Why One Comes First

Every value in the list is a positive whole number. Write it out in base 10 and look only at the digit on the left. That digit is one of 1 to 9, never 0, because a number is not written with a leading zero. The figure counts how many of the listed values begin with each digit and draws one bar per digit.

Across each bar sits a notch. That notch is the share predicted for that digit, which is log₁₀(1 + 1/d): about 30.1% for digit 1, 17.6% for 2, and 4.6% for 9. It is computed from the digit alone and does not know anything about the list, so it is a prediction placed beside the counts rather than a line fitted through them.

Beneath the bars the first thirty values are printed in full, with the first digit of each marked. At the smallest setting those thirty values are the entire list, so the whole tally can be checked by eye before any larger count is asked for.

Start at 1 and keep doubling. To move the first digit off 1 you need one doubling, from 1 to 2. To move it off 2 you need one more. But once you reach 8, a single doubling takes you to 16 and the first digit is 1 again, having skipped 9 entirely.

That asymmetry is not about doubling. It is about the ruler. On a scale where equal distances mean equal multiplications, the stretch occupied by the numbers beginning with 1, from 1 to 2, is a factor of 2 wide. The stretch beginning with 9, from 9 to 10, is a factor of 10/9 wide, which is about 1.11. The first stretch is more than six times the second.

So if a process moves along that multiplying scale without preferring any particular place on it, it will spend more of its time in the wide stretch near the front than in the narrow one at the back. The powers of two move along it in equal strides. The uniform draws do not move along it at all, and that is exactly why they behave differently.

Take logarithms and the multiplying becomes adding. Write 2ⁿ = 10^(n log₁₀ 2). The exponent n log₁₀ 2 splits into two parts: the whole number in front of the decimal point, which says only how many digits 2ⁿ has, and the part after the decimal point, which says what those digits are.

Throw away the whole part and keep the fraction. Every power of two is now one point in the interval from 0 to 1, and that is the strip the mechanism view draws. The powers beginning with a 1 are exactly the points below log₁₀ 2 ≈ 0.301, because a number whose logarithm has fractional part below 0.301 sits between 10^k and 2 × 10^k. That stretch is shaded, and Run drops the points onto the strip one at a time in order.

Now the question has changed shape. It is no longer "why does 1 come first" but "how do these points sit on the strip". They are the fractional parts of 0.301, 0.602, 0.903, 1.204, and so on, each one 0.301 further along and wrapping when it passes the end. Because log₁₀ 2 is irrational, no point ever lands where an earlier one did and the sequence never falls into a cycle. What it does instead is spread out evenly, and the share of the points inside a stretch becomes the length of that stretch.

Fix a base $b\ge 2$ and a first digit $d$ with $1\le d\le b-1$. A positive integer $N$ written in base $b$ begins with $d$ exactly when

for some integer $k\ge 0$. Taking logarithms in base $b$ and writing $\{x\}$ for the fractional part of $x$, this says

an interval of length ${\log }_b\left(1+\frac{1}{d}\right)$ inside $[0,1)$.

A sequence $(x_n)$ in $[0,1)$ is equidistributed when, for every subinterval $I$, the share of the first $n$ terms lying in $I$ tends to the length of $I$ as $n$ grows. Weyl proved that for irrational $\alpha$ the sequence $\{n\alpha \}$ is equidistributed. Since ${\log }_b{2}^n=n{\log }_{b}2$, the leading digits of the powers of two are governed by $\{n{\log }_{b}2\}$, and ${\log }_{b}2$ is irrational precisely when $b$ is not a power of $2$.

Put the two paragraphs above together. For a base $b$ that is not a power of $2$, the number ${\log }_{b}2$ is irrational, so by Weyl’s theorem the fractional parts $\{n{\log }_{b}2\}$ are equidistributed in $[0,1)$. The powers of two beginning with digit $d$ are exactly those whose fractional part lies in an interval of length ${\log }_b(1+1/d)$. The share of them therefore tends to that length, which is the prediction. The Fibonacci numbers follow the same route: the closed form makes ${\log }_{10}{F}_n$ equal to $n{\log }_{10}\phi$ plus a constant plus an error that tends to zero, and ${\log }_{10}\phi$ is irrational.

That is a proof about a limit, and this page is careful about the difference. The bars in the figure are counts of a finite list. The first thousand powers of two give 301 values beginning with 1, and the prediction for the limit is 30.1%, and the agreement of those two numbers is a pleasant fact about a thousand powers rather than a demonstration of the theorem. No finite count could be one, because a limit says nothing about any particular initial segment.

The uniform case is settled differently and more completely. The integers from 1 to 999,999 that begin with $d$ in base 10 are the six blocks $[d\cdot 10^k,(d+1)\cdot 10^k-1]$ for $k=0,\dots ,5$, holding $1+10+\cdots +100,000=111,111$ integers in total, the same for every $d$. That is the entire population counted, so it settles the model outright: uniform integers on that range lead with each digit exactly one ninth of the time, and one ninth is not ${\log }_{10}(1+1/d)$ for any $d$: it falls between the 12.49% predicted for digit 3 and the 9.69% predicted for digit 4.

The failure preset is the important part of this page. Switch the source to uniform draws and the bars flatten. Nothing has gone wrong with the code, and nothing has gone wrong with the digits: a uniform draw from a range that ends at a power of ten really does give every first digit the same share, and the laboratory computes that exact one-ninth distribution and draws it beside the sample rather than leaving the flatness to be inferred.

So the law is not a fact about numbers. It is a fact about processes that move multiplicatively over many orders of magnitude, and about how such a process distributes itself along a logarithmic ruler. A process that does not do that has no reason to obey it, and this page ships one that does not, on purpose.

There is a second failure on the page, and it is quieter. Write the powers of two in hexadecimal and they begin only with 1, 2, 4 or 8, each of them exactly a quarter of the time. Nothing has broken here either. Sixteen is itself a power of two, so the logarithm of 2 in base 16 is the exact fraction 1/4, and the wrapping points land on four positions and stay there. The theorem in Layer 5 asks for an irrational logarithm, and this base does not supply one. Bases 4 and 8 fail the same way. The condition in a theorem is not decoration, and this is what it looks like when it is not met.

That distinction is why the claim about real-world data is recorded here as a heuristic rather than a result. Benford tabulated twenty thousand collected values in 1938 and found the logarithmic shares; Hill showed in 1995 that samples drawn from randomly chosen distributions converge to them. Neither is a theorem about any dataset in front of you, and the behaviour is known to fail for data with a fixed number of digits, an imposed floor or ceiling, or values assigned rather than measured. This page computes nothing about collected data and therefore establishes nothing about it.

The open questions are about which processes qualify. There is no simple test that decides whether a given real-world quantity should follow the law, and the popular use of first-digit tallies to flag suspect accounts rests on assumptions about the underlying process that the tally itself cannot check. What this page can settle, it settles by counting; what it cannot, it labels.