The Number Behind Continuous Growth

In the compound-growth view, n tells us how many times growth is added. Each time, the current amount is multiplied by 1 + 1/n. For n = 2, that multiplier is 1.5, applied twice. For n = 4, it is 1.25, applied four times. We hold the nominal total rate at 100%, so increasing n does not mean increasing the rate assigned to each interval.

Run and Step compare n = 1, 2, 4, 8 and so on. A dot represents the final amount from one complete experiment, not the balance halfway through it. The transcript records every calculation, including steps that have not yet been drawn.

The exponential function gives meaning to e raised to different powers. In the series view, start with 1, add x, then x²/2!, then x³/3!, and continue. A factorial such as 3! means 3 × 2 × 1 = 6. At x = 1, the first sums are 1, 2, 2.5 and approximately 2.666667. They approach the same e that appeared in compounding.

At x = 2 the sum instead approaches e², about 7.389. At x = −1 it approaches e⁻¹ = 1/e, about 0.367879. Negative powers here describe decay rather than a negative amount. The early partial sums can overshoot or even be negative, especially when x is negative; only the infinite sum is the positive exponential value.

The imaginary unit i is defined by i² = −1. A complex number a + bi can be drawn at horizontal coordinate a and vertical coordinate b. On Euler’s circle, start at 1, the point (1, 0), and move counterclockwise. A radian measures angle by arc length divided by radius, so a half-turn is π radians and a full turn is 2π radians.

Euler’s formula states that e raised to iθ has coordinates cos θ and sin θ:

$e^{i\theta }=\cos \theta +i\sin \theta .$

At a half-turn, those coordinates are (−1, 0), so e^(iπ) = −1. Adding 1 gives the familiar Euler’s identity, e^(iπ) + 1 = 0. This relates e, π, i, 1 and 0 without claiming that an ordinary positive real power has become negative; the exponent is imaginary.

The expression 1/(1 − e) is a perfectly well-defined negative number, about −0.581977. It is not the ordinary sum of 1 + e + e² + ⋯. That series keeps adding positive terms which grow larger, so its partial sums cannot approach that negative number.

The relevant formula is 1 + r + r² + ⋯ = 1/(1 − r), and it requires |r| less than 1. Use r = 1/2 to approach 2, or r = 1/e to approach 1/(1 − 1/e), about 1.581977. The comparison view lets you try both convergent ratios and r = e, where the condition fails. Neither series is the factorial-weighted series that defines eˣ.

For the geometric partial sum Sₘ = 1 + r + ⋯ + rᵐ, multiplying by r and subtracting cancels all the middle terms. Thus (1 − r)Sₘ = 1 − rᵐ⁺¹. If |r| is less than 1, the final power tends to zero, giving the claimed limit. If |r| is at least 1, the terms themselves do not tend to zero, which already rules out ordinary convergence.

For compounding, the binomial theorem gives

${\left(1+\frac{1}{n}\right)}^n={\sum }_{k=0}^n\frac{1}{k!}{\prod }_{j=0}^{k-1}\left(1-\frac{j}{n}\right).$

Every product lies between zero and one, and for each fixed k it tends to one. The tail is bounded by the corresponding tail of ∑ 1/k!, a convergent series: after the first few terms, each new term is at most half its predecessor. Given any desired error, first choose a small factorial tail, then make n large enough for the remaining finitely many products to be close to one. This proves that the compounding limit is ∑ 1/k! = e.

For a complex exponent iθ, the exponential series is absolutely convergent, so its even and odd powers can be collected separately. Using i² = −1 produces the cosine series in the real part and the sine series in the imaginary part, yielding Euler’s formula.

The laboratory uses ordinary floating-point arithmetic. Its dashed references come from the browser’s exponential function, and the circle coordinates come from sine and cosine. These are numerical illustrations, not certified intervals or independently verified digits. The 16-term series can still visibly differ from eˣ, especially near the ends of the allowed range.

The statements above concern limits and exact identities, supported by their arguments and references. Increasing the number of drawn points cannot replace those arguments. A useful next investigation is to compare how quickly the compounding and factorial constructions approach e, then ask what error bound would certify a chosen number of decimal places.