaching an attractor means eventually entering its fixed point or cycle. In a continuous system, a path may instead get arbitrarily close without ever arriving exactly. The rule and allowed starting states determine the basin.

Example The 6174 basin contains the starts that reach 6174.

Connected ideas Cycle, Fixed point, Attractor, State space

Explore it The 6174 Attractor, Happy Numbers

Benford’s law is the observation that a first digit d turns up about log(1 + 1/d) of the time, the logarithm being taken in the base the numbers are written in.

Frank Benford tabulated first digits across twenty thousand collected values in 1938 and found them weighted towards the small ones. The reason is visible on a multiplying scale: the numbers beginning with 1 occupy a stretch a factor of two wide, while those beginning with 9 occupy one only a ninth wider than itself. Provable cases exist, including the powers of two and the Fibonacci numbers, where the argument reduces to how evenly a sequence wraps a unit interval. There is no theorem that a given dataset must comply, and collections with a fixed width, a floor or an assigned origin routinely do not.

Example Of the first thousand powers of two, 301 begin with a 1 and 45 begin with a 9.

Connected ideas Logarithm, Equidistribution, Uniform distribution, Law of large numbers, Proof by exhaustion

Explore it Why One Comes First

An estimator’s bias is the difference between its expected estimate and the quantity it aims to estimate.

Imagine repeating the same measurement procedure many times with fresh data. If its average estimate differs from the true target, the procedure has statistical bias. Unbiased means that difference is zero, not that each answer is correct or especially accurate. A biased estimator can still have lower mean squared error if it reduces variability enough. The shrinkage exploration demonstrates this trade-off rather than treating unbiasedness as the only possible measure of quality.

Example If a procedure averages 12 when the true target is 10, its bias is +2.

Connected ideas Estimator, Variance, Squared error, Chebyshev bias, Consecutive-prime bias

Explore it Three Guesses Beat One

A bijection is a correspondence in which every item on each side matches exactly one item on the other.

It is both injective and surjective, so the correspondence has an inverse. Bijections prove that two finite sets have the same size without listing a numerical formula directly. They are also useful for translating a difficult object into a representation with a simpler rule. For infinite collections this becomes a definition of equal size. Pairing each non-negative integer n with the even number 2n shows that these two collections have the same cardinality.

Example Reordering a list gives a bijection between positions before and after the reorder.

Connected ideas Function, Injective, Surjective, Countable

Explore it A Sentence About Its Own Number, Counting Around the Circle, Pascal Modulo m

Binary is a way to write numbers using only 0 and 1, with place values that double to the left.

Decimal notation uses ones, tens and hundreds. Binary uses ones, twos, fours, eights and so on. A 1 includes that place value and a 0 leaves it out: 1101 means one eight, one four, no twos and one one. The number is still thirteen; only its spelling has changed. Pascal’s triangle and the Josephus problem reveal patterns that are easier to see in this spelling.

Example 1101₂ = 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 13.

Connected ideas Binary expansion, Decimal, Exponent, Number base, Power of two

Explore it Counting Around the Circle, Pascal Modulo m

A binary expansion expresses a number as a sum of powers of two.

For non-negative integers, the places from right to left are 1, 2, 4, 8 and so on, and each digit is zero or one. This finite representation is unique apart from leading zeroes. Fractional places use halves, quarters and eighths. Fractions introduce a subtlety: a terminating expansion can also end in repeating ones, just as 1.000... and 0.999... denote the same decimal number.

Example 0.101₂ = 1/2 + 0/4 + 1/8 = 5/8.

Connected ideas Binary, Decimal, Exponent

Explore it Counting Around the Circle, Pascal Modulo m

A binomial coefficient counts selections of a given size when order does not matter.

Suppose five friends want to form a pair. There are five choices for the first person and four for the second, but this counts each pair twice, once in each order. Dividing twenty by two gives ten pairs. We write this as C(5,2), read “five choose two”. More generally, C(n,k) counts k-item selections from n distinct items and appears in row n of Pascal’s triangle.

Example C(5,2) = 5 × 4 / 2 = 10 different pairs.

Connected ideas Pascal's triangle, Combination, Coefficient, Triangular number, Combinatorics

Explore it Pascal Modulo m

The birthday problem asks how likely it is that at least two people in a group share a birthday under a stated model.

With independent birthdays uniformly assigned to 365 dates, a group of 23 people already has slightly more than a fifty percent chance of a match. The comparison includes every pair, not just pairs involving you. There are 253 pairs in that group. Counting pairs helps explain the surprise, but adding their match probabilities overcounts rooms with several matches. Multiplying the conditional probabilities of avoiding earlier dates gives the correct calculation.

Example Under the 365-day model, 23 people give a match probability of about 50.73%.

Connected ideas Binomial coefficient, Conditional probability, Complementary event, Uniform distribution, Pigeonhole principle

Explore it Birthday Matches in a Small Room

A bit is a binary digit, either 0 or 1.

A bit can record a two-way choice, such as off or on, without requiring that the symbols form a number. In the Thue-Morse sequence, each position holds one bit and the construction flips zeroes to ones and ones to zeroes. When a row of bits is interpreted as a binary integer, position supplies place value. The same row can instead be treated simply as an ordered pattern of two symbols.

Example 0110 contains four bits; its bitwise complement is 1001.

Connected ideas Binary, Bitwise complement, Sequence, Concatenation

Explore it A Pattern Built by Flipping Its Copy, Pascal Modulo m

The bitwise complement of a fixed string replaces every 0 with 1 and every 1 with 0, preserving the positions.

Starting from 0110, flip each symbol in place to obtain 1001. This is not reversal: reversal changes the order of positions, while complementation changes the symbols at those positions. Applying the complement twice restores the original string. The string length must be fixed when interpreting the operation numerically, since leading zeroes become ones. The Thue-Morse construction appends the complement of its current row at each doubling.

Example 0110 → 1001 → 0110 after two complements.

Connected ideas Bit, Concatenation, Binary, Thue-Morse sequence

Explore it A Pattern Built by Flipping Its Copy

Cantor pairing is a bijection that encodes an ordered pair of non-negative integers as one integer.

It walks the integer lattice along diagonal bands, assigning one number to each pair. The construction shows that a two-dimensional countable collection can be listed in one dimension without collisions. That idea is a practical ingredient in the toy formal language used by the Gödel exploration. The first pairs are (0,0), (1,0), (0,1), (2,0), (1,1) and (0,2). Reading along successive diagonals eventually visits every non-negative pair exactly once.

Example The pairing function π(a,b) = (a+b)(a+b+1)/2 + b is one standard form.

Connected ideas Countable

Explore it A Sentence About Its Own Number

A cardioid is a heart-shaped curve with one pointed indentation, called a cusp.

In the circle exploration, joining each numbered position to twice its number produces straight chords. With enough positions, their pattern suggests a curved boundary: a cardioid. The curve is an envelope of the chords, not one of the lines you drew. Its single inward point distinguishes it from a circle. Another construction traces a point on a circle rolling without slipping around the outside of an equally sized circle.

Example Multiplier 2 gives a cardioid; multiplier 3 gives the two-cusped nephroid.

Connected ideas Cusp, Envelope, Nephroid, Epicycloid

Explore it Times Tables on a Circle

A cellular automaton is a grid of cells updated in parallel by a local rule.

Each cell has a state and a fixed neighbourhood. The rule reads the old neighbourhood states and gives each cell its next state, with all cells updated together. Simple decisions can produce travelling shapes or apparently disordered patterns. Langton’s ant is more specifically a moving agent, often called a turmite: it changes one square at a time. It can be encoded as a cellular automaton by including the ant in cell states.

Example In Conway’s Game of Life, an empty cell with exactly three live neighbours becomes live.

Explore it The Ant That Builds a Road

Cesàro summation assigns a value to a series by taking the limit of the running averages of its partial sums.

Ordinary convergence asks whether the partial sums themselves settle. Cesàro summation asks whether their averages settle, which can succeed when the sums keep oscillating. For Grandi’s series 1 − 1 + 1 − 1 + ..., the partial sums flap between 1 and 0 while the averages close on 1/2. The method still fails for 1 + 2 + 3 + ..., whose averages grow without bound. A Cesàro value is therefore a property of the averaged sequence, not a claim that the original series converges.

Example The Cesàro sum of 1 − 1 + 1 − 1 + ... is 1/2.

Connected ideas Partial sum, Divergent series, Regularisation, Analytic continuation, Abel summation

Explore it The Sum That Is Not a Sum

Chebyshev bias is the tendency of primes leaving one remainder to stay ahead in the count for very long stretches.

Count the primes up to ten million by their remainder on division by four, and the remainder-3 team is ahead almost the whole way, even though the two counts finish within a few hundred of each other. The lead does change hands. Littlewood proved the difference changes sign infinitely often, and the first change modulo four happens at the prime 26,861. Rubinstein and Sarnak measured how much of the time each side is ahead, in a logarithmic sense, obtaining roughly 0.9959 for the remainder-3 team, conditionally on two unproved hypotheses.

Example Below ten million, the remainder-1 team leads over only two short stretches.

Connected ideas Prime counting function, Bias, Natural density, Consecutive-prime bias

Explore it The Prime Race That Fixes Its Own Result, Primes Dislike Repeating Themselves

A chord is a line segment whose endpoints lie on a circle.

A diameter is the special chord passing through the centre. Chords divide a circle into arcs and, when several are drawn, into regions. Their lengths depend on the endpoint separation, which makes a simple “connect every pair” rule a useful bridge between geometry and counting. Moving one endpoint around the rim changes the chord’s length even when the circle stays fixed. Doubling an endpoint label does not double the physical length.

Example A diameter is the longest chord of a circle.

Connected ideas Polygon, Envelope, Radius, Diameter, Circumference

Explore it Times Tables on a Circle, 1, 2, 4, 8, 16, and Then Not 32, Two Ways to Corner π

A circular arc is a portion of a circle’s rim; its arc length measures distance along that curved portion.

Choose two points on a circle. Following the rim from one to the other gives an arc, while going straight between them gives a chord. There are two possible arcs between distinct endpoints unless you specify a direction or choose the shorter one. For a circle of radius r, an arc sweeping through an angle θ radians has length rθ. Arc length is also used more generally for distance along other curves.

Example A quarter-circle arc has one quarter of the full circumference.

Connected ideas Chord, Circumference, Radian, Angle

Explore it Times Tables on a Circle, Two Ways to Corner π

The circumference is the distance all the way around a circle, equal to 2π times its radius.

Imagine wrapping a string once around a circular lid, then straightening it. The string length is the circumference. Unlike a diameter, it measures along the curved rim rather than straight across. The ratio of circumference to diameter is always π for a Euclidean circle, whatever its size. Numbering positions around a circle divides this rim into equal arcs, while the connecting chords take straight shortcuts through the interior.

Example A circle of radius 1 has circumference 2π, approximately 6.28.

Connected ideas Radius, Diameter, Circular arc, Chord, Pi

Explore it Times Tables on a Circle, Two Ways to Corner π

A coefficient is a number or expression multiplying a chosen variable or term.

In 3x + 5, the coefficient of x is three: each copy of x contributes three times its value. In 2x² + 7x + 1, the coefficients of x² and x are two and seven. A missing power can be treated as having coefficient zero. Pascal’s triangle supplies the binomial coefficients appearing when powers of a sum are expanded, connecting a visual arrangement of numbers with algebraic multiplication.

Example In (1 + x)² = 1 + 2x + x², the coefficient of x is 2.

Connected ideas Variable, Polynomial, Binomial coefficient

Explore it Pascal Modulo m, The Spiral, and What It Is Not

A combination is a selection of items where order does not matter.

Choosing a committee is a combination; seating that committee is a permutation. The number of k-element combinations from n items is the binomial coefficient C(n,k). Keeping the distinction clear prevents a common counting error and explains why Pascal’s triangle contains so many familiar totals. Choosing Alice and Ben gives the same pair as choosing Ben and Alice. Counting both orders would count the same selection twice, so that duplication must be removed.

Example There are ten pairs in a five-item set.

Connected ideas Pascal's triangle, Binomial coefficient, Permutation, Factorial, Combinatorics

Explore it Pascal Modulo m, 1, 2, 4, 8, 16, and Then Not 32

Combinatorics studies discrete arrangements, selections and patterns, including how many possibilities there are.

Choosing a committee, arranging objects in order and counting paths through a grid are combinatorial problems. The first task is often to decide whether order matters and whether repetitions are allowed. Counting with the wrong interpretation can give a perfectly correct answer to a different question. Pascal’s triangle records counts of selections, while circle-region formulas count geometric arrangements under particular hypotheses. Large counts can arise even when the individual objects and rules are simple.

Example Choosing 2 of 4 people gives 6 possible unordered pairs.

Connected ideas Combination, Permutation, Binomial coefficient

Explore it Pascal Modulo m, 1, 2, 4, 8, 16, and Then Not 32, A Number With No Room to Be Written

The complementary event consists of every allowed outcome in which the original event does not occur.

Either at least two people share a birthday or all their birthdays are different. These possibilities cannot both happen, and together they cover every room in the model. Their probabilities therefore add to one. It is often easier to count the all-different rooms and subtract their probability from one than to count every possible kind of match, especially when a room can contain several matching pairs.

Example P(at least one match) = 1 − P(all birthdays different).

Connected ideas Probability, Sample space, Conditional probability, Birthday problem

Explore it Birthday Matches in a Small Room

A complex number has the form a + bi, where a and b are real numbers and i² = −1.

Instead of locating a number only along a line, use a plane. The real part a gives its horizontal coordinate and the imaginary part b its vertical coordinate. Adding complex numbers adds their corresponding coordinates. Multiplication follows the usual distributive rule together with i² = −1. This makes multiplication capable of describing both scaling and turning. Real numbers are included as the points with imaginary part zero, so complex arithmetic extends rather than replaces real arithmetic.

Example (2 + i) + (1 + 3i) = 3 + 4i, represented by the point (3, 4).

Connected ideas Imaginary unit, Vector, Euler’s formula

Explore it The Number Behind Continuous Growth

A composite number is an integer greater than one with a divisor other than one and itself.

Composite numbers can be split into smaller factors, and their prime factorisation records those building blocks. One is neither prime nor composite. In experiments with primes, composites form the background against which gaps, patterns and failed candidates become visible. A rectangle of fifteen counters can have three rows of five. That arrangement exhibits the factors directly, whereas seven counters allow only one row or one column.

Example 15 is composite because 3 divides it.

Connected ideas Prime number, Factorisation, Divisor, Fermat number

Explore it Primes That Survive Every Shuffle, Every Even Number, Two Primes

Compound growth applies each percentage increase to the updated amount, including previous increases.

Begin with 100 counters and increase the pile by ten percent. You now have 110. A second ten percent increase adds eleven counters, not ten, because it uses the new total. Repeating a fixed percentage therefore multiplies by the same factor at every step. For a fixed total rate and duration, dividing the growth into more frequent updates changes the result. The ideal limit of continuously applied proportional growth is expressed using powers of e.

Example Two increases of 10% give 100 × 1.1 × 1.1 = 121, not 120.

Connected ideas Euler’s number, Exponential function, Ratio

Explore it The Number Behind Continuous Growth

Concatenation joins strings end to end without adding their numerical values.

Putting the string 0110 immediately before 1001 gives 01101001. The result has eight positions because the lengths add, but the symbols themselves are not added. Order matters: putting 1001 first produces a different string. This distinction is useful in digit procedures, substitution rules and computer programs, where an operation may manipulate a representation rather than the number it could represent. An empty string changes nothing when concatenated.

Example Concatenating 12 and 34 gives 1234, not 46.

Connected ideas Bit, Substitution, Sequence, Bitwise complement, Cube-free

Explore it A Pattern Built by Flipping Its Copy, The Sequence That Reads Itself Aloud

Conditional probability is the chance of an event when some other event is known to have happened.

Suppose birthdays have only three equally likely dates. If the first two people have different birthdays, only one date remains available for a third person to avoid a match. The conditional chance of that third person avoiding both is therefore one third. Without the condition, the first two might already match. Conditioning changes which possibilities you consider; it does not mean that learning something caused the event you learned about.

Example With three possible dates, P(no match among three) = 1 × 2/3 × 1/3.

Connected ideas Probability, Statistical independence, Sample space, Complementary event, Birthday problem

Explore it Birthday Matches in a Small Room

A confidence interval is a range calculated from data by a method with a stated rate of capturing the unknown value over repeated samples.

The familiar 95% label describes repeated use of the procedure, not a 95% probability that one fixed interval contains a parameter. Different assumptions produce different intervals. Reporting the method, data and target keeps the range from being mistaken for a guarantee about this one sample. Taking more independent observations usually narrows the interval. An interval can still miss the target, and systematic measurement bias is not repaired merely by collecting more data.

Example A 95% interval procedure covers the true value in about 95% of repeated samples when its assumptions hold.

Connected ideas Estimator, Squared error, Shrinkage

Explore it Three Guesses Beat One, Two Ways to Corner π

Two integers are congruent modulo m when they leave the same remainder on division by m.

Writing a ≡ b (mod m) says that m divides a−b. Congruence is an equality inside modular arithmetic, where many different integers represent the same residue class. It lets a proof replace an unbounded collection of cases with a finite set of behaviours. For instance, 2, 7 and 12 all sit in the same class modulo 5. The symbol does not say that these are equal as ordinary integers.

Example 17 ≡ 2 (mod 5).

Connected ideas Parity, Residue, Digit sum

Explore it Pascal Modulo m, Times Tables on a Circle

A conjecture is a precise mathematical claim supported by evidence but not yet proved.

A conjecture can be very well tested and still be open. The label keeps a promising pattern separate from a theorem and invites attempts to prove or disprove it. A counterexample would settle the conjecture negatively, while more examples alone only strengthen its evidence. Testing many starts of the Collatz rule suggests that they eventually reach 1. Since the positive integers never run out, those tests cannot cover every possible start.

Example Goldbach’s strong conjecture concerns sums of two primes.

Connected ideas Proof, Theorem, Counterexample, Hypothesis, Prime k-tuples conjecture

Explore it Every Even Number, Two Primes, The Collatz Orbit Atlas, Reverse, Add, and 196

Consecutive-prime bias is the tendency of a prime and the next prime to avoid having the same last digit, or the same remainder on division by the modulus.

Among primes below a hundred million, a prime ending in 1 is followed by a prime ending in 1 only about 17.7% of the time, although each ending is equally common overall and independence would give 25%. The effect was brought to wide attention by Lemke Oliver and Soundararajan in 2016, with a conjectured explanation from the Hardy–Littlewood conjectures that predicts it fades only very slowly as the primes grow. It concerns pairs, not single primes, which is why the even split of endings cannot rule it out.

Example A prime ending in 1 is followed by a prime ending in 1 about 17.7% of the time below a hundred million.

Connected ideas Chebyshev bias, Bias, Statistical independence, Prime k-tuples conjecture

Explore it Primes Dislike Repeating Themselves, The Prime Race That Fixes Its Own Result

A constructible polygon is a regular polygon that can be drawn exactly with an unmarked straightedge and a compass.

The classical tools allow only straight lines through two points and circles about a point, so which regular shapes they can produce is a genuine question. Gauss and Wantzel settled it: a regular polygon with an odd number of sides is constructible exactly when that number is a product of distinct Fermat primes. Three, five, fifteen and seventeen sides are therefore drawable, and seven, nine and eleven are not. Doubling the sides of a constructible polygon by bisecting angles keeps it constructible.

Example A regular pentagon is constructible; a regular heptagon is not.

Connected ideas Fermat number, Regular polygon

Explore it Five Primes and Then Never Again

A contraction mapping moves any two points strictly closer together, by at least a fixed factor.

The factor has to serve every pair at once and has to be strictly below one, so a map that merely shortens some distances does not qualify. Moving half the way to a fixed corner qualifies with factor one half, and moving two thirds of the way qualifies with factor one third. Banach proved that a contraction of a complete space has exactly one fixed point and that iterating from anywhere converges to it. Applying the same argument to whole sets rather than to single points is what pins down the picture a chaos game draws.

Example Moving half the way towards a fixed corner halves every distance in the plane.

Connected ideas Iterated function system, Limit

Explore it The Chaos Game

A sequence converges when its terms approach a definite limit as the index grows.

Approaching does not require ever landing exactly on the limit. The distance can shrink monotonically, oscillate while shrinking, or behave irregularly. A visual approximation suggests convergence; a proof specifies the limit and controls the error for every sufficiently late term. The terms 1, 1/2, 1/3 and so on never equal zero, yet eventually stay within any positive distance you specify from zero. That is the key requirement.

Example 1/n converges to 0.

Connected ideas Limit, Divergent series, Partial sum, Geometric series, Harmonic series

Explore it Two Ways to Corner π, Tiles That Never Repeat, The Sum That Is Not a Sum

The convolution of two functions slides one across the other and records the overlap at each position.

The convolution of f and g at t is the integral of f(s) times g(t − s) over all s. The density of the sum of two independent random numbers is the convolution of their two densities, so adding a uniform number on (−1, 1) to one on (−2, 2) gives a flat-topped trapezoid. Fourier transforms turn a convolution into an ordinary product, and a product into a convolution, which is the bridge between multiplying waves and adding random numbers.

Example Two uniform numbers on (−1, 1) have a triangular sum on (−2, 2).

Connected ideas Uniform distribution, Sinc function

Explore it π/2, π/2, π/2 … and Then Not

A coordinate is one of the numbers used to locate a point relative to chosen axes or another reference system.

On an ordinary two-dimensional grid, the pair (3, 2) means three units along the horizontal axis and two along the vertical axis, starting from the origin. The order matters. In three dimensions a third coordinate records depth or height. Coordinates describe locations, while vectors can describe changes between locations. Screen coordinates often increase downward, unlike the upward-positive vertical axis usually used in mathematical diagrams; drawings must account for that difference.

Example Moving from (1, 2) to (4, 6) gives displacement (3, 4).

Connected ideas Vector, Unit circle, Lattice

Explore it Two Ways to Corner π, Three Guesses Beat One, The Spiral, and What It Is Not

Two integers are coprime when their greatest common divisor is one.

Coprime numbers need not be prime themselves: 8 and 15 are a pair. They share no prime factor, so many modular equations involving them have a unique solution. Coprimality is a relationship between numbers, not a label attached to one number in isolation. A fraction such as 8/15 is already in lowest terms because there is no common factor to cancel. By contrast, 8/12 simplifies by cancelling 4.

Example 8 and 15 are coprime.

Connected ideas Greatest common divisor

Explore it Times Tables on a Circle

The cosine of an angle is the horizontal coordinate reached by turning through that angle anticlockwise from (1, 0) on the unit circle.

A point begins at the rightmost edge of the unit circle, where its horizontal coordinate is one. After a quarter-turn that coordinate is zero, and after a half-turn it is minus one. Cosine records this horizontal motion as the angle changes. In a right triangle it also equals the adjacent side divided by the hypotenuse. Together, cosine and sine specify the entire circle point; the sum of their squares is one because the radius remains one.

Example cos(0) = 1, cos(π/2) = 0, and cos(π) = −1.

Connected ideas Sine, Unit circle, Euler’s formula

Explore it The Number Behind Continuous Growth, Two Ways to Corner π

A set is countable if its elements can be listed using natural-number positions, with finite sets included by convention here.

An infinite list need not finish; it must ensure that every element appears at some finite position. The positive integers are countable, and pairs of non-negative integers can also be listed by traversing successive diagonals of a grid. Cantor pairing gives an explicit number for each such pair. Countable does not mean small or practically enumerable in a short time. It describes whether this kind of mathematical listing is possible at all.

Example The integers can be listed as 0, 1, −1, 2, −2, 3, −3, ... .

Connected ideas Natural number, Cantor pairing, Bijection

Explore it A Sentence About Its Own Number

A counterexample is one valid case that makes a universal claim false.

Finding a counterexample ends a conjecture as stated, even if the pattern holds for millions of earlier cases. It can also reveal which hypothesis was doing hidden work. Good explorations make boundary cases easy to try because failure often explains a rule more clearly than a successful run. The case must meet the assumptions first. Trying a negative input does not refute a claim stated only for positive integers, even when the calculation fails.

Example An even integer greater than 2 that is not a sum of two primes would refute Goldbach’s strong conjecture.

Connected ideas Proof, Theorem, Conjecture, Prefix, Proof by exhaustion

Explore it Every Even Number, Two Primes, 1, 2, 4, 8, 16, and Then Not 32, Tiles That Never Repeat

In the study of strings, cube-free means containing no non-empty block repeated three times consecutively.

The string 010101 contains three consecutive copies of 01, so it contains a cube. A cube need not consist of three identical individual symbols; the repeated block may be longer. The Thue-Morse sequence is cube-free because it has the stronger overlap-free property: any triple repetition contains an overlap at its start. Here cube refers to repetition of a string. In number theory, cube-free has a different meaning involving divisibility by integer cubes.

Example 000 and 010101 are cubes; 0101 is only a double repetition.

Connected ideas Overlap-free, Concatenation, Period

Explore it A Pattern Built by Flipping Its Copy

A cusp is a sharp point where a curve folds back with its two branches sharing a tangent direction.

Follow the smooth outline of a cardioid until you reach its inward tip. The curve doubles back there, making a cusp rather than a rounded bend. This differs from an ordinary polygon corner, where two straight sides meet in different directions. In the circle laboratory, the cardioid cusp lies inside the surrounding circle. Where the curve touches the circle rim it is smooth, so counting rim contacts is not identifying the cusps.

Example A cardioid has one cusp, while a nephroid has two.

Connected ideas Cardioid, Tangent, Vertex, Nephroid, Epicycloid

Explore it Times Tables on a Circle

A cycle is a repeating loop of states reached by a rule.

If a state returns to one already seen, deterministic iteration repeats the same future forever. The number of distinct states in the loop is its period. A fixed point is the special case with period one, while longer cycles show that “does not settle” need not mean “has no structure.” The initial path leading into a loop is not part of the cycle. A run can therefore have many steps before settling into a short repeating pattern.

Example 0 → 1 → 0 is a cycle of length two.

Connected ideas Fixed point, Attractor, Basin, State space, Amicable numbers

Explore it The 6174 Attractor, The Collatz Orbit Atlas, Differences Around a Circle

Decimal notation writes numbers in base ten, using digits 0 through 9 and place values based on powers of ten.

In 243, the positions mean two hundreds, four tens and three ones. To the right of the decimal point come tenths, hundredths and smaller places. A decimal expansion may stop, repeat forever or continue without repeating. Writing a number in decimal instead of binary changes its representation, not its value. Digit-based rules can behave differently when the same value is written in a different base.

Example 2.43 = 2 + 4/10 + 3/100.

Connected ideas Binary, Fraction, Irrational number, Binary expansion, Exponent

Explore it How Long Can a Number Resist?, Two Ways to Corner π, Happy Numbers

The denominator is the bottom part of a fraction and specifies what the numerator is divided by.

In 3/4, the denominator is four. Picture a unit divided into four equal pieces; the numerator says to take three of them. With the same positive numerator, a larger positive denominator makes smaller pieces and a smaller fraction. A denominator cannot be zero in ordinary arithmetic. To add fractions with different denominators, first express them using a common piece size, such as sixths for halves and thirds.

Example 1/2 + 1/3 = 3/6 + 2/6 = 5/6.

Connected ideas Fraction, Numerator

Explore it Two Ways to Corner π, The Sum That Is Not a Sum

A derivative measures a function’s instantaneous rate of change, where that rate exists.

An average speed divides distance travelled by elapsed time. An instantaneous speed asks what that ratio approaches as the time interval becomes smaller and smaller around one moment. Derivatives use the same limiting idea. For a graph y = f(x), the derivative gives the tangent slope. A parametrized curve has coordinate derivatives describing its velocity; both vanish at the cusp parameters in the circle laboratory, although zero velocity alone does not always imply a cusp.

Example For f(x) = x², the derivative is 2x, giving slope 6 at x = 3.

Connected ideas Limit, Tangent, Function, Exponential function, Taylor series

Explore it Times Tables on a Circle, Three Guesses Beat One

A diagonal joins non-neighbouring polygon vertices, or follows matching row-and-column steps through an array or grid.

A square has two diagonals, both passing through its centre. A triangle has none because every pair of its vertices is already joined by a side. Diagonals help divide a polygon into smaller regions, though several may meet at one interior point. On a square grid, diagonal positions advance equally along both axes, as in the Ulam spiral. The word also describes diagonal entries in matrices. A diagonal argument in logic is a different idea, explained separately.

Example A pentagon has 5 diagonals in addition to its 5 sides.

Connected ideas Vertex, Polygon, Intersection

Explore it 1, 2, 4, 8, 16, and Then Not 32, The Spiral, and What It Is Not

The diagonal lemma builds a sentence equivalent to a specified property of its own code, inside a formal system with enough arithmetic.

It is a general self-reference theorem, not a picture of a diagonal line. A formal system capable of representing enough arithmetic can use the lemma to create a sentence that talks about the sentence built from it. The construction underlies Gödel’s incompleteness results, although the toy laboratory only illustrates the mechanism. More precisely, for a formula P with one free variable it supplies a sentence G equivalent, within the system, to P applied to the code of G.

Example A sentence can be made to say “the sentence with this code has property P.”

Connected ideas Self-reference, Gödel numbering, First incompleteness theorem

Explore it A Sentence About Its Own Number

A diameter is a chord passing through the centre of a circle; its length is twice the radius.

Draw a straight line from one side of a circle to the opposite side through its centre. The part inside the circle is a diameter. It consists of two radii placed end to end and is the longest possible chord of that circle. Other chords can be shorter, depending on their endpoints. Doubling a numbered endpoint in modular arithmetic does not mean doubling the physical length of a chord.

Example With radius 5, every diameter is 10 units long.

Connected ideas Radius, Chord, Circumference, Pi

Explore it Times Tables on a Circle, 1, 2, 4, 8, 16, and Then Not 32

Digit reversal writes a chosen string of digits in the opposite order.

Reversing 532 produces 235: the last digit comes first, the middle stays in place and the first comes last. The rule acts on a written string, not just an abstract value. Reversing 099 gives 990, whereas reversing 99 gives 99. Those starting strings represent the same number but have different lengths. A digit-reversal procedure therefore needs to specify its base and whether it keeps leading zeroes.

Example 099 reversed is 990, but 99 reversed is still 99.

Connected ideas Place value, Palindrome, Decimal

Explore it Why the Sum Becomes 1089, Reverse, Add, and 196

A digit sum adds the individual digits of a number written in a specified base.

For decimal 407, add four, zero and seven to get eleven. This is an operation on the written digits, so changing the base can change the answer. Rearranging the digits leaves their sum unchanged, which helps explain divisibility properties of the Kaprekar subtraction. A digit sum is not a digit product, and it is not the sum of squared digits used for happy numbers. These similar-looking rules define different processes.

Example The digit sum of 9831 is 9 + 8 + 3 + 1 = 21.

Connected ideas Number base, Congruence, Permutation

Explore it The 6174 Attractor, Happy Numbers, How Long Can a Number Resist?

A divergent series is an infinite sum whose ordinary partial sums do not approach a finite limit.

The terms can shrink and the series can still diverge, as with the harmonic series. Some summation methods assign useful regularised values, but those values are not ordinary sums and must be labelled as such. Keeping the definitions separate prevents a formal continuation from being mistaken for convergence. By contrast, 1 + 1/2 + 1/4 + ... converges to 2. It is the accumulated totals, rather than just the individual terms, that decide convergence.

Example 1 + 1/2 + 1/3 + ... diverges.

Connected ideas Convergence, Limit, Regularisation, Analytic continuation, Cesàro summation

Explore it The Sum That Is Not a Sum

A divisor of n is an integer that divides n with no remainder.

If d divides n, then n = d × q for another integer q. Divisors arrive in pairs except at a square root, which makes their count and sum useful signals in several sequences. The word factor is often used interchangeably, although factor can also mean one component of a product in a broader algebraic setting. Dividing 12 by 5 leaves something over, so 5 is not a divisor of 12. In divisor sums here, only positive divisors are included.

Example 3 is a divisor of 12 because 12 = 3 × 4.

Connected ideas Prime number, Composite number, Factorisation, Remainder, Proper divisor

Explore it Sum the Divisors and Go Again, Every Even Number, Two Primes, Primes That Survive Every Shuffle

An edge is a connection between two nodes in a graph.

A directed edge can mean “moves to,” while an undirected edge only records mutual connection. Multiple edges may share endpoints, and a loop can connect a node to itself. In a state map, the edges are the rule made visible: follow one to see exactly what changes in one step. In geometry, an edge can also be a side of a polygon. An arc of a circle is a curved piece of its circumference, not a synonym in that setting.

Example An arrow from 12 to 16 is an edge in an aliquot graph.

Connected ideas Graph, Node, Vertex

Explore it The 6174 Attractor, Sum the Divisors and Go Again, The Collatz Orbit Atlas

An eigenvalue is the scale factor by which a linear transformation changes a corresponding non-zero eigenvector.

A matrix may turn and stretch most vectors, but some vectors stay on their original line. For such a vector v, the equation Av = λv says that applying the matrix A has the same effect as multiplying by λ. That factor is an eigenvalue. Under suitable conditions, an eigenvalue of largest absolute size controls long-run growth when the matrix is repeatedly applied. Those conditions matter; not every initial vector exhibits that growth.

Example The matrix with diagonal entries 2 and 3 scales horizontal vectors by 2 and vertical ones by 3.

Connected ideas Eigenvector, Matrix, Vector

Explore it The Sequence That Reads Itself Aloud, Tiles That Never Repeat

An eigenvector is a non-zero vector that a linear transformation only scales, without moving it off its original line.

Picture a transformation that stretches a sheet horizontally by two and vertically by three. Horizontal and vertical arrows stay on their own lines, so they are eigenvectors; most slanted arrows change direction. Their scale factors are eigenvalues. A negative factor reverses direction, and a zero factor collapses the vector to zero. The original eigenvector itself must be non-zero, otherwise the defining equation would work for every scale factor and carry no information.

Example If Av = 2v for a non-zero v, then v is an eigenvector with eigenvalue 2.

Connected ideas Eigenvalue, Matrix, Vector

Explore it Tiles That Never Repeat, The Sequence That Reads Itself Aloud

An envelope is a curve tangent to each member of a changing family of curves or lines.

Instead of tracing one moving line, an envelope records the boundary touched by the whole family. Dense drawings can make an envelope appear as a smooth curve even though it is assembled from many straight pieces. The construction and the limiting curve should be distinguished when interpreting a plot. In the circle laboratory, a finite set of chords suggests an envelope. The smooth boundary belongs to the continuous family that the drawing samples, not to any single chord.

Example The tangent lines to a parabola have the parabola as their envelope.

Connected ideas Polygon, Chord, Cardioid, Nephroid, Epicycloid

Explore it Times Tables on a Circle, Two Ways to Corner π

An epicycloid is the path of a point on a circle rolling without slipping around the outside of another circle.

Imagine marking the rim of a small wheel and rolling it around a fixed circular track. The mark traces an epicycloid rather than a circle. When the fixed radius is a positive integer multiple of the rolling radius, that integer counts the cusps. Equal radii give a cardioid; a ratio of two gives a nephroid. In the multiplication laboratory, whole-number multiplier k gives an envelope with k − 1 cusps.

Example Multiplier 4 gives a three-cusped epicycloid in the circle laboratory.

Connected ideas Cardioid, Nephroid, Radius, Cusp, Envelope

Explore it Times Tables on a Circle

An equation states that two expressions have equal values.

The equals sign makes a claim about the expressions on its two sides. An equation such as x + 2 = 5 is true for some choices of x and false for others; solving it means finding the allowed choices. An identity, such as x + 0 = x, holds for every allowed x. Rearranging an equation must preserve its solutions, so dividing by an expression that might be zero needs care.

Example The equation x² = 9 has two real solutions, x = 3 and x = −3.

Connected ideas Variable, Polynomial, Square root

Explore it The Spiral, and What It Is Not, Times Tables on a Circle

A sequence is equidistributed in an interval when the share of its terms falling in any subinterval tends to the length of that subinterval.

Take the fractional parts of 0.301, 0.602, 0.903, 1.204 and so on, keeping only what follows the decimal point. The points march rightward and wrap at the end, and Weyl proved in 1916 that when the step is irrational they eventually fill the interval evenly: the share falling in any stretch tends to the length of that stretch. Irrationality is the whole condition. A rational step would cycle through finitely many positions and leave the rest of the interval permanently empty, so no amount of running would spread the points out.

Example The fractional parts of n times log 2 land below 0.301 about 30.1% of the time.

Connected ideas Uniform distribution, Irrational number, Law of large numbers, Benford’s law, Proof by exhaustion

Explore it Why One Comes First, The Chaos Game

An estimator is a rule that uses observed data to approximate an unknown quantity.

The estimator is the procedure; its estimate is the number produced on one dataset. Bias, variance and squared error describe how the procedure behaves across repeated samples. A simulation can compare estimators under its chosen model, but it cannot make a universal claim beyond that design. For instance, average ten measurements to estimate a quantity. Taking another ten gives a different estimate, but the averaging rule used to produce it is the same estimator.

Example The sample mean estimates a population mean.

Connected ideas Squared error, Shrinkage, Confidence interval, Bias, Standard error

Explore it Three Guesses Beat One

Euler’s formula e^(iθ) = cos(θ) + i sin(θ) connects the complex exponential to a turn of θ radians on the unit circle.

Start at 1 on the complex plane and rotate anticlockwise through an angle θ. The new point has horizontal coordinate cos(θ) and vertical coordinate sin(θ). Euler’s formula says that the same point is e^(iθ). Its distance from the origin stays one: a purely imaginary exponent produces a turn rather than real growth. The angle must be in radians. Here the name refers to the complex-exponential formula, not Euler’s separate relation between vertices, edges and faces.

Example e^(iπ/2) = i: a quarter-turn takes (1, 0) to (0, 1).

Connected ideas Euler’s identity, Complex number, Radian, Sine, Cosine

Explore it The Number Behind Continuous Growth, Two Ways to Corner π

Euler’s identity is e^(iπ) + 1 = 0, the half-turn case of Euler’s formula.

On the unit circle in the complex plane, begin at 1 and turn through π radians, which is half a revolution. You arrive at −1. Euler’s formula expresses that motion as e^(iπ) = −1; adding one to both sides gives the identity. This relates e, π, i, one and zero through precise operations. It does not say that e times π equals −1, and the imaginary unit is essential: e raised to a real power is always positive.

Example e^(iπ) = cos(π) + i sin(π) = −1 + 0i.

Connected ideas Euler’s formula, Euler’s number, Pi, Imaginary unit

Explore it The Number Behind Continuous Growth, Two Ways to Corner π

Euler’s number, written e, is approximately 2.71828 and is the limit of (1 + 1/n)ⁿ as positive integers n increase.

Start with one unit and split a total growth rate of one into n equal updates. Each update multiplies the current amount by 1 + 1/n. One update gives 2, two give 2.25, and four give about 2.4414. Increasing the number of updates approaches e rather than growing without bound. The same constant appears in the exponential function, whose rate of change equals its value. Euler’s number is different from the Euler-Mascheroni constant, usually written γ.

Example (1 + 1/10)¹⁰ ≈ 2.59374, while e ≈ 2.71828.

Connected ideas Compound growth, Exponential function, Limit, Natural logarithm, Euler’s identity

Explore it The Number Behind Continuous Growth

A sequence is eventually periodic if, after some finite beginning, it repeats with a fixed positive period forever.

The sequence 9, 4, 2, 7, 2, 7, ... has an initial part followed by the repeating block 2, 7. It is eventually periodic even though the repetition does not start at the first entry. An orbit in a finite deterministic state space must eventually repeat a state and then follow a cycle. Infinite state spaces do not provide that guarantee. The Thue-Morse sequence is not even eventually periodic.

Example A trajectory that enters 4 → 2 → 1 → 4 has an eventually periodic tail.

Connected ideas Periodic sequence, Cycle, State space, Thue-Morse sequence, Overlap-free

Explore it A Pattern Built by Flipping Its Copy, The Collatz Orbit Atlas, Happy Numbers

The expected value is the probability-weighted average outcome of a random variable.

It describes the centre of a distribution over many repetitions, not necessarily an outcome that can occur in one trial. Linearity of expectation often makes it easier to calculate than the full distribution. When a simulation estimates it, the sample size and random seed are part of the evidence. For a fair die, add the six face values and divide by six. The result lies between faces, illustrating why expected does not mean predicted for the next roll.

Example A fair six-sided die has expected value 3.5.

Connected ideas Probability, Arithmetic mean, Random variable, Law of large numbers, Stationary distribution

Explore it Three Guesses Beat One, Two Ways to Corner π

A positive whole-number exponent tells you how many copies of a base to multiply together.

In 2³, two is the base and three is the exponent, giving 2 × 2 × 2 = 8. This is different from 2 × 3. For a non-zero base, exponent zero gives one and a negative integer exponent takes a reciprocal. Fractional exponents connect to roots, subject to the number system in use. Exponent towers need explicit grouping because changing the order changes the value.

Example 2³ = 8, while 3² = 9.

Connected ideas Binary expansion, Up-arrow notation, Logarithmic scale, Binary, Decimal

Explore it A Number With No Room to Be Written, Pascal Modulo m

An exponential function has a fixed positive base raised to a variable power; the natural exponential is exp(x) = eˣ.

In 2ˣ, changing x by one multiplies the output by two. The variable belongs in the exponent, unlike x², where the exponent stays fixed. The natural exponential eˣ has the special property that its derivative equals itself. It equals one at x = 0 and remains positive for every real x. Negative inputs give reciprocals: e⁻¹ = 1/e. Complex inputs extend this function beyond real growth, allowing e raised to an imaginary multiple to describe rotation.

Example e⁰ = 1, e¹ ≈ 2.71828, and e⁻¹ ≈ 0.36788.

Connected ideas Euler’s number, Exponent, Derivative, Euler’s formula, Compound growth

Explore it The Number Behind Continuous Growth

The factorial n! is the product of every positive integer from 1 through n.

Factorials count permutations of n distinct objects and grow faster than any fixed power of n. The convention 0! = 1 keeps counting formulas consistent. Because factorials become large quickly, exact integer arithmetic and careful notation matter in computations that use them. With three different books, there are three choices for the first position, two for the next and one for the last, giving 3 × 2 × 1 arrangements.

Example 5! = 5 × 4 × 3 × 2 × 1 = 120.

Connected ideas Permutation, Combination, Taylor series

Explore it Pascal Modulo m, Counting Around the Circle

Factorisation rewrites a number or expression as a product of simpler factors.

For integers, prime factorisation is unique apart from the order of its factors. That uniqueness turns multiplication into a kind of address system for numbers. It also explains why divisor sums, coprimality and many tests for primality can be reduced to the primes appearing in a factorisation. For example, replacing 12 with 2 × 2 × 3 tells you at once that it is divisible by 4 and 6, but not by 5.

Example 60 = 2² × 3 × 5.

Connected ideas Prime number, Composite number, Divisor, Trial division

Explore it Sum the Divisors and Go Again, Primes That Survive Every Shuffle

A Fermat number is one more than a power of two whose exponent is itself a power of two.

Writing F_n for 2 raised to the power 2^n, plus one, gives 3, 5, 17, 257, 65,537 and then 4,294,967,297. Each one squares the power of two used by the one before, so the numbers double in digit length at every step. The first five are prime; the sixth is 641 times 6,700,417. A Fermat number that happens to be prime is called a Fermat prime, and only those five are known.

Example F4 = 2^16 + 1 = 65,537, the largest known Fermat prime.

Connected ideas Prime number, Composite number, Constructible polygon

Explore it Five Primes and Then Never Again

The Fibonacci numbers start 0, 1, and each later number is the sum of the previous two.

Beginning with zero and one gives 0, 1, 1, 2, 3, 5, 8, 13 and so on. The rule needs two starting values, not just the instruction to add the previous terms. Ratios of consecutive positive Fibonacci numbers approach the golden ratio. Fibonacci-related counts occur in some substitution patterns, but a picture resembling a spiral is not by itself evidence that it follows this sequence or has golden-ratio proportions.

Example After 5 and 8 comes 13, because 5 + 8 = 13.

Connected ideas Recurrence, Golden ratio, Ratio

Explore it Tiles That Never Repeat, Pascal Modulo m

Gödel’s first incompleteness theorem says that a consistent, effectively axiomatized system rich enough for arithmetic has undecidable statements.

Undecidable here means that the system proves neither the statement nor its negation, under the theorem’s precise hypotheses. The result is about formal systems, not about every question humans can ask. A toy encoding can demonstrate self-reference, but it does not reproduce the full proof or establish the theorem by itself. Consistent means the system cannot prove both a sentence and its negation. Effectively axiomatized means its axioms can be generated by an algorithm, a real restriction on the theorem’s scope.

Example The theorem applies to systems that can express a suitable amount of elementary arithmetic.

Connected ideas Self-reference, Diagonal lemma, Gödel numbering, Axiom

Explore it A Sentence About Its Own Number

A fixed point is a state that the rule leaves unchanged.

For a function f, a fixed point satisfies f(x) = x. It is the smallest possible cycle and often acts as a destination for nearby states. Finding one is not the same as showing that every start reaches it; that stronger statement concerns the basin of attraction and may require a proof. For example, multiplying by two leaves zero unchanged, but nearby positive values move away. Being fixed therefore does not, on its own, make a point attracting.

Example 6174 is fixed by the four-digit Kaprekar routine.

Connected ideas Cycle, Attractor, Basin, State space, Perfect number

Explore it The 6174 Attractor, Happy Numbers, The Sequence That Reads Itself Aloud

A fractal has intricate structure at arbitrarily small scales, often generated by repeating a construction.

Zooming into a fractal can reveal new detail instead of a smooth straight piece. In the Sierpiński gasket, smaller copies repeat the whole triangular structure exactly. Other fractals need not have exact copies, and there is no single definition covering every use of the word. A recursive instruction alone does not make its result fractal. Any screen drawing shows only finitely many stages of the mathematical object.

Example The Sierpiński gasket is a recursively built fractal.

Connected ideas Self-similar, Sierpiński gasket, Iterated function system

Explore it Pascal Modulo m, Gilbreath’s Difference Field

A fraction writes a division as a numerator over a non-zero denominator.

In 3/4, the denominator four tells you to divide a unit into four equal pieces, and the numerator three counts how many of those pieces to take. Fractions can exceed one: 7/4 is one whole unit and three quarters. Multiplying the top and bottom by the same non-zero number changes the writing without changing the value. Thus 3/4 and 6/8 name the same amount.

Example 3/4 = 6/8 = 0.75.

Connected ideas Numerator, Denominator, Rational number, Ratio, Integer

Explore it Two Ways to Corner π, The Sum That Is Not a Sum

A function assigns exactly one output to each allowed input.

Think of a machine that takes an input and follows a specified rule. The function f(x) = x + 2 takes 3 to 5 and 10 to 12. Two inputs may share an output, but one input cannot have two different outputs under the same function. The allowed inputs form its domain. Iteration means feeding the output back in, which requires that output to be an allowed input too.

Example For f(x) = x², both 3 and −3 give 9.

Connected ideas Iteration, Injective, Surjective, Bijection, Variable

Explore it The Collatz Orbit Atlas, A Sentence About Its Own Number

The gambler’s ruin problem asks for the chance that a player who bets one unit a round is wiped out before winning the opponent’s whole bankroll.

Both sides stake one unit on a bet that wins with a fixed chance, and the game stops when one of them has nothing. For a fair bet the chance of ruin is the opponent’s share of the total money, so a player holding 10 against 90 is ruined nine times in ten, although the average final bankroll is still 10. A bet that favours the casino makes ruin more likely again. The answer comes from a short equation about the first round, solved exactly.

Example Holding 10 units against 90 and betting fairly, ruin has chance 90%.

Connected ideas Random walk, Absorbing barrier, Expected value

Explore it The Gambler’s Ruin

General position means an arrangement avoids specified special coincidences, with the exact conditions depending on the problem.

In the circle-regions problem, distinct points are placed on the rim so that no three of the connecting chords meet at one interior point. These conditions make a particular region-counting formula apply. Equally spaced points can violate them because symmetry forces crossings together. General position does not simply mean random or untidy: the required conditions must be stated. A drawing that looks irregular is not by itself proof that every coincidence has been excluded.

Example The three long diagonals of a regular hexagon meet at its centre, violating this condition.

Connected ideas Intersection, Hypothesis, Symmetry

Explore it 1, 2, 4, 8, 16, and Then Not 32

A geometric series adds terms obtained by repeatedly multiplying by the same ratio.

The series 1 + 1/2 + 1/4 + 1/8 + ... uses ratio one half. Its partial sums approach two as the missing amount is halved at each stage. For a non-zero first term a, the infinite series has ordinary sum a/(1 − r) when the ratio satisfies |r| < 1. Outside that condition it does not converge ordinarily, even if an algebraic manipulation suggests assigning a value.

Example 1/2 + 1/4 + 1/8 + ... converges to 1.

Connected ideas Partial sum, Ratio, Convergence, Cesàro summation, Abel summation

Explore it The Sum That Is Not a Sum, Two Ways to Corner π

Gödel numbering assigns natural numbers to symbols and formulas so syntax can be discussed arithmetically.

A numbering must be decodable and, for the useful constructions, preserve enough structure to recognise valid formulas. Once a formula has a number, arithmetic can represent statements about formulas, proofs and substitution. The number is an encoding, not a measure of mathematical importance or truth. The choice of numbering is not unique. What matters is that manipulating formulas and checking finite proofs can be represented through effective operations on their assigned numbers.

Example A finite string of symbols can be mapped to one integer and decoded again.

Connected ideas Self-reference, Diagonal lemma, First incompleteness theorem

Explore it A Sentence About Its Own Number

The golden ratio is approximately 1.618, the proportion in which whole/longer equals longer/shorter.

Its value is (1 + √5)/2, and its defining proportion appears in Fibonacci growth, continued fractions and Penrose geometry. The symbol is a constant, while the ratio of two measured lengths only approximates it. A numerical match should be paired with the equation or construction that explains it. Divide a length into a longer piece and a shorter one so whole/longer equals longer/shorter. This common ratio satisfies φ² = φ + 1.

Example φ is approximately 1.618.

Connected ideas Tiling, Aperiodic, Ratio, Fibonacci number

Explore it Tiles That Never Repeat, Pascal Modulo m

A graph is a collection of nodes joined by edges that record relationships or transitions.

Edges can be directed, weighted or labelled, depending on the question. A state-space graph uses nodes for complete configurations and arrows for the rule’s next move. Graphs turn a process into a structure that can be searched for paths, cycles, components and attractors. The word also names a function plot, such as y = x². Here the emphasis is on connections: moving the drawn nodes without changing the edges gives the same abstract graph.

Example A road map is a graph with places as nodes and roads as edges.

Connected ideas Node, Edge

Explore it The 6174 Attractor, The Collatz Orbit Atlas, The Ant That Builds a Road

The greatest common divisor of two integers is the largest positive integer dividing both.

The GCD measures the shared arithmetic structure of two numbers. Euclid’s algorithm finds it by repeatedly replacing a pair with the smaller number and the remainder. When the GCD is one, the numbers are coprime, a condition that often makes modular inverses and counting arguments possible. For positive inputs this process eventually stops at a zero remainder; the last non-zero remainder is the GCD. The pair (0,0) needs a separate convention.

Example gcd(18, 30) = 6.

Connected ideas Coprime

Explore it Times Tables on a Circle

The harmonic series is the sum 1 + 1/2 + 1/3 + 1/4 + … of the reciprocals of the whole numbers, and it grows without bound even though its terms shrink to zero.

Grouping the terms from 1/(2^j + 1) to 1/2^(j+1) shows each group adds at least one half, so the partial sums pass every number, although slowly: past 10 at the 12,367th term and past 100 only after more than 10^43 terms. Because it diverges so gently, it is the standard test of how much can be removed from a series. Removing the terms whose denominators contain a given digit leaves a series that converges, which is the opposite of what the slow growth suggests.

Example The sum of the first 12,367 terms is just over 10.

Connected ideas Partial sum, Divergent series, Convergence, Kempner series

Explore it Kempner’s Series, The Sum That Is Not a Sum

A heuristic is a method or explanation that guides discovery without guaranteeing a proof.

Heuristics can be extraordinarily effective: they suggest patterns, estimates and promising routes. Their conclusions depend on approximations or assumptions that may fail in edge cases. Marking the boundary lets a reader use the idea productively without mistaking plausibility for certainty. For instance, treating nearby primes as approximately random can predict average counts. It does not make their actual positions independent, so exceptional arithmetic structure still needs attention.

Example A random model for prime gaps can be a useful heuristic.

Explore it Every Even Number, Two Primes, The Spiral, and What It Is Not, Two Ways to Corner π

In a theorem, a hypothesis is an assumption that must hold for the stated conclusion to be guaranteed.

A statement of the form if A, then B assumes A and concludes B. Without A, the theorem gives no guarantee about B, even if B sometimes still happens. In the circle-regions exploration, conditions on how chords intersect are hypotheses of the counting formula. In other contexts, hypothesis can also mean a proposed explanation to investigate. Distinguishing an assumption inside a theorem from a claim awaiting evidence prevents two different uses from getting mixed together.

Example If an integer is divisible by 4, then it is even; divisibility by 4 is the hypothesis.

Connected ideas Theorem, Conjecture, General position

Explore it 1, 2, 4, 8, 16, and Then Not 32, A Sentence About Its Own Number

The imaginary unit i is a complex number satisfying i² = −1; multiplication by i turns the complex plane a quarter-turn anticlockwise.

No real number squares to a negative number. Complex arithmetic introduces i so that its square is −1 while retaining familiar addition and multiplication rules. Multiplying 1 by i gives i, multiplying again gives −1, then −i, and then 1. These four locations are equally spaced around the unit circle. The word imaginary is historical: these numbers are precise mathematical objects, not guesses. Their geometric interpretation is particularly useful when describing oscillations and rotations.

Example i × (2 + 3i) = −3 + 2i, a quarter-turn of the point (2, 3).

Connected ideas Complex number, Unit circle, Euler’s identity

Explore it The Number Behind Continuous Growth

A function is injective when different inputs always produce different outputs.

No two inputs collide. An injective map can lose some possible outputs, but it never loses the identity of an input. Proving injectivity often establishes that an encoding is reversible on its image, which is why it appears in pairing functions and formal numbering schemes. The function n ↦ n² is not injective on all integers because 2 and −2 collide at 4. Restricting the inputs to non-negative integers removes that collision.

Example f(n) = 2n is injective on the integers.

Connected ideas Function, Surjective, Bijection

Explore it A Sentence About Its Own Number, Counting Around the Circle

An integer is a whole-number value, including zero and negative whole numbers, with no fractional part.

The integers run in both directions: ..., −3, −2, −1, 0, 1, 2, 3, ... . They are useful for counting changes as well as objects: a balance can increase by three or decrease by three. Adding, subtracting or multiplying integers always gives another integer. Division may not, since 3 divided by 2 is 1.5. Many number laboratories restrict their starting values to positive integers.

Example −4, 0 and 17 are integers; 2.5 is not.

Connected ideas Fraction, Prime number, Parity, Lattice, Natural number

Explore it The Collatz Orbit Atlas, The 6174 Attractor

An intersection consists of the points or elements that two or more objects have in common.

Two crossing lines share a point, so that point is their intersection. Two overlapping discs share a whole region instead. For sets, intersection means the elements belonging to every set under discussion. In the circle-regions laboratory, several chords can pass through the same location. Counting each pair as a separate crossing would overcount the distinct intersection points, which is why the arrangement of the chords matters as well as their number.

Example The intersection of {1, 2, 3} and {2, 3, 4} is {2, 3}.

Connected ideas Chord, General position, Diagonal

Explore it 1, 2, 4, 8, 16, and Then Not 32

An invariant is a quantity or property that remains unchanged under a rule.

An invariant gives a stable thread through changing states. Parity, a conserved sum or a geometric relationship can rule out destinations before the full orbit is known. To prove an invariant, show that the rule preserves it at one step; iteration then preserves it for every later step. If a move always adds two, odd starts remain odd and even starts remain even. That immediately rules out reaching an even target from an odd start.

Example The parity of a chessboard square can be invariant under some moves.

Explore it Differences Around a Circle, The Ant That Builds a Road, Tiles That Never Repeat

An irrational number is a real number that cannot be expressed as a ratio of two integers.

Its decimal expansion neither terminates nor falls into a repeating cycle. Square roots of non-square integers and constants such as π are familiar examples. Irrationality is a statement about exact representation, not about whether a finite decimal approximation is useful for drawing or calculation. The decimal 1.414 is rational even when used to approximate √2. No finite display of digits can, by itself, establish that the intended exact number is irrational.

Example √2 is irrational.

Connected ideas Square root, Real number, Equidistribution, Decimal

Explore it Two Ways to Corner π, Tiles That Never Repeat

An iterated function system is a finite collection of shrinking maps, applied over and over in any order.

The collection matters and the order does not. Three halvings towards the corners of a triangle, applied in whatever sequence a die dictates, fill out one particular set: the only non-empty compact set that equals the union of its own three images. That is what makes the chaos game work, since picking a map at random each step and plotting the landing place draws the set without ever computing it. Replace one map and the set changes. Reshuffle the draws and only the order the dots appear in changes. Systems whose images overlap have solid regions for attractors rather than dust.

Example Three half-size copies of a triangle, one at each corner, give the Sierpiński gasket.

Connected ideas Fractal, Attractor, Self-similar, Contraction mapping

Explore it The Chaos Game

Iteration applies the same rule repeatedly, feeding each output into the next step.

One application reveals the rule; iteration reveals the behaviour created by repetition. A process can settle, cycle, grow, wander or become too large to compute. The step count is part of the experiment because two starts following the same rule can take very different routes. For example, doubling from 3 gives 6, then 12, then 24. The multiplier remains two throughout, even though the amount added gets larger.

Example Applying f(x) to x, then to f(x), is iteration.

Connected ideas Sequence, Recurrence, Orbit, Substitution, Function

Explore it The 6174 Attractor, The Collatz Orbit Atlas, Happy Numbers

A Kempner series is the harmonic series with every term deleted whose denominator contains a chosen digit or string of digits, and it converges.

Kempner showed in 1914 that deleting every term with a 9 leaves a finite sum. The proof counts: of the whole numbers with k digits, only 8 · 9^(k−1) have no 9, each has a reciprocal below 10^−(k−1), and the resulting geometric series adds to at most 80. The actual value, about 22.92067, was computed by Baillie in 1979. The series converges so slowly that a million terms reach only about half of it.

Example Deleting 1/9, 1/19, 1/29 … leaves a series whose sum is about 22.92067.

Connected ideas Harmonic series, Geometric series, Convergence

Explore it Kempner’s Series

A geometric lattice is a regular array of points obtained by taking integer combinations of independent direction vectors.

The intersections of square graph paper form a familiar two-dimensional lattice. Starting at one intersection, whole-number steps right or left and up or down reach every other lattice point. Other lattices use slanted directions and need not form squares. In the Ulam spiral, integers occupy positions on a square lattice, while the spiral determines their order. The lattice is the underlying geometry, not the rule deciding which numbers are prime.

Example (2, −3) is a point of the standard square lattice.

Connected ideas Coordinate, Vector, Integer

Explore it The Spiral, and What It Is Not, The Ant That Builds a Road

The law of large numbers says the average of independent repeated draws approaches the underlying mean as the number of draws grows.

It explains why a long simulation settles down, and it is routinely asked to do more than it can. The law describes a limit, so it promises nothing about any particular run and gives no schedule for how quickly a run gets close. A million tosses of a fair coin can still sit a thousand heads away from half. It also says nothing about the model being right: a simulated average converges on the mean of whatever process was simulated, which is a different thing from the quantity anyone cares about unless the two were matched deliberately.

Example Switching wins about two thirds of a long seeded run, without any run being obliged to.

Connected ideas Expected value, Probability, Benford’s law, Equidistribution, Proof by exhaustion

Explore it The Door You Should Not Keep, Birthday Matches in a Small Room, Why One Comes First

A limit is the value that a sequence or function approaches under a stated limiting process.

The notation records a relationship between closeness and progress, not a final term that must be reached. Limits can describe infinite series, derivatives, curves and asymptotic growth. Different approaches can lead to different limits, or no limit at all, which is why the limiting process must be named. A sequence that alternates 0 and 1 has no single limit: it keeps leaving any sufficiently small neighbourhood of either candidate, however long you wait.

Example As n grows, 1/n has limit 0.

Connected ideas Convergence, Divergent series, Derivative, Ratio, Euler’s number

Explore it Two Ways to Corner π, Tiles That Never Repeat, The Sum That Is Not a Sum

A logarithm answers which exponent is needed to produce a number from a specified positive base other than one.

Because 10³ = 1000, the base-ten logarithm of 1000 is three. Taking a logarithm undoes exponentiation when its real-number inputs satisfy the required restrictions. Equal steps on a logarithmic scale therefore correspond to equal multiplicative factors, not equal additions. This is useful when a trajectory ranges from tiny values to enormous ones. The displayed height can then show orders of magnitude without pretending that all visible gaps are equal numerical differences.

Example log₂(8) = 3 because 2³ = 8.

Connected ideas Exponent, Logarithmic scale, Natural logarithm, Prime counting function, Benford’s law

Explore it The Collatz Orbit Atlas, Sum the Divisors and Go Again, The Sequence That Reads Itself Aloud

On a logarithmic scale, equal visual distances represent equal multiplication factors rather than equal additions.

An axis marked 1, 10, 100 and 1000 at equally spaced positions uses a base-ten logarithmic scale. Each step multiplies by ten, so a value of 1000 does not need a thousand times the screen height of 1. This helps compare processes with very different magnitudes. Zero and negative values cannot appear on an ordinary real logarithmic axis; any special handling must be stated by the chart.

Example The distance from 10 to 100 equals the distance from 100 to 1000 on a log scale.

Connected ideas Exponent, Sequence, Orbit, Logarithm

Explore it The Collatz Orbit Atlas, Sum the Divisors and Go Again, The Sequence That Reads Itself Aloud

A Lychrel number never reaches a palindrome through repeated reverse-and-add in a specified base; a finite unsuccessful search finds only a candidate.

Reverse a number’s digits and add that reversal to the original, then repeat. A palindrome reads the same in both directions, such as 121. If this process never produces one, the start is called a Lychrel number. A computation that stops after a chosen number of steps cannot establish “never”. The familiar decimal start 196 is therefore a candidate, not a proved example; the base is part of the question.

Example 47 + 74 = 121, so 47 is not Lychrel in base ten.

Connected ideas Palindrome

Explore it Reverse, Add, and 196

A Markov chain is a process that moves between a set of states, where the chance of the next state depends only on the present one.

Nothing before the present state matters, so the whole process is described by a table giving, for each state, the chance of moving to each other state. That table makes questions about the long run answerable by arithmetic instead of by simulation. Parrondo’s game B looks only at the capital modulo 3, which is why the capital modulo 3, and not the capital itself, is a three-state chain. The rule has to be checked, not assumed: a game that remembered earlier rounds would need a bigger state.

Example Capital modulo 3 is a three-state chain: from 0 you reach 1 or 2, and so on round the ring.

Connected ideas Probability, Modular arithmetic, Stationary distribution, Absorbing barrier

Explore it Two Losing Games That Win Together

Mathematical induction proves a claim for all successive integers by proving a starting case and a step from each case to the next.

Picture an endless row of dominoes. You need the first one to fall, and you need a reason each falling domino knocks down the next. In a proof, the starting case supplies the first fact and the induction step proves that whenever the claim holds at n, it also holds at n + 1. Checking several starting values cannot replace that general step, and the step cannot replace the starting case.

Example To prove 1 + ... + n = n(n+1)/2, start at n = 1 and add n + 1 for the next case.

Connected ideas Proof, Recurrence, Integer

Explore it Pascal Modulo m, Counting Around the Circle

A matrix is a rectangular array of numbers or symbols used to represent a transformation or system of relationships.

Rows and columns give a compact way to compose linear operations, solve equations and count transitions. Multiplication combines the relationships encoded by two matrices, and powers can describe repeated steps. A matrix model is exact only relative to the states and quantities it was chosen to represent. A substitution matrix can count tile types without storing their positions. That makes the count efficient, but the same matrix alone does not specify the full geometry of the tiling.

Example A 2 by 2 matrix can rotate a vector in the plane.

Connected ideas Vector, Eigenvalue, Eigenvector

Explore it Tiles That Never Repeat, Pascal Modulo m, Counting Around the Circle

Modular arithmetic performs ordinary operations while identifying numbers with the same remainder.

A modulus sets the size of the repeating world. Addition, multiplication and powers can then be studied as motion among finitely many residues. This is why a simple rule can produce a circle, a cycle or a striking picture even though the original integers never end. For example, a clock showing 10 moves to 2 after four hours. The arithmetic wraps at 12, so 14 and 2 occupy the same position.

Example In modulo 10 arithmetic, 8 + 5 is 3.

Connected ideas Modulus, Modulo, Remainder, Markov chain

Explore it Times Tables on a Circle, Pascal Modulo m, The Spiral, and What It Is Not

The modulo operation gives the remainder after division by a chosen positive integer.

Share seventeen counters into groups of five. You can make three complete groups, with two counters left over, so 17 mod 5 = 2. The answer is a remainder, not the number of groups. We use remainders from zero up to one less than the modulus. Working modulo five treats 2, 7 and 12 alike, since moving by five leaves the remainder unchanged.

Example 17 mod 5 = 2, because 17 = 3 × 5 + 2.

Connected ideas Modulus, Modular arithmetic, Remainder

Explore it Times Tables on a Circle, Pascal Modulo m

The modulus is the number you divide by when keeping only the remainder.

Imagine placing seventeen counters into groups of five. Three complete groups use fifteen counters, leaving two. The modulus is five, and the remainder is two. Choosing a different modulus changes which numbers share a remainder. In the circle exploration, it is the number of labelled positions; in Pascal’s triangle, it determines the remainder colours. Elsewhere in mathematics, modulus can also mean absolute value, so context matters.

Example 17 = 3 × 5 + 2: the modulus is 5 and the remainder is 2.

Connected ideas Modulo, Modular arithmetic, Remainder, Absolute value

Explore it Pascal Modulo m, Times Tables on a Circle, A Number With No Room to Be Written

Monte Carlo methods use repeated random samples to estimate a quantity or explore a model.

Sampling turns a difficult exact calculation into a statistical one with an error that usually shrinks as more samples are taken. A seed makes a run reproducible, but it does not remove randomness from the method. The reported range, estimator and uncertainty matter as much as the displayed decimal. For independent samples with finite variance, a typical sampling error scales like 1/√n. Reducing that error by half therefore usually needs four times as many samples.

Example Random points can estimate a circle’s area relative to its square.

Connected ideas Sample size

Explore it Two Ways to Corner π, Three Guesses Beat One

Multiplicative persistence counts how many rounds of multiplying decimal digits are needed to reach one digit.

The process repeatedly replaces a number with the product of its digits. Zero and one act as immediate stopping cases, while long records come from carefully arranged digit patterns. The persistence belongs to the written base as well as the integer, so changing base changes the question. Any single digit already has persistence zero. A multi-digit number containing zero reaches zero in one multiplication, because one zero factor makes the entire product zero.

Example 39 → 27 → 14 → 4 has persistence 3.

Explore it How Long Can a Number Resist?

The natural density of a property is the limit, if it exists, of the fraction of the first n cases that have it.

Count how many of the first n terms are 1s, divide by n, and see where the fraction goes as n grows. If it settles on a single value, that value is the density; if it oscillates without settling, there is no density to report, only limit points. The distinction matters because a computation can only ever show a fraction at one finite n. In the Kolakoski sequence the fractions hug one half, yet neither that value nor the existence of the limit has been proved.

Example The even numbers have natural density one half.

Connected ideas Limit, Probability, Chebyshev bias

Explore it The Kolakoski Sequence, The Prime Race That Fixes Its Own Result

The natural logarithm ln(x) tells you which real exponent of e gives a positive number x.

Because e² is approximately 7.389, the natural logarithm of that exact value is two. In real arithmetic, ln accepts only positive inputs. It reverses the natural exponential: ln(eˣ) = x, and e raised to ln(x) returns x. Multiplication becomes addition under a logarithm, so ln(ab) = ln(a) + ln(b) for positive a and b. This makes logarithms useful for comparing multiplicative changes and for solving questions about how long proportional growth takes.

Example ln(1) = 0 and ln(e) = 1.

Connected ideas Logarithm, Exponential function, Euler’s number

Explore it The Number Behind Continuous Growth

Natural numbers are the counting numbers; conventions differ on whether zero is included.

Counting three stones, four stones and five stones uses natural numbers. Some authors write them as 1, 2, 3, ...; others begin at 0. Neither convention is universal, so a statement involving natural numbers should make its starting point clear. Negative numbers and fractions are not natural numbers under either convention. In algorithms, an explicit domain such as non-negative integers or positive integers avoids the ambiguity and states exactly which inputs are allowed.

Example The positive natural numbers begin 1, 2, 3; the non-negative convention also includes 0.

Connected ideas Integer, Countable

Explore it Counting Around the Circle, A Sentence About Its Own Number

A nephroid is a rounded, kidney-shaped curve with two pointed indentations called cusps.

Choose multiplier three in the circle exploration and the chords outline a nephroid. It belongs to the same family as the cardioid, but has two cusps rather than one. You can also generate it by tracing a point on a circle rolling without slipping around the outside of a circle twice its radius. The traced point, the rolling circle and the outer boundary are different objects in that construction.

Example The continuous times-three chord family has a nephroid as its envelope.

Connected ideas Cardioid, Epicycloid, Cusp, Envelope

Explore it Times Tables on a Circle

A node is one object or state represented as a point in a graph.

The node contains the information needed to identify a state; edges describe what can happen next. In a visual map, a node’s position may be an informative layout choice rather than a physical coordinate. Selecting one should reveal the state and its relationships, not only a decorative dot. In geometry, vertex can instead mean a corner of a polygon. Both usages identify an important point, but a polygon’s corner need not represent a changing state.

Example Each four-digit state is a node in the Kaprekar state graph.

Connected ideas Graph, Edge, Vertex

Explore it The 6174 Attractor, The Collatz Orbit Atlas, The Ant That Builds a Road

The normal distribution is the bell-shaped probability model determined by a mean and a variance.

Its density is symmetric around the mean and decreases smoothly in both directions. Sums of many small independent effects often approach this shape, but real data need not be normal. A normal approximation is a model choice that should be checked against the question and the available evidence. The mean moves the bell left or right, while the standard deviation stretches it. Neither parameter makes all observations fall inside a fixed interval around the centre.

Example The standard normal distribution has mean 0 and variance 1.

Connected ideas Variance, Standard deviation, Uniform distribution

Explore it Three Guesses Beat One

A number base determines the place values in positional notation, with each place worth a fixed multiple of its neighbour.

In base ten, places from right to left are worth 1, 10, 100 and so on. In base two they are worth 1, 2, 4, 8 and so on. Changing the base changes the written digits, not the quantity being represented. A digit-based procedure can nevertheless change its behaviour, because it acts on that representation. This distinction matters in digit sums, palindromes and multiplicative persistence.

Example The quantity thirteen is written 13 in decimal and 1101 in binary.

Connected ideas Binary, Decimal, Digit sum

Explore it How Long Can a Number Resist?, Happy Numbers, Reverse, Add, and 196

The numerator is the top part of a fraction, or the part before the slash.

For an ordinary positive fraction, the denominator names the size of each equal piece and the numerator counts the pieces. In 5/8, five is the numerator: you have five pieces, each one eighth of a unit. Increasing the numerator while keeping a positive denominator fixed increases the value. A numerator can be zero, negative or an algebraic expression, depending on the calculation you are doing.

Example In 5/8, the numerator is 5.

Connected ideas Fraction, Denominator

Explore it Pascal Modulo m, Two Ways to Corner π

The orbit of a starting value is the sequence of states produced by repeated iteration.

An orbit records the route, not just the eventual destination. Two starting values can share an attractor while having different lengths, peaks or detours. Plotting an orbit turns an abstract recurrence into a path that can be inspected step by step and compared with another start. In a deterministic process, once two orbits reach the same state, their futures coincide. This explains why separate paths can merge in the Kaprekar map.

Example The orbit of 3 under x ↦ x + 1 is 3, 4, 5, ...

Connected ideas Sequence, Iteration, Recurrence, Rotational symmetry, Logarithmic scale

Explore it The Collatz Orbit Atlas, The 6174 Attractor, Happy Numbers

A string is overlap-free when it contains no consecutive block of the form aXaXa, where a is one symbol and X may be empty.

An overlap repeats a block twice and then repeats its first symbol once more. For example, 01010 has this form with a = 0 and X = 1. Taking X empty gives three identical symbols, such as 111. Thue-Morse is overlap-free, a theorem about the whole infinite sequence. This rules out eventual periodicity: a sufficiently long periodic tail would contain two copies of its period block followed by their first symbol.

Example 01010 is an overlap, while the shorter square 0101 is allowed.

Connected ideas Cube-free, Period, Eventual periodicity

Explore it A Pattern Built by Flipping Its Copy

A palindrome is a string that reads the same from left to right and right to left.

For numbers, compare the first digit with the last, then work inward. Every pair must agree, while a middle digit in an odd-length string can be anything. A palindrome is a property of the written representation: a number may be palindromic in one base but not another. The reverse-and-add exploration repeatedly adds a digit reversal and checks whether the resulting string has this symmetry.

Example 121 and 4884 are palindromes; 123 is not.

Connected ideas Lychrel number, Decimal, Iteration, Digit reversal

Explore it Reverse, Add, and 196

Parity records whether an integer is even or odd.

An even number is divisible by two; an odd number leaves remainder one. Many rules simplify when viewed only through this two-class lens. Parity is the smallest non-trivial modular system, and it often predicts whether a construction can continue before the exact size of a number matters. Adding two odd numbers gives an even number, so an odd result immediately tells you that the inputs could not both have been odd.

Example 17 is odd, while 18 is even.

Connected ideas Congruence, Residue, Power of two, Thue-Morse sequence, Integer

Explore it The Collatz Orbit Atlas, Pascal Modulo m, Counting Around the Circle

A partial sum adds only the first finitely many terms of a sequence or series.

For 1 + 1/2 + 1/4 + ..., the first three partial sums are 1, 1.5 and 1.75. Each is an ordinary finite addition. An infinite series converges only when its sequence of partial sums approaches a finite limit. Displaying many partial sums can suggest what happens, but a finite display does not establish the infinite conclusion. A regularised value for a divergent series is not the ordinary limit of these partial sums.

Example For 1 + 2 + 3 + ..., the first four partial sums are 1, 3, 6 and 10.

Connected ideas Convergence, Divergent series, Geometric series, Cesàro summation, Abel summation

Explore it The Sum That Is Not a Sum, Two Ways to Corner π

Pascal’s triangle starts with 1; each new row has 1 at both ends and adds neighbouring entries above for its interior.

Write 1 at the top, then 1 and 1 below it. Add neighbouring entries to build the next interiors: the rows begin 1; 1, 1; 1, 2, 1; and 1, 3, 3, 1. We number the top row zero. Each entry counts selections of a given size, which explains why the triangle also appears in algebra. Colouring just odd entries reveals the repeating triangular pattern explored here.

Example Row 4 is 1, 4, 6, 4, 1; its middle 6 is the sum of the two 3s above.

Connected ideas Binomial coefficient, Combination

Explore it Pascal Modulo m

A perfect number is a positive integer equal to the sum of its positive proper divisors.

The positive divisors of six smaller than itself are one, two and three. Their sum is six, making six perfect. Twenty-eight works too, since 1 + 2 + 4 + 7 + 14 = 28. Under the aliquot rule, a perfect number repeats forever as a fixed point. Perfect is a technical name here, not a claim that these numbers are better or more complete than others.

Example 6 → 6 → 6 is an aliquot trajectory already at a fixed point.

Connected ideas Proper divisor, Fixed point, Amicable numbers, Square number

Explore it Sum the Divisors and Go Again

For a sequence, a period is a positive shift that leaves every comparable entry unchanged.

In 010101..., shifting by two positions preserves every symbol, so two is a period. Four is also a period, while two is the smallest positive one. For a finite string, only positions that remain inside the string can be compared. Finding a mismatch rules out that shift for the string and any infinite continuation. Agreement across a short displayed prefix does not guarantee that the next comparison will also agree.

Example The finite string 0110 agrees at shift 3, but 01101001 fails that shift at positions 2 and 5.

Connected ideas Periodic sequence, Prefix, Cycle, Overlap-free, Cube-free

Explore it A Pattern Built by Flipping Its Copy, Differences Around a Circle

A periodic sequence repeats after a fixed positive number of positions, from its beginning onward.

The sequence 2, 7, 2, 7, ... repeats a block of length two. Every entry equals the entry two places later, and this statement holds indefinitely. A sequence can contain many repeated small blocks without having one period that works everywhere. This is the distinction explored by Thue-Morse: local repetition coexists with a proof that no fixed shift preserves the entire infinite sequence.

Example 001001001... is periodic with smallest period 3.

Connected ideas Period, Eventual periodicity, Aperiodic

Explore it A Pattern Built by Flipping Its Copy, Happy Numbers

A permutation is an arrangement of items where order matters.

The same collection can have many permutations, and swapping two positions usually creates a new one. Counting permutations explains factorials and the size of search spaces built from digits. In a laboratory, enumerating every permutation is a finite computation only when the input length is explicitly bounded. Repeated items need care: the word AAB has only three distinct arrangements, not six, because exchanging the two copies of A makes no visible change.

Example ABC has six permutations.

Connected ideas Combination, Factorial, Digit sum, Combinatorics

Explore it Primes That Survive Every Shuffle, Counting Around the Circle

Pi, written π, is the circumference of a Euclidean circle divided by its diameter, approximately 3.14159.

Measure straight across a circle through its centre, then measure around its rim. Dividing the second length by the first gives the same ratio for every circle in ordinary flat geometry. Doubling the diameter doubles the circumference, so the ratio stays fixed. Pi is not exactly 3.14 or 22/7; those are approximations. Its decimal expansion never ends or eventually repeats. Pi also measures rotation: half a turn is π radians, which explains its appearance in Euler’s identity.

Example A circle with diameter 10 cm has circumference 10π cm, approximately 31.416 cm.

Connected ideas Circumference, Diameter, Radian, Euler’s identity

Explore it Two Ways to Corner π, Times Tables on a Circle, The Number Behind Continuous Growth

If more objects than boxes are placed into those boxes, at least one box must contain more than one object.

Imagine eight people assigned birthdays using only seven possible dates. Even with perfect effort to keep the dates different, someone must reuse a date. This conclusion needs no randomness or simulation: it follows from counting. More generally, placing N objects into a positive number k of available boxes forces at least one box to contain at least the smallest integer not below N/k. The boxes themselves may start empty.

Example Eight people and seven possible dates guarantee a shared date.

Connected ideas Proof, Integer, Birthday problem

Explore it Birthday Matches in a Small Room

Place value is the contribution a digit makes because of its position in a written number.

In 532, the five contributes five hundreds, the three contributes three tens and the two contributes two ones. Moving a digit changes its contribution even when the symbol stays the same. Reversing 532 gives 235, so the outside contributions exchange places while the tens stay unchanged. Leading zeroes contribute nothing to the original value, but preserving their positions matters if a later operation reverses the written digits.

Example 532 = 5 × 100 + 3 × 10 + 2 × 1.

Connected ideas Decimal, Binary, Digit reversal

Explore it Why the Sum Becomes 1089, The 6174 Attractor

A polygon is a closed plane figure made from finitely many straight line segments.

Its vertices are the corners and its edges are the segments between them. Regular polygons have equal sides and angles, making them convenient approximations to a circle. Increasing the number of sides creates a polygon ladder that can certify bounds without drawing a mathematically perfect curve. An inscribed polygon has its vertices on the circle, while a circumscribed polygon has sides tangent to it. Their perimeters provide lower and upper bounds for the circumference.

Example A triangle and a hexagon are polygons.

Connected ideas Chord, Envelope, Regular polygon, Vertex, Diagonal

Explore it Two Ways to Corner π, 1, 2, 4, 8, 16, and Then Not 32, Tiles That Never Repeat

A polynomial is a finite sum of terms made from coefficients and variables raised to non-negative integer powers.

Polynomials are built from addition and multiplication, with non-negative whole-number exponents. Their degree is the largest exponent with a non-zero coefficient. They are flexible enough to approximate curves and rigid enough for exact algebra, which is why they appear in interpolation, recurrences and proofs. Terms such as 1/x or √x fall outside the usual polynomial definition because their exponents are negative or fractional. Constants count as polynomials too.

Example 3x² − 2x + 1 is a quadratic polynomial.

Connected ideas Coefficient, Equation

Explore it Pascal Modulo m, Two Ways to Corner π

Here, a power of two is a number obtained by starting at one and doubling a whole number of times.

The sequence begins 1, 2, 4, 8, 16, 32. These are 2⁰, 2¹, 2² and so on. Each has a binary representation consisting of one followed by zeros, because binary place values are powers of two. Powers with negative exponents also exist, but they are fractions and are not what a positive-integer input means by this phrase. Several laboratories behave differently at powers of two because binary structure enters their rules.

Example 32 = 2⁵ is written 100000 in binary.

Connected ideas Binary, Exponent, Parity

Explore it Differences Around a Circle, Counting Around the Circle, Pascal Modulo m

A prefix is an initial segment of a string or sequence, beginning at its first position.

The strings 0, 01 and 0110 are prefixes of 01101001, because each stops at some point without changing or skipping any earlier symbol. A prefix of an infinite sequence contains only finitely much information. It can disprove a universal claim by exposing a mismatch, but apparent agreement may fail later. When every construction round preserves the previous row as a prefix, those rounds consistently specify one infinite sequence.

Example 0110 is a prefix of 01101001; 110 is a substring but not a prefix of that string.

Connected ideas Sequence, Period, Counterexample

Explore it A Pattern Built by Flipping Its Copy, The Sequence That Reads Itself Aloud

The prime counting function reports how many primes are less than or equal to a given number.

Written π(x), it counts rather than produces: π(10) is 4, because 2, 3, 5 and 7 are the primes up to ten, and π(100) is 25. The value steps up by one exactly at each prime and stays flat in between, so its graph is a staircase whose risers are the primes themselves. Counting only the primes that leave a chosen remainder gives the two separate staircases that a prime race compares. The prime number theorem says π(x) grows roughly like x divided by the natural logarithm of x.

Example π(10) = 4 and π(100) = 25.

Connected ideas Prime number, Logarithm, Chebyshev bias

Explore it The Prime Race That Fixes Its Own Result, Every Even Number, Two Primes

A prime gap is the difference between consecutive prime numbers.

The gap after p is the distance to the next prime, so it measures local irregularity rather than the density of all primes. Small gaps recur often, while unusually large gaps can appear far out. A plotted sample shows a range of behaviour, not a proof about every future gap. Apart from the gap from 2 to 3, every prime gap is even because both neighbouring primes are odd. That explains one regular feature inside an irregular sequence.

Example The gap between 11 and 13 is 2.

Connected ideas Prime k-tuples conjecture

Explore it Gilbreath’s Difference Field, The Spiral, and What It Is Not

The Hardy–Littlewood prime k-tuples conjecture predicts how often a fixed pattern of offsets from a number are all prime at once.

Take the pattern n and n + 2. The conjecture gives a formula for how many n up to x make both prime, with a constant that depends on which small primes the offsets are forced to avoid. A pattern such as n, n + 2 and n + 4 cannot occur beyond 3, 5, 7, because one of the three is always a multiple of 3, and the formula gives zero for it. The conjecture was stated in 1923 and is unproved, even for pairs. Work on consecutive primes uses it as an assumption, so what follows from it is conditional.

Example It predicts the count of twin primes, n and n + 2 both prime, and says nothing for n, n + 2, n + 4 beyond 3.

Connected ideas Conjecture, Prime number, Prime gap, Consecutive-prime bias

Explore it Primes Dislike Repeating Themselves

A prime number is an integer greater than one whose only positive divisors are one and itself.

Try arranging seven counters into equal rows. Only one row of seven, or seven rows of one, works without leftovers. Twelve counters also make three rows of four, so twelve is composite. The number one is neither prime nor composite. Every integer greater than one can be broken into a product of primes, uniquely apart from order. This makes primes fundamental to divisibility and factorisation.

Example 7 is prime; 12 = 3 × 4 is composite; 1 is neither.

Connected ideas Composite number, Factorisation, Divisor, Prime counting function, Arithmetic progression

Explore it The Spiral, and What It Is Not, Primes That Survive Every Shuffle, Every Even Number, Two Primes

Probability measures how strongly an outcome is expected within a specified random model.

A probability is not a promise about one trial. It is a number from zero to one attached to an event and a model for generating outcomes. Repeated experiments can reveal the model’s typical behaviour, but sampling variation and model assumptions must remain visible in the interpretation. For a finite collection of equally likely outcomes, count the favourable outcomes and divide by the total. That shortcut fails when the outcomes have different probabilities.

Example A fair coin has probability 1/2 of landing heads.

Connected ideas Expected value, Arithmetic mean, Random variable, Sample size, Conditional probability

Explore it Two Ways to Corner π, Three Guesses Beat One, Every Even Number, Two Primes

A proof is a finite argument showing that a conclusion follows from stated assumptions.

A proof can be formal, geometric, algebraic or computational when the finite scope is explicit. It must account for every case covered by the claim, not only the examples that inspired it. A proof sketch is useful when it marks which routine steps are omitted and which ideas carry the argument. A computer can also help prove an infinite claim when a mathematical argument reduces it to finitely many cases and each case is checked correctly.

Example Induction proves a statement for every natural number by anchoring and advancing one step.

Connected ideas Theorem, Conjecture, Counterexample, Axiom, Pigeonhole principle

Explore it Pascal Modulo m, A Sentence About Its Own Number, Tiles That Never Repeat

A proof by exhaustion splits a statement into finitely many cases and settles every one of them.

The method is only available when the cases genuinely run out. Checking a rule for the first million numbers is not exhaustion, because a million is not all of them; checking all four-digit starting values, or all three positions a car can occupy behind three doors, is. Two conditions have to hold together: the list of cases must cover the statement with nothing left over, and each case must actually be settled rather than merely tried. Where both hold, a computer that walks every case has produced a proof, and the range it walked is the theorem.

Example Three car positions settle the three-door game outright, since there is no fourth arrangement.

Connected ideas Counterexample, Law of large numbers, Benford’s law, Equidistribution

Explore it The Door You Should Not Keep, Five Primes and Then Never Again, The 6174 Attractor

Here, a proper divisor is a positive divisor smaller than the number itself.

The positive divisors of 12 are 1, 2, 3, 4, 6 and 12. Excluding 12 leaves its proper divisors, whose sum is 16. Keeping this exclusion clear matters: including the number itself would define a different process. A prime has only one proper divisor, namely 1. The number 1 has none, so the sum of its proper divisors is zero, using the empty-sum convention.

Example The proper divisors of 6 are 1, 2 and 3; they sum to 6.

Connected ideas Divisor, Aliquot sequence, Fixed point, Perfect number, Amicable numbers

Explore it Sum the Divisors and Go Again

A quadratic expression or equation has a highest power of two.

A quadratic graph is a parabola, and its coefficients control the opening, position and roots. Quadratics are the first setting where algebraic formulas, symmetry and geometry meet in a visibly rich way. The term describes degree, not a particular shape or sign pattern. The equation x² = 9 has two real solutions, 3 and −3. A squared variable therefore cannot generally be undone by keeping only the positive square root.

Example x² − 5x + 6 = 0 is quadratic.

Explore it Two Ways to Corner π, Pascal Modulo m

One radian is the angle that cuts off a circular arc whose length equals the circle’s radius.

Instead of dividing a full turn into 360 degrees, measure the arc length and divide it by the radius. The resulting ratio is the angle in radians. A full circle has circumference 2πr, so a full turn is 2π radians. Radians make formulas for circular motion and calculus simpler because they come directly from the circle’s geometry. They measure an angle, not the radius or a distance across the circle.

Example A half-turn is π radians, and a quarter-turn is π/2 radians.

Connected ideas Angle, Circular arc, Unit circle, Pi, Euler’s formula

Explore it Times Tables on a Circle, Two Ways to Corner π

A radius is a straight segment from the centre of a circle to its rim, or the length of that segment.

Every point on a circle is the same distance from its centre. That common distance is the radius, often written r. Doubling it gives the diameter, the distance across the circle through its centre. A chord need not pass through the centre and need not equal the radius. In a times-table drawing, the multiplier changes which rim positions are joined, not the radius or a sequence of line lengths.

Example A circle of radius 5 has diameter 10.

Connected ideas Diameter, Chord, Circumference, Epicycloid, Tangent

Explore it Times Tables on a Circle, Two Ways to Corner π

A random variable assigns a numerical value to each possible outcome of a random experiment.

Before a die is rolled, its score is uncertain. The random variable records that score, while its probability distribution says how likely the possible values are. Different random variables can be defined on the same experiment: one may record the score and another whether it is even. An observed value is one outcome, not the whole distribution. Simulations generate repeated observations to investigate features such as averages, variability and estimation error.

Example For a fair die score X, each of 1 through 6 has probability 1/6.

Connected ideas Probability, Expected value, Variance, Statistical independence, Uniform distribution

Explore it Three Guesses Beat One, Two Ways to Corner π

A random walk is a path made of steps whose direction is chosen by chance, such as up one unit or down one unit on a coin toss.

Take a position on a number line and move it up one with chance p and down one otherwise, again and again. The position after many steps is not predictable, but its behaviour is: for a fair coin it wanders as far as about the square root of the number of steps, and it keeps returning to places it has visited. A walk can be stopped when it reaches a chosen place, which turns questions about the long run into questions about where and when it stops.

Example Your bankroll in a bet staking one unit a round is a random walk on the whole numbers.

Connected ideas Probability, Expected value, Absorbing barrier, Gambler’s ruin

Explore it The Gambler’s Ruin, Two Losing Games That Win Together

A ratio compares two quantities by division, describing how large one is relative to the other.

If a recipe uses two cups of water for one cup of rice, the water-to-rice ratio is 2:1. Doubling both amounts keeps that ratio unchanged. Ratios differ from differences: ten is five more than five but twice as large. In a growing sequence, dividing each term by the previous one measures a growth factor. A few similar ratios suggest a pattern but do not establish that a limiting ratio exists.

Example The ratio 6:3 is the same as 2:1.

Connected ideas Fraction, Golden ratio, Limit, Fibonacci number, Geometric series

Explore it The Sequence That Reads Itself Aloud, Tiles That Never Repeat

A rational number can be written as a fraction of two integers with a non-zero denominator.

Rational decimals terminate or repeat because long division eventually revisits a remainder. They are closed under addition, subtraction, multiplication and division by a non-zero rational. Exact rational arithmetic is valuable when a decimal approximation would blur a bound or make a claim look stronger than it is. Writing 1/3 as 0.333 truncates the value, whereas the fraction keeps it exact. This is why some laboratories keep fractions throughout their calculations.

Example −3/4 and 2 are rational.

Connected ideas Real number, Fraction

Explore it Two Ways to Corner π, The Sum That Is Not a Sum, Three Guesses Beat One

Real numbers are the numbers represented by positions on a continuous number line, including rational and irrational numbers.

Integers such as −2, fractions such as 3/4, and irrational numbers such as √2 and π all belong to the real numbers. Between any two distinct real numbers there are more real numbers, so this is not a discrete counting system. A computer usually stores only finite approximations to arbitrary real values. An experiment using floating-point values therefore samples or approximates a real-number problem rather than representing every possible real number exactly.

Example −2, 0.75 and π are real; a number whose square is −1 is not real.

Connected ideas Rational number, Irrational number, Square root

Explore it Differences Around a Circle, Two Ways to Corner π, Three Guesses Beat One

A recurrence defines later terms from earlier terms together with initial conditions.

The rule describes change locally, while the starting values anchor the whole sequence. Solving a recurrence can reveal a closed formula, a growth rate or a conserved quantity. Recurrences are especially useful when a visual process is easier to state as “what happens next” than as one large expression. For the displayed rule, choosing a₀ = 1 gives 1, 1, 2, 4, 7. A different starting value produces a different sequence from exactly the same recurrence.

Example aₙ₊₁ = aₙ + n is a recurrence.

Connected ideas Sequence, Iteration, Orbit, Fibonacci number, Mathematical induction

Explore it Counting Around the Circle, Recamán’s Sequence, The Sequence That Reads Itself Aloud

Regression models a relationship between variables so one can be predicted or explained from others.

A regression line summarises a chosen model and loss function; it does not automatically reveal causation. Residuals show what the model misses, and uncertainty describes how much the estimate would vary under repeated data. The same language applies to simple lines and much richer polynomial or probabilistic models. A line relating height to shoe size could help predict one from the other. It would not imply that changing someone’s shoes changes their height.

Example Least-squares regression chooses the line with smallest total squared error.

Explore it Three Guesses Beat One

A regular polygon has all side lengths equal and all interior angles equal.

An equilateral triangle and a square are regular polygons. A rectangle that is not a square has equal angles but unequal side lengths, so it is not regular. Joining neighbouring, equally spaced points around a circle produces a regular polygon. Joining every pair adds diagonals and can create crossings that coincide because of symmetry. That is why a highly regular drawing may behave differently from a slightly perturbed one in the circle-regions exploration.

Example Six equally spaced points joined in order form a regular hexagon.

Connected ideas Polygon, Vertex, Symmetry, Constructible polygon

Explore it 1, 2, 4, 8, 16, and Then Not 32, A Number With No Room to Be Written

Regularisation is a method for assigning a controlled value or object when an ordinary calculation diverges.

A regularisation changes the question or adds a limiting procedure so that a useful finite quantity can be extracted. The result can preserve algebraic relationships without being the usual sum or integral. Any honest account states the method, its domain and which operations remain justified. For 1 + 2 + 3 + ..., every ordinary partial sum increases. Giving a regularised value of −1/12 does not change any of those positive totals.

Example Zeta regularisation assigns a value to some divergent expressions.

Connected ideas Analytic continuation, Divergent series, Cesàro summation, Abel summation, Riemann zeta function

Explore it The Sum That Is Not a Sum

The remainder is what is left after making as many complete groups of a chosen positive integer size as possible.

If seventeen counters are packed into groups of five, fifteen counters make three complete groups and two remain. We write 17 = 3 × 5 + 2. For division by a positive integer m, the remainder is at least zero and smaller than m. A zero remainder means the division is exact. Keeping just the remainder discards size information but retains patterns of divisibility.

Example 17 divided by 5 has quotient 3 and remainder 2.

Connected ideas Modulus, Modulo, Divisor, Modular arithmetic

Explore it Pascal Modulo m, Times Tables on a Circle, A Number With No Room to Be Written

A repunit is a number written as a string of repeated one digits in a chosen base.

In base ten, 111 is the third repunit, often written R₃. Repunits connect digit patterns to geometric sums and divisibility, and some are prime while most are not. The base must be stated because the same string has a different value in another numeral system. For a base b greater than one, a length-n repunit equals (bⁿ − 1)/(b − 1). Changing the length therefore changes its factors in structured ways.

Example R₄ = 1111 in base ten.

Explore it Primes That Survive Every Shuffle, The Spiral, and What It Is Not

A residue is the remainder, or the class of integers sharing that remainder, modulo a chosen modulus.

Modulo 7 has seven residue classes, represented by 0 through 6. Adding or multiplying residues stays inside that finite set. A diagram of residues is a compressed view of integer arithmetic, useful when a rule repeats or when a large number matters only through its remainder. Think of sorting all integers into labelled boxes. The box marked 2 modulo 5 contains 2, 7, 12 and also −3, because their differences are multiples of 5.

Example The residue of 29 modulo 6 is 5.

Connected ideas Parity, Congruence

Explore it Times Tables on a Circle, Pascal Modulo m, The Spiral, and What It Is Not

The Riemann zeta function agrees with the series of n to the power −s where that series converges, and continues uniquely beyond that region as an analytic function.

For real s greater than 1, ζ(s) is the sum 1^(−s) + 2^(−s) + 3^(−s) + .... That series diverges at s = −1, so the symbol ζ(−1) does not mean the ordinary sum 1 + 2 + 3 + .... It means the value at −1 of the analytic continuation of zeta, which equals −1/12. Keeping the series and the continued function apart is the whole point of careful accounts of that famous number.

Example ζ(2) = π²/6, while ζ(−1) = −1/12 by analytic continuation.

Connected ideas Analytic continuation, Divergent series, Regularisation, Cesàro summation, Abel summation

Explore it The Sum That Is Not a Sum

A shape has non-trivial rotational symmetry when a turn smaller than a full revolution makes it coincide with itself.

An unmarked square matches itself after turns of 90, 180, 270 and 360 degrees, giving rotational symmetry of order four. A non-square rectangle has order two. Counting includes the full turn, so order one means no smaller turn works. In the circle laboratory, the symmetry order describes the entire chord arrangement. It should not be confused with the number of steps in one point’s repeated-multiplication orbit.

Example An equilateral triangle has rotational symmetry of order 3.

Connected ideas Symmetry, Orbit, Angle

Explore it Times Tables on a Circle, 1, 2, 4, 8, 16, and Then Not 32

Run-length encoding rewrites a row of symbols as the length of each block of identical neighbours, in order.

A block of identical neighbours is called a run. The row 1, 2, 2, 1, 1, 2 has four runs, so its encoding is 1, 2, 2, 1. Nothing is lost when the symbols alternate between two values, because the encoding plus the first symbol rebuilds the row exactly. Two laboratories turn on this operation and use it differently: the look-and-say rule writes each count beside its symbol and feeds the result back in, while the Kolakoski sequence is the one row whose encoding is the row itself.

Example The row 1, 2, 2, 1, 1, 2 encodes as 1, 2, 2, 1.

Connected ideas Self-reference

Explore it The Kolakoski Sequence, The Sequence That Reads Itself Aloud

Sample size is the number of observations or trials included in a statistical calculation.

Estimating a probability from ten trials usually gives a less stable result than using ten thousand comparable independent trials. The count is the sample size, often written n. Its usefulness depends on how the observations were obtained: repeated copies of one observation do not provide the information of independent measurements. A larger simulation reduces random sampling fluctuations under its model, but cannot establish that the model matches reality or that an untested universal claim is true.

Example Counting successes in 1000 simulated coin tosses uses sample size 1000.

Connected ideas Standard error, Monte Carlo, Probability

Explore it Three Guesses Beat One, Two Ways to Corner π

A sample space is the collection of all possible outcomes specified by a probability model.

For two people and three possible birthdays, an outcome records both dates in order, such as (1, 3). There are nine possible ordered pairs. Three are matches: (1, 1), (2, 2) and (3, 3). If assignments are independent and uniform, all nine outcomes are equally likely, giving match probability three ninths. A different model may keep the same sample space while assigning different probabilities to its outcomes.

Example Three people with three possible dates give 3³ = 27 ordered outcomes.

Connected ideas Probability, Uniform distribution, Statistical independence, Conditional probability, Complementary event

Explore it Birthday Matches in a Small Room, Two Ways to Corner π

Self-reference occurs when a statement, program or object refers to itself.

Everyday paradoxes show how unstable unrestricted self-reference can be. Formal mathematics controls it by encoding syntax as numbers or strings, then applying a carefully defined construction. The result can expose limits of a system without requiring the system to contain a literal mirror or an infinite loop. A phrase such as “this sentence contains five words” points back to the sentence containing it. Formal encodings replace that informal pointing with operations whose behaviour can be proved.

Example A sentence that describes its own code is self-referential.

Connected ideas Diagonal lemma, Gödel numbering, First incompleteness theorem, Run-length encoding

Explore it A Sentence About Its Own Number, The Sequence That Reads Itself Aloud

A self-similar object contains smaller parts that reproduce its overall pattern after rescaling.

The copies may be exact, approximate or arranged by a rule that repeats at each stage. Self-similarity explains why a small local construction can fill an entire figure with related detail. It is a description of structure, not a guarantee that every visible patch is identical. For exact copies, lengths change by one scale factor while angles stay the same. A photo that merely resembles another at a different size need not satisfy this definition.

Example Each corner of the Sierpiński gasket contains a smaller gasket.

Connected ideas Fractal, Sierpiński gasket, Substitution, Iterated function system

Explore it Pascal Modulo m, Tiles That Never Repeat

A sequence is an ordered list of values, usually indexed by whole numbers.

A sequence may be defined by a formula, a recurrence or a process that produces one term from another. Order matters, and repeated values can still carry information about time or position. Looking at initial terms suggests patterns; a proof must explain why the pattern persists for every index in scope. The repeated 1 in the Fibonacci sequence occupies two different positions. Knowing a few values does not determine a unique rule for all the later terms.

Example 1, 1, 2, 3, 5, ... is a sequence.

Connected ideas Iteration, Recurrence, Orbit, Triangular number, Bit

Explore it The Collatz Orbit Atlas, Recamán’s Sequence, The Sequence That Reads Itself Aloud

Shrinkage pulls an estimate toward a shared centre or model instead of leaving it fully independent.

The pull trades a little bias for a potentially larger reduction in variance. James–Stein estimation is surprising because this trade can lower total squared error when estimating several means at once. The benefit depends on dimension and loss function, so the context is part of the result. A lower expected total error does not promise a better answer in each trial or for every individual coordinate. The improvement is an average property under the specified model.

Example A noisy estimate can be shrunk toward zero.

Connected ideas Estimator, Squared error, Confidence interval

Explore it Three Guesses Beat One

The Sierpiński gasket is the triangular fractal left after recursively removing central triangles.

Start with a filled triangle, remove its open middle, and repeat on the three corner copies. The area shrinks toward zero while the boundary and detail remain richly structured. Pascal’s triangle modulo two produces the same pattern when its odd entries are kept, linking a number array to a geometric construction. At each stage, every remaining triangle is replaced by three copies with half its side length. Counting odd entries in Pascal’s triangle reveals the same threefold replication.

Example The first stage has three smaller corner triangles.

Connected ideas Fractal, Self-similar

Explore it Pascal Modulo m

The sinc function is sin(x) divided by x, a wave that starts at 1 and ripples away to nothing.

The curve sinc(x) = sin(x)/x equals 1 at x = 0, crosses zero at π, 2π, 3π and so on, and its ripples shrink as x grows. Its area from 0 to infinity is π/2, a result that needs care because the positive and negative ripples cancel and the curve is not absolutely integrable. Stretching it, as in sinc(x/3), gives the same shape three times as wide. Its Fourier transform is a flat rectangle, which is why products of sinc functions turn into sums of uniform random numbers.

Example The area under sin(x)/x from 0 to infinity is exactly π/2.

Connected ideas Convolution, Uniform distribution

Explore it π/2, π/2, π/2 … and Then Not

The sine of an angle is the vertical coordinate reached by turning through that angle anticlockwise from (1, 0) on the unit circle.

As the point travels around the circle, its height rises from zero to one, falls through zero to minus one, then returns to zero. That repeated height pattern is a sine wave. In a right triangle, sine also equals the length of the opposite side divided by the hypotenuse. The circle definition extends the idea to any real angle. Be explicit about units: sin(π/2) uses radians and equals one, while an input of ninety means the same turn only in degrees.

Example sin(0) = 0, sin(π/2) = 1, and sin(π) = 0.

Connected ideas Cosine, Unit circle, Radian, Euler’s formula

Explore it The Number Behind Continuous Growth, Two Ways to Corner π

A square number is an integer obtained by multiplying an integer by itself.

Nine objects can form a three-by-three square, and sixteen can form a four-by-four square. This arrangement explains the name and the notation n². Consecutive non-negative square numbers differ by successive odd numbers: 1, 3, 5, 7 and so on. Every square number is non-negative, even when the integer being squared is negative. A perfect square is unrelated to a perfect number, which is defined using sums of proper divisors.

Example 25 = 5² is a square number; 26 is not.

Connected ideas Square root, Exponent, Perfect number

Explore it Happy Numbers, The Spiral, and What It Is Not

A square root of a number is a value that gives that number when multiplied by itself.

Since 3 × 3 = 9, three is a square root of nine. Negative three also squares to nine, but the symbol √9 means the non-negative square root, namely three. Every non-negative real number has one non-negative square root. Some, such as √2, cannot be written exactly as a fraction of integers. Negative real numbers have no real square roots, although complex numbers extend the number system to handle them.

Example √25 = 5, while the equation x² = 25 has solutions 5 and −5.

Connected ideas Exponent, Irrational number, Real number, Equation, Square number

Explore it Two Ways to Corner π, Tiles That Never Repeat, Three Guesses Beat One

Squared error is the square of the difference between an estimate and the value it aims to measure.

Squaring removes the sign and makes larger mistakes count disproportionately. Averaging squared errors across repeated samples gives mean squared error, which combines variance and squared bias. It is one useful criterion, not a universal definition of what makes an estimate good. An error of four contributes sixteen, while an error of two contributes four. A few large errors can therefore dominate a comparison using this measure.

Example Estimating 10 by 8 gives squared error (8−10)² = 4.

Connected ideas Estimator, Shrinkage, Confidence interval, Bias, Absolute value

Explore it Three Guesses Beat One

Standard deviation measures spread around the mean and is the square root of variance.

Subtract the mean from each possible value, square those differences and average them with the appropriate weights to get variance. Taking its square root returns to the original units: centimetres rather than square centimetres, for example. A small standard deviation means values cluster near the mean. Sample formulas can use different denominators, so the convention matters. No fixed percentage falls within one standard deviation for every distribution.

Example Equally likely values −2 and 2 have mean 0, variance 4 and standard deviation 2.

Connected ideas Variance, Normal distribution, Arithmetic mean, Standard error

Explore it Three Guesses Beat One, Two Ways to Corner π

The standard error is the standard deviation of an estimator’s sampling distribution, often itself estimated from data.

The standard deviation of observations describes how individual measurements vary. The standard error describes how an estimate, such as their average, would vary across repeated samples. For independent observations with a common finite variance, the standard error of the sample mean is the population standard deviation divided by the square root of sample size. More observations can reduce that variability, but they do not automatically remove bias or repair a badly chosen sampling procedure.

Example With population standard deviation 10 and 100 independent observations, the mean’s standard error is 1.

Connected ideas Standard deviation, Sample size, Estimator

Explore it Three Guesses Beat One, Two Ways to Corner π

A state space is the collection of all configurations a rule can occupy.

Each point in a state space is a complete state, not merely a summary statistic. A deterministic rule draws an arrow from every state to its successor, making a directed graph. Finite state spaces can be enumerated; infinite ones need bounds, abstractions or proofs to support claims. For Langton’s ant, the state includes every square’s colour together with the ant’s position and direction. Its position alone does not determine the next move.

Example The four-digit strings 0000 through 9999 form a finite state space.

Connected ideas Cycle, Fixed point, Attractor, Basin, Eventual periodicity

Explore it The 6174 Attractor, Differences Around a Circle, The Ant That Builds a Road

A stationary distribution of a Markov chain is a set of state probabilities that one more step leaves unchanged.

If the chain starts with these probabilities, it has them after every step, so they describe the share of time spent in each state over a long run. For a finite chain in which every state can be reached from every other there is exactly one, and it can be found by solving a system of linear equations, exactly if the transition probabilities are fractions. Reading averages off it replaces a very long simulation with a short calculation, and it does not depend on where the chain began.

Example The three residues are equally likely, at one third each, when only game A is played.

Connected ideas Markov chain, Expected value, Law of large numbers

Explore it Two Losing Games That Win Together

Two events are statistically independent when the probability that both occur equals the product of their individual probabilities.

If two fair coin tosses are independent, the chance of two heads is one half times one half, or one quarter. Knowing the first outcome does not change the probabilities for the second. The birthday model treats each person’s date as an independent assignment. However, the events that different pairs match are not all mutually independent: if A matches B and B matches C, then A necessarily matches C.

Example Independent fair tosses give P(heads then heads) = 1/2 × 1/2 = 1/4.

Connected ideas Conditional probability, Probability, Random variable, Sample space, Consecutive-prime bias

Explore it Birthday Matches in a Small Room, Two Ways to Corner π

A stopping time is the first step at which an iterated process meets a chosen condition.

The condition might be reaching one, hitting a fixed point or dropping below the starting value. It is a measurement of one orbit, not a universal guarantee that the condition will ever occur. Comparing stopping times reveals how similar starts can have very different difficulty or delay. In Collatz terminology, stopping time usually means first dropping below the start, while total stopping time means reaching 1. These can give different counts for the same orbit.

Example The stopping time to 6174 counts the changes before the fixed point.

Explore it The Collatz Orbit Atlas, The 6174 Attractor, Happy Numbers

Substitution replaces specified symbols or expressions according to a stated rule.

In algebra, substituting x = 3 into x + 2 gives 5. For a sequence of symbols, a substitution rule can instead replace each symbol by a whole string. The rules 0 → 01 and 1 → 10 turn 01 into 0110 when applied to both original symbols simultaneously. Do not replace newly written symbols again during that same round. Repeating complete rounds generates longer strings with structure inherited from the replacements.

Example Under 0 → 01 and 1 → 10, the string 10 becomes 1001.

Connected ideas Concatenation, Iteration, Self-similar

Explore it A Pattern Built by Flipping Its Copy, Tiles That Never Repeat

A function is surjective when every output in its stated target collection is reached by some input.

Imagine drawing an arrow from each input to its output. Surjectivity says that no member of the declared target collection is left without an incoming arrow. Several inputs may still reach the same output. The target matters: n ↦ 2n is not surjective from the integers to all integers, because it misses the odd ones, but it is surjective when the target is the even integers.

Example Squaring real numbers reaches every non-negative real number, but no negative one.

Connected ideas Function, Injective, Bijection

Explore it A Sentence About Its Own Number, 1, 2, 4, 8, 16, and Then Not 32

A symmetry is a transformation that leaves an object looking exactly the same.

A square still occupies the same outline after a quarter-turn, and a butterfly-like outline can match itself after reflection in a vertical mirror line. These are different types of symmetry. In mathematics, the transformation and the object both need to be specified: a coloured square may lose a symmetry that its uncoloured outline has. The symmetry of a whole chord drawing is also different from the route followed by one numbered point.

Example An unmarked square has both reflection and quarter-turn rotational symmetries.

Connected ideas Rotational symmetry, Regular polygon, General position

Explore it Times Tables on a Circle, 1, 2, 4, 8, 16, and Then Not 32, The Ant That Builds a Road

A tangent line follows a curve’s local direction at a point; for a circle it touches the rim without cutting through it.

For a circle, the tangent at a rim point is perpendicular to the radius ending there. A line cutting through the circle at two points is instead a secant. For more general curves, a tangent can cross the curve, so touching without crossing is not a universal definition. In the chord exploration, the chords are tangent to an inner envelope even though they cut across the surrounding circle.

Example At the top of a circle, its tangent line is horizontal.

Connected ideas Radius, Envelope, Derivative, Cusp

Explore it Times Tables on a Circle

A Taylor series uses a function’s derivatives at one point to form an infinite power series; a Maclaurin series centres it at zero.

A constant can match a function’s value at a point. Adding a linear term can match its slope, and further powers can match higher derivatives. Stopping after finitely many terms gives a polynomial approximation whose error must be considered. A Taylor series does not automatically converge or equal its original function everywhere. For the exponential function it does converge to the function for every real or complex input, which provides a bridge between growth, sine, cosine and rotation.

Example eˣ = 1 + x + x²/2! + x³/3! + ...; at x = 1 this approaches e.

Connected ideas Derivative, Factorial, Partial sum, Exponential function

Explore it The Number Behind Continuous Growth, Two Ways to Corner π

Termination means that a process reaches an accepted end condition after finitely many steps.

A program can terminate at a fixed point, a target state or an explicit failure. Showing termination requires more than observing many successful runs when the inputs are unbounded. A decreasing quantity, a finite state argument or a well-founded order can provide the missing proof. A repeated state alone proves a loop for a deterministic rule, not termination. It becomes an end condition only if the experiment explicitly stops when it detects repetition.

Example A loop that reaches 0 and stops terminates.

Explore it The Collatz Orbit Atlas, The 6174 Attractor, A Sentence About Its Own Number

A theorem is a mathematical statement established by a proof from accepted definitions and earlier results.

The word signals a level of justification, not simply a statement that sounds certain. A theorem has a scope: its hypotheses say where the conclusion applies. A page may cite a theorem without reproducing its proof, but it should not quietly broaden the result beyond those hypotheses. For example, the uniqueness statement concerns positive integers greater than one and disregards factor order. Specifying those conditions is part of saying exactly what the theorem proves.

Example The fundamental theorem of arithmetic says prime factorisation is unique.

Connected ideas Proof, Conjecture, Counterexample, Axiom, Hypothesis

Explore it A Sentence About Its Own Number, Every Even Number, Two Primes, Tiles That Never Repeat

The Thue-Morse sequence starts at 0 and grows by repeatedly appending a copy with every bit flipped.

The first completed rows are 0, 01, 0110 and 01101001. Each row remains the beginning of the next, which defines an infinite sequence. Another way to find the bit at position n, counting from zero, is to count the ones in the binary writing of n and keep their parity. The sequence has exact structure but never settles into repetition of a fixed block. A finite displayed prefix cannot establish that last theorem by itself.

Example Position 5 has binary index 101, with two ones, so its bit is 0.

Connected ideas Bitwise complement, Binary, Parity, Eventual periodicity

Explore it A Pattern Built by Flipping Its Copy

A tiling covers a region with shapes whose interiors do not overlap and leave no gaps.

The shapes are called tiles, and rules may constrain which edges can meet. Periodic tilings repeat a finite pattern; aperiodic tilings cover the plane without any translational repeat. Drawing a patch cannot by itself establish what happens on the whole infinite plane. Floor tiles offer a finite example you can check directly: each interior point should lie in a tile, with no area covered twice. Infinite tilings require the same condition everywhere.

Example A square grid is a periodic tiling.

Connected ideas Aperiodic, Golden ratio

Explore it Tiles That Never Repeat

A torus is a surface shaped like a doughnut, or a rectangle whose opposite edges are identified.

On a torus, leaving the right edge brings you back at the left and the top connects to the bottom. A finite grid can therefore have wraparound boundaries without corners. This changes paths and long-term behaviour while keeping the number of cells finite, which is why it is a useful laboratory setting. A drawn seam is a display convention, not a boundary of the surface. A path crossing it continues immediately on the opposite side without bouncing or disappearing.

Example A 20 by 20 wraparound board is a discrete torus.

Explore it The Ant That Builds a Road, Differences Around a Circle

A relation is transitive when a link from a first thing to a second and from the second to a third always gives a link from the first to the third.

Being greater than is transitive: if 6 is greater than 4 and 4 is greater than 3, then 6 is greater than 3. A comparison that counts something instead of reading off one number need not be. If one die beats another when it wins more of the pairings between their faces, three dice can each beat the next and close a loop, and such a set is called intransitive. The comparison is between two lists of faces, not two numbers, and a ranking needs one number for each thing ranked. Dice whose faces are all equal are only numbers, so for them the loop cannot happen.

Example Three dice can each beat the next, which no three numbers can do.

Connected ideas Probability

Explore it Three Dice, Each Beaten by the Next, The Coin Bet You Can Always Beat

Trial division tests whether a number is prime by dividing it by every possible divisor in turn.

To decide whether 257 is prime, divide it by 2, 3, 4 and so on up to 16, which is the largest whole number whose square does not exceed 257. Any larger divisor would pair with a smaller one already tried. If no division comes out exactly, the number is prime, and that verdict is complete rather than probable. The cost grows with the square root of the number, so the method settles 65,537 quickly and gives up long before a twenty-digit number.

Example Dividing 91 by 2, 3, 5 and 7 finds 91 = 7 × 13, so 91 is not prime.

Connected ideas Prime number, Factorisation

Explore it Five Primes and Then Never Again, Primes That Survive Every Shuffle

The nth triangular number counts dots in rows of lengths 1 through n, giving n(n + 1)/2.

Arrange one dot in the first row, two in the second and three in the third. There are six dots altogether, forming a triangular arrangement. Adding a row of four makes ten. Pairing two copies of the arrangement creates a rectangle with n rows and n + 1 columns, explaining the division by two in the formula. Triangular numbers also appear as a diagonal of Pascal’s triangle.

Example The first five positive triangular numbers are 1, 3, 6, 10 and 15.

Connected ideas Sequence, Binomial coefficient

Explore it Pascal Modulo m, A Sentence About Its Own Number

A finite uniform distribution assigns the same probability to every possible outcome; the continuous version assigns equal probability to equal-length subintervals.

A fair six-sided die gives each face probability one sixth. In the simple birthday model, every one of 365 dates receives probability 1/365. This is a modelling assumption, not a claim that real births occur equally often on every date. For a continuous uniform distribution on an interval, individual points have probability zero, while subinterval probabilities depend on their lengths relative to the whole interval.

Example In a seven-day uniform model, each numbered day has probability 1/7.

Connected ideas Probability, Random variable, Normal distribution, Sample space, Birthday problem

Explore it Birthday Matches in a Small Room, Two Ways to Corner π, π/2, π/2, π/2 … and Then Not

The unit circle has radius one and is usually centred at the coordinate origin.

Choosing radius one removes an unnecessary size parameter, letting you study a shape without choosing centimetres or pixels. In Cartesian coordinates its points satisfy x² + y² = 1. A point at angle θ from the positive horizontal axis has coordinates (cos θ, sin θ). A screen drawing can scale these coordinates by any radius without changing the underlying geometry. The circle laboratory uses this separation between mathematical coordinates and display size.

Example (1, 0), (0, 1) and (−1, 0) lie on the unit circle.

Connected ideas Radius, Coordinate, Angle, Radian, Imaginary unit

Explore it Times Tables on a Circle, Two Ways to Corner π

Up-arrow notation is Knuth’s compact language for repeated exponentiation and still faster operations.

One arrow means exponentiation, two means iterated exponentiation and additional arrows repeat the same idea at a higher level. The notation makes Graham’s number writable, but it does not make the resulting quantities intuitively small. Expanding a few low cases is the safest way to see what a new arrow means. Exponent towers are evaluated from the top down: 3 ↑↑ 3 is 3²⁷, not 27³. Even this small example shows why grouping cannot be left implicit.

Example 3 ↑ 3 means 3³, while 3 ↑↑ 3 means 3^(3^3).

Connected ideas Exponent

Explore it A Number With No Room to Be Written

A variable is a symbol representing a quantity whose value can vary or has not yet been specified.

Writing x + 2 lets one expression describe many additions at once. If x is three, the result is five; if x is ten, it is twelve. A problem may instead ask which value of x makes an equation true. Letters have no built-in numerical meaning: their role and allowed values come from the surrounding definition. One variable might count points while another describes a multiplier or a changing angle.

Example In 2x = 10, solving for the variable gives x = 5.

Connected ideas Equation, Coefficient, Function

Explore it Times Tables on a Circle, The Spiral, and What It Is Not

Variance is the expected squared distance of outcomes from their mean.

It measures spread while giving extra weight to large deviations. Its square root is the standard deviation, in the same units as the data. Variance belongs to a distribution or a stated sample calculation, so comparing two values requires matching definitions and assumptions. If every measurement doubles, its distance from the mean doubles and the variance becomes four times larger. This is why variance is expressed in squared units.

Example A constant random variable has variance zero.

Connected ideas Standard deviation, Normal distribution, Random variable, Bias, Arithmetic mean

Explore it Three Guesses Beat One, Two Ways to Corner π

A vector is an ordered quantity with components that can represent direction, displacement or coordinates.

Vectors can be added and scaled, and a basis describes how their components encode an object. In geometry they describe arrows; in data analysis they can hold several measurements; in a matrix calculation they are transformed as a unit. The interpretation depends on the space and operations declared. In the pair (3,4), the first component can mean three units right and the second four units up. The order carries information, so (4,3) describes a different displacement.

Example (3, 4) is a vector of length 5 in the plane.

Connected ideas Matrix, Coordinate, Lattice, Eigenvalue, Eigenvector

Explore it Two Ways to Corner π, Three Guesses Beat One, Tiles That Never Repeat

A vertex is a corner of a geometric shape, or a node in a graph.

A triangle has three vertices, one where each pair of neighbouring sides meets. In a graph, a vertex need not be a physical corner: it is an object that edges connect to other objects. When counting regions in a drawing, crossings may become additional graph vertices even though they were not original polygon corners. Keep track of which model is being counted, since the same picture can support several different collections of vertices.

Example A square has 4 vertices; adding a crossing of its diagonals introduces another graph vertex.

Connected ideas Node, Edge, Polygon, Cusp, Regular polygon

Explore it 1, 2, 4, 8, 16, and Then Not 32, A Number With No Room to Be Written