In Mathematical, the constant of Champernowne , noted $C\_\; \{10\}\; \backslash ,$ is a Real number, named thus in the honor of the Mathématicien D.G. Champernowne. It is a number simple to build, which has certain important properties.

Normality

We will say that a real number X is normal bases B of it if, by examining his figures, we are certain that, whatever the figure whom we choose, it will appear as much as the different ones, or in an equivalent way, the probability of finding a certain chain of figures among the figures of X is the same one if we had sought it among a chain of figure taken randomly. See the article on the normal numbers for fuller explanations.

If we note a chain of figures by $\backslash ,$, then in bases 10, we would expect to see appearing each chain,…, with a probability equalizes at 1/10, each chain,…, with a probability equalizes to 1/100, and so on, in a normal number.

According to is the definition, possible to build a normal number? Naturally, one could consider the concatenation of the chains,…, to satisfy the first condition, then the concatenation of the chains,…, to satisfy the second condition, and so on.

It is precisely in this way that the constant of Champernowne was defined.

In base 10, we have

$C\_\; \{10\}\; =\; 0,12345678910111213141516\; \backslash \; dots$

It is, in a clear way, normal bases 10 of them. We can create constants of Champernowne which are normal in the other bases, in a similar way, for example,

$C\_2\; =\; 0,1\; \backslash ,\; 10\; \backslash ,\; 11\; \backslash ,\; 100\; \backslash ,\; 101\; \backslash ,\; 110\; \backslash ,\; 111\; \backslash \; dowries\; \{\}\; \_2$

$C\_3\; =\; 0,1\; \backslash ,\; 2\; \backslash ,\; 10\; \backslash ,\; 11\; \backslash ,\; 12\; \backslash ,\; 20\; \backslash ,\; 21\; \backslash ,\; 22\; \backslash \; dowries\; \{\}\; \_3$

and so on.

Calculations

The calculation of the constant of Champernowne can be made by the concatenation of the chains of bits on a Ordinateur, but it is not necessarily the fastest manner for calculation. Often, the calculation of the constant is faster if it is made in a purely numerical way.

One of the methods includes calculation by the continuous fractions of a number. To calculate this form can also help us to analyze the number.

Normally by considering a fraction, we take a certain real number there which we divide into a quotient of two entireties has and B thus there = has / B , the continued fraction takes a real number there and

$divides\; it\; in\; the\; following\; way\; there\; =\; a\_0+\; \{1\; \backslash \; over\; a\_1\; +\; \{1\; \backslash \; over\; a\_2\; +\; \{1\; \backslash \; over\; a\_3\; +\; \{1\; \backslash \; over\; a4\; +\; \backslash \; ddots\}\}\}\}$

that we can write in a more compact way by '' has '' ₁, '' has '' ₂,….

For example, if one considers the number $e\; \backslash ,$:

$e=2,718281828\; \backslash \; cdots\; =\; 2\; +\; \{1\; \backslash \; over\; 1\; +\; \{1\; \backslash \; over\; 2\; +\; \{1\; \backslash \; over\; 1\; +\; \{1\; \backslash \; over\; 1\; +\; \backslash \; ddots\}\}\}\}\; =\; 1,2,1,1,4,1,1,6,1,1,\; \backslash \; cdots$

The terms in the fraction continues stop after a certain point if the number is rational, and continue indefinitely if the number is irrational. In a clear way, the constant of Champernowne is irrational, since the rational numbers have a repetitive or finished decimal development. The fraction continues constant of Champernowne does not finish.

If we stop the fraction continues after a certain point for a rational number, we would obtain an approximation of this number in the form of a simple fraction. The more we would take terms, the more the approximation would be precise. For example,

E - 1,2,1,1 = 2,714285714, E - 1,2,1,1 = 0,003996114

E - 1,2,1,1,4,1,1,6,1,1 = 2,718281718, E - 1,2,1,1,4,1,1,6,1,1 ~ 1.10 ×10^-7

If we examine the fraction continues constant of Champernowne, we obtain a certain erratic behavior. In base 10,

$C\_\; \{10\}\; =\; 8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15,\; \backslash ,$ : $45754\; \backslash ,\; 01113\; \backslash ,\; 91031\; \backslash ,\; 07648\; \backslash ,\; 36466\; \backslash ,\; 28242\; \backslash ,\; 95611\; \backslash ,\; 85996\; \backslash ,\; 03939\; \backslash ,\; 71045\; \backslash ,\; 75550\; \backslash ,\; 00662\; \backslash ,\; 00439\; \backslash ,\; 30902\; \backslash ,\; 62659\; \backslash ,\; 25631\; \backslash ,\; 49379\; \backslash ,\; 53207\; \backslash ,\; 74712\; \backslash ,\; 86563$

$1\; \backslash ,\; 38641\; \backslash ,\; 20937\; \backslash ,\; 55035\; \backslash ,\; 52094\; \backslash ,\; 60718\; \backslash ,\; 30899\; \backslash ,\; 84575\; \backslash ,\; 80146\; \backslash ,\; 98631\; \backslash ,\; 48833\; \backslash ,\; 59214\; \backslash ,\; 17830\; \backslash ,\; 10987$

$6,1,1,21,1,9,1,1,2,3,1,7,2,1,83,1,156,4,58,8,54,\; \backslash \; cdots]$

We obtain other extremely large numbers as left the continuous fraction if we continue. The next term of the continuous fraction is enormous, it has 2.504 digits. This can pose problems in the calculation of the terms of the continuous fraction, and can disturb the weak algorithms of calculation of continuous fraction. Nevertheless, the fact that there exist great numbers as left the development of the continuous fraction wants to say that if we take the terms above and below these great numbers, we obtain a good surplus approximation compared to the great number which we did not include. By calling K the great number above in position 19 in the fraction continues, then, for example,

C₁₀ - 8,9,1,149083,1,1,1,4,1,1,1,3,4,1,1,1,15 ~ -9 ×10^-190

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

who is an improvement of precision by 166 orders of magnitude.

References

• D.G. Champernowne, normal The construction off decimals in the scale off ten , Newspaper off the London Mathematical Society, vol. 8 (1933), p. 254-260
• Rytin, Mr. Constant Champernowne and Its Continued Fraction Expansion , (1999), http://library.wolfram.com/infocenter/MathSource/2876/

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