Pi Digit Distribution

This animated chart tracks how often each digit (0 through 9) appears in the decimal expansion of π across the first 1,000 decimal places. At first the frequencies fluctuate wildly; by 1,000 decimal places they converge toward 10% each — a behaviour predicted by the normal number conjecture.

π and the Normal Number Conjecture

What does this chart show?

Each coloured line tracks the running frequency of one digit (0 through 9) as we read through the first 1,000 decimal places of π. After 10 decimal places the frequencies are erratic; by 1,000 they have already settled into a tight spread around 10%. The convergence is not a coincidence — it is exactly what is predicted if π is a normal number.

Why do the lines start so spread out?

With very few decimal places, any single digit that is over- or under-represented has a large percentage effect. For example, the first 10 decimal places of π are 1415926535 — digit 5 appears three times (30%) while digit 0 appears zero times (0%). Those extreme values are pure small-sample noise; they wash out as N grows.

What is a normal number?

A real number is said to be normal in base 10 if every digit (0–9) appears with equal asymptotic frequency 1⁄10, every pair of digits appears with frequency 1⁄100, and so on for every finite block of digits. The concept was introduced by Émile Borel in 1909. Almost all real numbers are normal in a measure-theoretic sense — yet proving that any specific "natural" constant is normal has proved extraordinarily difficult. Liouville's constant and Champernowne's number (0.123456789101112…) are known to be normal, but for π, e, or √2, no proof exists.

Is π proven to be normal?

No — and this is one of the deepest open questions in mathematics. Every statistical test applied to the known digits of π (now exceeding 200 trillion) is consistent with normality: no digit, pair, or block appears significantly more or less often than expected. But passing statistical tests is not the same as a proof. A rigorous proof of the normality of π would require understanding the deep arithmetic nature of π in a way that current mathematics cannot reach. Explore the digits yourself with our Pi Digits tool or search for specific sequences with Find in Pi.

Frequently asked questions

Is π proven to be a normal number?

No. Despite enormous numerical evidence, no mathematical proof exists that π is normal. It remains one of the most celebrated open problems in number theory.

What would it mean if π were NOT normal?

If some digit were systematically rarer or more frequent in π, it would overturn a widely held belief and imply a deep, hidden arithmetic structure in π. It would also have implications for pseudorandom number generation algorithms that rely on the digits of π.

Why does digit 0 seem absent at the start of π?

Digit 0 first appears at the 32nd decimal place of π. This may seem surprising, but it is not statistically exceptional. With a small number of decimal places, some digits can be absent or over-represented. As more decimals are observed, all frequencies gradually approach 10%.

Where can I see or download the raw digits of π?

Use our Pi Digits tool to view, copy, or download up to 100 million consecutive decimal digits of π. To search for a specific number within the digits, try Find in Pi.