Champernowne constant

In mathematics, the Champernowne constant C₁₀ is a certain real number, named after mathematician D. G. Champernowne. It is a simple number to construct, which has some important properties.

Normality

Say we have some real number x. We call x normal in base b if, in examining its digits, we are certain that any digit we choose is equally likely, or equivalently, the probability of finding some digit string among the digits of x is the same as if we were to search amongst some random sequence of digits. See normal number for a more detailed explanation.

If we denote a digit string as [a₀,a₁,...], then, in base ten, we would expect strings [0],[1],[2],...,[9] to occur 1/10 of the time, strings [0,0],[0,1],...,[9,8],[9,9] to occur 1/100 of the time, and so on, in a normal number.

Given this definition, is it possible to construct a normal number? Naturally, one would consider to concatenate strings [0],[1],[2],...,[9] which would satisfy the first condition, then strings [0,0],[0,1],...,[9,8],[9,9] so as to satisfy the second condition, and so on.

This is precisely how the Champernowne constant is defined.

In base 10, we have

<math>C_{10} = 0.12345678910111213141516\dots<math>

This is clearly normal in base ten. We can create Champernowne constants that are normal in other bases, similarly, for example,

<math>C_2 = 0.1\,10\,11\,100\,101\,110\,111\dots {}_2<math>

<math>C_3 = 0.1\,2\,10\,11\,12\,20\,21\,22\dots {}_3<math>

and so on.

Another way of understanding the normality of Champernowne's constant is to realize that the act of counting itself will ultimately explore every combination of base b digits. That's what counting is all about.

Computations

Computing Champernowne's constant can be done by concatenation of bit strings on a computer, but this may not necessarily be the fastest way of computation. Often computing the constant is much faster if one does so purely numerically.[1] (http://library.wolfram.com/infocenter/MathSource/2876/)

One method of computation includes the calculation of the continued fraction form of a number. Computing this form can help us analyse the number also.

Normally in considering a fraction, we take some real number y and split it into the quotient of two integers a and b so y = a/b, the continued fraction takes a real number y and splits in the following way

<math> y = a_0+ {1 \over a_1 + {1 \over a_2 + {1 \over a_3 + {1 \over a4 + \ddots } } } }<math>

which we can write more compactly as [a₀; a₁, a₂, ...].

For example, consider the exponential constant:

<math>e=2.718281828\cdots = 2 + {1 \over 1 + {1 \over 2 + {1 \over 1 + {1 \over 1 + \ddots } } } } = [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, \cdots]<math>

The terms in the continued fraction stop after some point if the number is rational, and continue indefinitely if the number is irrational. Clearly Champernowne's constant is irrational, since rational numbers have a repeating or terminating expansion into digits to the right of the "decimal" (radix) point. So the continued fraction of Champernowne's constant does not terminate.

If we were to stop the continued fraction after a certain point in an irrational number, we get an approximation to that number by means of a simple fraction. The more terms we take, the more accurate the approximation. For example,

e - [2; 1, 2, 1, 1] = 2.714285714, e - [2; 1, 2, 1, 1] = 0.003996114

e - [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1] = 2.718281718, e - [2; 1, 2, 1, 1, 4, 1, 1, 6, 1, 1] ~ 1.10 ×10^-7

If we are to examine the continued fraction of Champernowne's constant, we get some erratic behaviour. In base 10,

<math>C_{10} = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, \,<math>

<math>45754\,01113\,91031\,07648\,36466\,28242\,95611\,85996\,03939\,71045\,75550\,00662\,00439\,30902\,62659\,25631\,49379\,53207\,74712\,86563<math>

<math>1\,38641\,20937\,55035\,52094\,60718\,30899\,84575\,80146\,98631\,48833\,59214\,17830\,10987<math>

<math> 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54,\cdots ]<math>

We get other extremely large numbers as part of the continued fraction if we were to continue. The next term of the continued fraction is huge, having 2504 digits. This can pose problems in computing the terms of the continued fraction, and may stress weak algorithms for computing the continued fraction. However, the fact that there are such large numbers as part of the continued fraction expansion means that if we take terms up to and onward from these large numbers, we get an exceedingly good approximation in comparison to the terms that did not include the large number. Calling this large number above at position 19 in the continued fraction K, then, for example,

C₁₀ - [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15] ~ -9 ×10^-190

C₁₀ - [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, K] ~ 3 ×10^-356

which is an improvement in accuracy by 166 orders of magnitude.

A direct computation of Champernowne's constant for a base b is given by

<math> C_b = \sum_{n=1}^\infty\frac{\sum_{k=b^{n-1}}^{b^n-1}kb^{-n(k-(b^{n-1}-1))}}{b^{\sum_{k=0}^{n-1}k(b-1)b^{k-1}}} <math>,

(Parkin).

References

• D. G. Champernowne, The construction of decimals normal in the scale of ten, Journal of the London Mathematical Society, vol. 8 (1933), p. 254-260
• Rytin, M. Champernowne Constant and Its Continued Fraction Expansion, (1999), http://library.wolfram.com/infocenter/MathSource/2876/
• Parkin, Spencer T., "An Identity for Champernowne's Constant", (Not published)