About

The Champernowne constant C₁₀ = 0.12345678910111213... is constructed by concatenating successive positive integers after the decimal point. It is one of the few explicitly constructible normal numbers - meaning every finite digit string of length n appears with limiting frequency 1÷10ⁿ in base 10. Champernowne proved normality in base 10 in 1933. Normality in one base does not imply normality in another. Misidentifying digit positions in this sequence leads to errors in pseudorandom sampling, statistical testing, and constructive analysis proofs. This generator computes exact digits using group-counting arithmetic rather than brute-force string concatenation, remaining accurate and efficient even at position 10¹⁵ and beyond.

The tool generalizes to any integer base b from 2 to 36, producing the Champernowne constant C_b. Note: computational limits apply. Browsers handle positions up to approximately 2⁵³ due to JavaScript's IEEE 754 double-precision integer ceiling. For cryptographic or high-precision research beyond that range, use an arbitrary-precision library outside the browser.

Formulas

The Champernowne constant in base b is defined as the real number formed by concatenating all positive integers written in base b after a decimal point.

C_b = 0. 1 2 3 … (b−1) (10)_b (11)_b …

To find the digit at position p (1-indexed from the first digit after the decimal point), we use the group-counting method. The k-digit numbers in base b contribute a total digit count of:

D_k = (b − 1) ⋅ b^k−1 ⋅ k

We subtract group sizes from p until the remainder falls within a k-digit group. Then we locate the exact number and digit within it:

n = b^k−1 + floor((p_rem − 1) ÷ k)

d = ((p_rem − 1) mod k)^th digit of n in base b

Where p_rem is the remaining position after subtracting all complete groups of fewer than k digits, n is the integer containing the target digit, and d is the specific digit index within that integer (0-indexed from left).

The total number of digits contributed by all integers from 1 up to and including all k-digit numbers:

T_k = k∑i=1 (b − 1) ⋅ bⁱ⁻¹ ⋅ i

Reference Data

Base b	Constant Name	First 30 Digits After Decimal	OEIS ID	Normality Proved
2	C₂	0.1 10 11 100 101 110 111 1000...	A030190	Yes (Champernowne, 1933)
3	C₃	0.1 2 10 11 12 20 21 22 100 101...	A054635	Yes
8	C₈	0.1 2 3 4 5 6 7 10 11 12 13 14...	A030373	Yes
10	C₁₀	0.1 2 3 4 5 6 7 8 9 1 0 1 1 1 2...	A033307	Yes (Original proof)
16	C₁₆	0.1 2 3 4 5 6 7 8 9 A B C D E F...	A030374	Yes
Digit Group Structure (Base 10)
Digits k	Range	Count of Numbers	Total Digits in Group	Cumulative Digits
1	1 - 9	9	9	9
2	10 - 99	90	180	189
3	100 - 999	900	2,700	2,889
4	1,000 - 9,999	9,000	36,000	38,889
5	10,000 - 99,999	90,000	450,000	488,889
6	100,000 - 999,999	900,000	5,400,000	5,888,889
7	1,000,000 - 9,999,999	9,000,000	63,000,000	68,888,889
8	10,000,000 - 99,999,999	90,000,000	720,000,000	788,888,889
Digit Group Structure (Base 2)
1	1	1	1	1
2	10 - 11	2	4	5
3	100 - 111	4	12	17
4	1000 - 1111	8	32	49

Frequently Asked Questions

Champernowne proved in 1933 that C₁₀ is normal in base 10 - every finite digit string appears with the expected asymptotic frequency. However, normality is base-dependent. A number normal in base 10 is not necessarily normal in base 2. Proving absolute normality (normality in every base simultaneously) remains an open problem for the Champernowne constant. The generalized constant C_b is normal in base b by the same argument, but cross-base normality is unresolved.

The algorithm exploits the regularity of the sequence structure. In base b, there are exactly (b−1)×b^(k−1) numbers with exactly k digits, contributing (b−1)×b^(k−1)×k digits to the sequence. By summing these group sizes, we identify which digit-length group contains position p, then compute the exact integer and digit offset using division and modular arithmetic. No concatenation string is ever built. This allows extraction at position 10¹² in microseconds.

JavaScript uses IEEE 754 double-precision floats, which can represent integers exactly up to 2⁵³ (approximately 9 × 10¹⁵). Beyond this threshold, integer arithmetic loses precision, and computed digit positions may be incorrect. This tool caps input at 9,007,199,254,740,991 (Number.MAX_SAFE_INTEGER). For positions beyond that, arbitrary-precision arithmetic (e.g., BigInt or external libraries) would be required.

In base b, each digit d (where 0 ≤ d < b) appears with asymptotic frequency 1/b in the Champernowne constant C_b, by the normality property. However, in finite prefixes, distribution varies. The digit 0 appears less frequently early in the sequence because leading zeros are not represented - the number 10 contributes a "1" and a "0", but there is no standalone "0" integer in the concatenation. The bias toward non-zero digits diminishes as position increases.

No. This tool is specific to the Champernowne constant, which concatenates all positive integers. The Copeland - Erdős constant concatenates successive primes (0.2357111317...), requiring a prime sieve. The Smarandache constant concatenates specific subsequences. Each requires different generation logic. The group-counting shortcut used here relies on the uniform structure of consecutive integers and does not apply to primes or other irregular sequences.

Yes. Kurt Mahler proved in 1937 that C₁₀ is transcendental - it is not a root of any non-zero polynomial with rational coefficients. This follows from the Thue - Siegel - Roth theorem and the fact that C₁₀ can be approximated by rationals "too well" for an algebraic irrational. The same argument extends to C_b for any base b ≥ 2.