Champernowne constant

The Champernowne constant, denoted C_{10}, is a transcendental real number whose decimal expansion is formed by concatenating the positive integers in order after the decimal point: $0.123456789101112131415\dots. Introduced by British mathematician and economist David Gawen Champernowne in 1933 while he was an undergraduate at the University of Cambridge, it serves as the first explicit artificial example of a number that is normal in base 10.^[1]^[2] Champernowne proved that C_{10} is normal in base 10, meaning that every finite sequence of p digits appears in its decimal expansion with asymptotic frequency $1/10^p, as expected for a "random" sequence.^[1] This property implies that C_{10} is irrational, and in 1937, Kurt Mahler established its transcendence, showing that it is not the root of any non-zero polynomial with rational coefficients.^[3] The constant is computable and has been calculated to billions of decimal places, with its continued fraction expansion exhibiting unusually large partial quotients that grow rapidly.^[4] The construction of C_{10} generalizes to other bases b \geq 2, yielding Champernowne constants C_b that are similarly normal and transcendental in base b; for example, the binary Champernowne constant is $0.1101110010111011\dots_2.^[3] These constants have applications in number theory, including studies of discrepancy, pair correlations, and irrationality measures, with C_{10} having an irrationality measure of exactly 10. Despite their constructed nature, they exhibit pseudorandom behavior in digit sequences, influencing research on algorithmic randomness and automatic sequences.^[4]

Definition and Construction

Formal Definition

The Champernowne constant, denoted C_{10}, is an irrational real number in the interval (0, 1) whose base-10 decimal expansion is formed by the infinite concatenation of the decimal representations of the positive integers in ascending order: C_{10} = 0.123456789101112131415\dots.^[1] This places it strictly between 0 and 1, as the expansion begins with a leading zero after the decimal point followed by digits from 1 onward, ensuring the value is positive but less than 1.^[1] Introduced by the British mathematician David Gawen Champernowne, the constant is formally defined as the limit of finite partial concatenations of these integers, where each partial sum approximates the infinite decimal by successively appending the digits of 1, then 2, up to increasingly larger integers.^[1] Champernowne presented this construction in his 1933 paper, emphasizing its systematic digit sequence to demonstrate specific properties of the resulting number.^[1] The definition extends naturally to any integer base b \geq 2, producing the generalized Champernowne constant C_b, whose base-b expansion concatenates the base-b representations of the positive integers.^[5] For b = 10, this yields the standard C_{10}.^[5]

Concatenation Process

The Champernowne constant in base 10 is constructed through a systematic concatenation of the decimal representations of all positive integers in ascending order, forming an infinite string of digits placed after the decimal point. This process begins with the single-digit integers from 1 to 9, which together provide exactly 9 digits: 123456789.^[4]^[1] Following this, the two-digit integers from 10 to 99 are appended; there are 90 such numbers, each contributing 2 digits, for a total of 180 digits.^[4] Next comes the block of three-digit integers from 100 to 999, consisting of 900 numbers each with 3 digits, adding 2700 digits to the expansion.^[4] This pattern continues indefinitely, with each subsequent block covering the n-digit integers from $10^{n-1} to $10^n - 1. For a general n-digit block, there are $9 \times 10^{n-1} numbers, each supplying n digits, resulting in a contribution of $9 n \times 10^{n-1} digits.^[4]^[1] To locate a specific digit or number within the expansion, one computes the cumulative digit count up to the relevant block. For instance, the total digits before the two-digit block is 9, from the single-digit numbers alone; before the three-digit block, it is 9 + 180 = 189 digits.^[4] This cumulative approach allows precise positioning: the first digit of the m-th n-digit number appears after summing the digits from all prior blocks plus the contributions from the preceding numbers in the current block.^[4] A concrete illustration of the early stages yields the first 20 digits as 0.1234567891011121314, where the sequence transitions seamlessly from 9 to 10, then 11, 12, 13, and 14.^[4]^[1] Since there are infinitely many positive integers, this concatenation process generates an unending decimal expansion that never terminates or repeats.^[1]

Mathematical Properties

Normality

A normal number in base 10 is defined as an irrational number whose digits in its decimal expansion are distributed such that every possible finite sequence of length k appears with asymptotic frequency $10^{-k}.^[6] This property ensures that the digits are statistically random, with no bias toward any particular pattern in the limit. For the Champernowne constant, this normality manifests in its concatenated decimal representation of the positive integers. The Champernowne constant was the first explicit example of a normal number in base 10, as proved by David G. Champernowne in 1933. His proof employs combinatorial arguments by partitioning the decimal expansion into blocks corresponding to the concatenations of 1-digit, 2-digit, up to n-digit numbers, and demonstrating that the frequency of any specific k-digit sequence approaches $10^{-k} as n increases, due to the uniform distribution within these blocks.^[1] This construction guarantees that single digits 0 through 9 each appear with limiting frequency $1/10, and similarly for longer sequences, establishing uniform digit distribution overall.^[4] The normality property extends to the generalized Champernowne constant C_b in any integer base b \geq 2, where the expansion concatenates representations of positive integers in base b. This was established by Yoshinobu Nakai and Iekata Shiokawa in 1992 through discrepancy estimates showing that digit blocks in the generalized construction satisfy the required frequency conditions for normality in base b.^[7] Their approach builds on Champernowne's method but accounts for the base-b digit sets, confirming asymptotic equidistribution. As a consequence of its normality, the Champernowne constant is disjunctive: every possible finite sequence of digits appears at least once in its decimal expansion.^[8] This disjunctiveness underscores the constant's richness, ensuring the presence of all patterns, though normality provides the stronger equidistribution guarantee.

Transcendence and Irrationality

The Champernowne constant C_{10} is irrational, as its decimal expansion is non-terminating and non-periodic, arising from the infinite concatenation of all positive integers without repetition. This construction ensures that no repeating block can encompass the entire expansion, precluding rationality.^[9] In 1937, Kurt Mahler proved that C_{10} is transcendental, meaning it is not algebraic over the rationals and thus satisfies no non-zero polynomial equation with integer coefficients. Mahler's approach analyzed the arithmetic properties of a class of decimal fractions, including the Champernowne constant, using methods from Diophantine approximation to derive a contradiction assuming algebraic dependence. This result established C_{10} as one of the first explicitly constructed transcendental numbers beyond simpler examples like e and \pi.^[10] The irrationality measure of the Champernowne constant C_b in base b \geq 2 is exactly b, indicating the quality of its rational approximations. Specifically, there are infinitely many rationals p/q satisfying |C_b - p/q| < q^{-b}, but for any \epsilon > 0, the stronger inequality |C_b - p/q| < q^{-(b + \epsilon)} holds for only finitely many such rationals. For the base-10 Champernowne constant, \mu(C_{10}) = 10. This precise value was established by Masaaki Amou in 1991 through bounds on approximations by algebraic numbers of bounded degree.^[11]

Representations

Infinite Series

The Champernowne constant in base 10, C_{10}, admits an infinite series representation that directly reflects its construction via concatenation of the decimal expansions of the positive integers. This double summation groups the contributions by the number of digits n in each integer k, accounting for the shifting positions of these blocks in the overall decimal expansion. The formula is

C_{10} = \sum_{n=1}^\infty \sum_{k=10^{n-1}}^{10^n - 1} \frac{k}{10^{s(n) + n(k - 10^{n-1} + 1)}},

where s(n) denotes the total number of digits from all integers with fewer than n digits, with s(1) = 0 and s(n) = 9 \sum_{m=1}^{n-1} m \, 10^{m-1} for n > 1.^[4] This expression arises because each n-digit integer k contributes its decimal value scaled by the appropriate power of 10, corresponding to the position of its units digit in the expansion. The term s(n) captures the cumulative digits preceding the block of n-digit numbers (specifically, 9 one-digit numbers contribute 9 digits, 90 two-digit numbers contribute 180 digits, and so on). Within the n-digit block, the starting position of k's digits is offset by n(k - 10^{n-1} + 1), ensuring the exponent s(n) + n(k - 10^{n-1} + 1) aligns with the location of k's units place relative to the decimal point. Thus, dividing k by $10 raised to this power yields the precise fractional contribution of k's digits.^[4] The representation generalizes naturally to arbitrary base b \geq 2, yielding the Champernowne constant C_b:

C_b = \sum_{n=1}^\infty \sum_{k=b^{n-1}}^{b^n - 1} \frac{k}{b^{s_b(n) + n(k - b^{n-1} + 1)}},

where s_b(1) = 0 and s_b(n) = (b-1) \sum_{m=1}^{n-1} m \, b^{m-1} for n > 1, adjusting for the (b-1) b^{m-1} numbers with m digits in base b. This mirrors the base-10 case, with the summation structure preserving the positional encoding from the concatenation process.^[4] In practice, this series facilitates high-precision computation of C_{10} (or C_b) by evaluating partial sums up to a finite N, as the terms diminish rapidly due to the exponential growth in the exponents (roughly on the order of n \cdot b^{n-1}). Truncating after a modest N (e.g., N=10) yields approximations accurate to hundreds of decimal places, making it suitable for numerical verification of properties like normality.^[4]

Continued Fraction Expansion

The continued fraction expansion of the Champernowne constant C_{10} takes the form [0; a_1, a_2, \dots], where the partial quotients a_i are positive integers forming a non-periodic and unbounded sequence.Champernowne Constant Continued Fraction This irregularity stems from the constant's construction via concatenated natural numbers, leading to sporadic alignments that produce exceptionally large quotients.Continued fraction expansion of the Champernowne constant 0.1234567891011121314... Notable examples include a_4 = 149083, a_{18} \approx 10^{166} (a 166-digit integer), and a_{40} \approx 10^{2504} (a 2504-digit integer), arising when segments of the decimal expansion mimic rational approximations closely.Continued fraction expansion of the Champernowne constant 0.1234567891011121314...Analysis of the High Water Mark Convergents of Champernowne's Constant Continued Fraction Expansion These "high-water marks"—positions where a quotient exceeds all predecessors—occur at indices such as 4, 18, and 40, with their sizes growing doubly exponentially due to patterns tied to powers of the base 10.On the High Water Mark Convergents of Champernowne's Constant in Base Ten Computation of the expansion involves iteratively deriving quotients from the decimal digits of C_{10}, generated on demand through the concatenation process, though large terms demand immense precision (e.g., millions of digits for later quotients), rendering it challenging.Champernowne Constant Continued Fraction The unbounded and irregularly large partial quotients underpin the Diophantine approximation behavior of C_{10}, manifesting in its irrationality measure of exactly 10, as established for base-b Champernowne constants.Numbers with known finite irrationality measure greater than 2

History and Developments

Discovery and Initial Proofs

The Champernowne constant was introduced by David Gawen Champernowne, a 21-year-old undergraduate at the University of Cambridge, in 1933 as part of his work on constructing explicit examples of normal numbers in base 10.^[2] Champernowne, born on July 9, 1912, developed this constant during his studies under the supervision of G. H. Hardy, presenting it through a straightforward concatenation of positive integers.^[12] The construction served to illustrate a simply defined irrational decimal whose digits exhibit the uniform distribution characteristic of normality, addressing a gap in earlier theoretical discussions of normal numbers by Borel.^[2] In his seminal paper, Champernowne proved that the constant is normal in base 10 by employing an elementary counting argument based on the asymptotic density of digit blocks within the concatenated sequences.^[13] He analyzed the frequency of arbitrary p-digit sequences (denoted y_p) across blocks of r-digit numbers for r \geq p, distinguishing between undivided occurrences—where the sequence spans a single number—and divided occurrences—where it crosses boundaries between numbers. The proof demonstrates that the total count G(x) of such sequences in the first x digits satisfies G(x) = 10^{-p} x + o(x) as x \to \infty, ensuring each p-digit block appears with limiting frequency $1/10^p.^[13] This rigorous asymptotic estimate established the constant as the first explicit example of a normal number with a constructive definition.^[14] The work was published in the Journal of the London Mathematical Society under the title "The construction of decimals normal in the scale of ten."^[13] Although Champernowne did not name the constant after himself in the original publication, it became retroactively known as the Champernowne constant in subsequent mathematical literature honoring his contribution.^[4] This early proof of normality laid foundational groundwork.

Subsequent Advances

In 1937, Kurt Mahler established the transcendence of the Champernowne constant using a method based on functional equations and Diophantine approximation techniques, constructing a sequence of rational approximations to show that it cannot satisfy any algebraic equation with integer coefficients.^[15] This proof extended earlier results on transcendental numbers and confirmed that the constant is not merely irrational but transcends the algebraic numbers entirely.^[4] During the 1970s, Yoshinobu Nakai and Iekata Shiokawa provided rigorous proofs of the normality of the Champernowne constant in arbitrary integer bases b \geq 2, generalizing Champernowne's original result for base 10 by analyzing the uniform distribution of digit blocks in the concatenated expansion.^[16] Their work employed discrepancy estimates and ergodic theory to demonstrate that every finite sequence of digits appears with the expected asymptotic frequency in base b.^[16] The irrationality measure of the Champernowne constant C_b in base b \geq 2 is exactly b, extending Émile Borel's 1909 results on the typical behavior of almost all real numbers, where normality ensures no better than quadratic Diophantine approximations on average.^[17] In 2012, John K. Sikora conducted a detailed computational analysis of the continued fraction expansion of C_{10}, identifying patterns in the "high-water mark" convergents—positions where the partial quotients reach new maxima—and proposing conjectures on their locations and the accuracy of preceding approximations, such as the error being approximately $9 \times 10^{-\text{NCD}(N) - N + 2} before the N-th high-water mark for N \geq 5.^[18] Recent theoretical advances include a 2024 discrete and elementary proof of discrepancy estimates for C_{10} by Verónica Becher and Nicole Graus, providing upper and lower bounds on the rate at which its digits approach uniformity and improving on earlier methods using exponential sums.^[12] Sikora's patterns have informed subsequent numerical studies.^[18] Key open problems include whether C_b is normal in bases other than b or powers thereof—for instance, the normality of C_{10} in base 9 remains unresolved—and obtaining finer Diophantine approximation bounds beyond the measure of b, such as explicit exponents for the growth of partial quotients in its continued fraction.^[19] Computational efforts have advanced the explicit evaluation of C_{10}, with the Online Encyclopedia of Integer Sequences providing its decimal expansion to over 1 million digits via efficient concatenation algorithms, and Eric Weisstein computing up to $6 \times 10^{10} digits in 2013 using high-precision arithmetic in the Wolfram Language.^[20]^[4]

Bases Other Than Ten

The Champernowne constant generalizes naturally to any integer base b \geq 2. The base-b Champernowne constant C_b is defined as the real number between 0 and 1 whose base-b expansion is obtained by concatenating the base-b representations of the positive integers in order, starting from 1. Unlike the base-10 case, the digit alphabet consists of symbols from 0 to b-1, and the representations of larger integers require more digits as their length increases with the number of digits d for numbers in [b^{d-1}, b^d - 1). For instance, in base 2, C_2 = 0.11011100101110111100010011010111\dots_2, formed by appending the binary strings 1, 10, 11, 100, 101, 110, 111, and so on. Similarly, in base 3, C_3 = 0.1201011122021100120\dots_3, concatenating ternary strings like 1, 2, 10, 11, 12, 20, 21, 100. In base 16, the hexadecimal version C_{16} uses digits 0-9 and A-F, beginning as 0.123456789ABCDEF101112\dots_{16}.^[4] Like its base-10 counterpart, C_b exhibits strong normality and transcendence properties. It is normal in base b, meaning that every finite string of k digits from the alphabet \{0, 1, \dots, b-1\} appears in its base-b expansion with asymptotic frequency exactly b^{-k}, ensuring a uniform distribution of blocks. This was originally shown for b=10 by Champernowne in 1933 and extended to arbitrary b \geq 2 by Stoneham in 1973 via a direct combinatorial argument on digit frequencies across blocks of increasing length. Additionally, C_b is transcendental for every b \geq 2, as established by Mahler using Diophantine approximation techniques to show it cannot satisfy any algebraic equation of finite degree. The irrationality measure of C_b is \mu(C_b) = b, reflecting the quality of rational approximations derived from truncating the concatenation at points where full blocks of d-digit numbers end, which provide denominators growing exponentially in base b.^[21] The generalization to base b alters certain distributional aspects compared to base 10, primarily due to the larger or smaller alphabet size affecting string probabilities directly via the b^{-k} frequency formula. The construction must account for the varying lengths of integer representations: for each digit length d \geq 1, there are exactly (b-1) b^{d-1} numbers with d digits in base b, contributing d (b-1) b^{d-1} digits to the expansion, which ensures the normality proof holds by balancing contributions from these blocks. However, while C_b is proven normal in its native base b, its normality in a different base m \neq b remains an open problem for most pairs (b, m) with m \geq 2; no constructed normal number is yet known to be normal across multiple distinct bases.^[22]

Champernowne Word and Disjunctive Sequences

The Champernowne word is an infinite sequence over the finite alphabet \{0, 1, \dots, 9\}, formed by concatenating the base-10 representations of the positive integers in ascending order: w = 123456789101112131415\dots. This sequence, also known as the Barbier word, is cataloged as OEIS A007376 and serves as the digit string underlying the decimal expansion of the Champernowne constant.^[23] The Champernowne word possesses the disjunctive property: it contains every possible finite word (substring) over its alphabet as a factor, and in fact infinitely often. This follows from its normality in base 10, which ensures that every finite digit sequence appears with the limiting frequency expected under uniform random distribution; Champernowne established this normality via a recursive block construction, where blocks of increasing length are concatenated to guarantee the appearance of all required substrings.^[1] The relation to the Champernowne constant C_{10} is direct: the fractional part \{C_{10}\} is obtained by interpreting the Champernowne word w = w_1 w_2 w_3 \dots as a base-10 decimal expansion, yielding \{C_{10}\} = 0.w_1 w_2 w_3 \dots. This construction embeds the sequence-theoretic properties of the word into the numerical framework of the constant.^[1] In combinatorics on words, the Champernowne word exemplifies maximal growth in subword complexity, with the function p_w(n) = 10^n for all n \geq 0, indicating that it contains exactly $10^n distinct substrings of length n—all possible ones over the alphabet. This extreme complexity contrasts with periodic or low-complexity words and aids in studying bounds on repetition and factor sets in infinite sequences.^[24] Due to its disjunctiveness and normality, the word is employed in randomness tests as a computable benchmark for evaluating pseudo-random generators, where deviations from its uniform substring frequencies signal non-randomness. Additionally, it approximates de Bruijn sequences by providing an infinite extension that includes all substrings of any fixed length, useful for constructing or analyzing cyclic sequences in algorithmic contexts.^[25] Generalizations of the Champernowne word extend to arbitrary finite alphabets and bases b \geq 2, formed by concatenating the base-b representations of positive integers over \{0, 1, \dots, b-1\}; for example, the binary Champernowne word is $110111001011101111000\dots, obtained by concatenating binary expansions starting from 1. These generalized words inherit disjunctiveness and are normal in base b, with proofs adapting the original block construction to the larger alphabet.^[24]^[1]