Champernowne constant
Encyclopedia
In mathematics
, the Champernowne constant C10 is a transcendental
real
constant
whose decimal expansion has important properties. It is named after mathematician
D. G. Champernowne
, who published it as an undergraduate in 1933.
In base 10, the number is defined by concatenating successive integers:
can be expressed exactly as an infinite series:
Champernowne constants can also be constructed in other bases, similarly, for example:
x is said to be normal
if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. x is said to be normal in base b if its digits in base b follow a uniform distribution.
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.
Champernowne proved that
is normal in base ten, although it is possible that it is not normal in other bases.
expansion of Champernowne's constant has been studied as well. Kurt Mahler
showed that the constant is transcendental
; therefore its continued fraction does not terminate (because it is not rational
) and is aperiodic (because it is not an irreducible quadratic).
The terms in the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing between many small ones. For example, in base 10,
The large number at position 19 has 166 digits.
We get other extremely large numbers as part of the continued fraction if we 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 terms of the continued fraction expansion implies that the convergents obtained by stopping before these large number provide an exceptionally good approximation
of the Champernowne constant. For example, truncating before the 4th partial quotient, we obtain the partial sum 10/81, which approximates Champernowne constant with an error of about 1×10-9, while truncating just before the 18th partial quotient, we get
that approximates Champernowne constant with an error of about 9×10-190.
This sum may be used as a tool to analyze the constant.
The naïve method of adding the digits one by one may also be slower to perform on a computer than other more sophisticated algorithms.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Champernowne constant C10 is a transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
whose decimal expansion has important properties. It is named after mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
D. G. Champernowne
D. G. Champernowne
David Gawen Champernowne was an English economist and mathematician.After academic work at Cambridge and the London School of Economics, he worked at the London School of Economics and Cambridge University...
, who published it as an undergraduate in 1933.
In base 10, the number is defined by concatenating successive integers:
- C10 = 0.12345678910111213141516...
can be expressed exactly as an infinite series:Champernowne constants can also be constructed in other bases, similarly, for example:
- C2 = 0.11011100101110111... 2
- C3 = 0.12101112202122... 3.
Normality
A real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
x is said to be normal
Normal number
In mathematics, a normal number is a real number whose infinite sequence of digits in every base b is distributed uniformly in the sense that each of the b digit values has the same natural density 1/b, also all possible b2 pairs of digits are equally likely with density b−2,...
if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. x is said to be normal in base b if its digits in base b follow a uniform distribution.
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.
Champernowne proved that
is normal in base ten, although it is possible that it is not normal in other bases.Continued fraction expansion
The simple continued fractionContinued fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on...
expansion of Champernowne's constant has been studied as well. Kurt Mahler
Kurt Mahler
Kurt Mahler was a mathematician and Fellow of the Royal Society.He was a student at the universities in Frankfurt and Göttingen, graduating with a Ph.D...
showed that the constant is transcendental
Transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...
; therefore its continued fraction does not terminate (because it is not rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
) and is aperiodic (because it is not an irreducible quadratic).
The terms in the continued fraction expansion exhibit very erratic behaviour, with huge terms appearing between many small ones. For example, in base 10,
- C10 = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15,
-
- 4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987,
- 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, ...].
-
The large number at position 19 has 166 digits.
We get other extremely large numbers as part of the continued fraction if we 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 terms of the continued fraction expansion implies that the convergents obtained by stopping before these large number provide an exceptionally good approximation
Diophantine approximation
In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers....
of the Champernowne constant. For example, truncating before the 4th partial quotient, we obtain the partial sum 10/81, which approximates Champernowne constant with an error of about 1×10-9, while truncating just before the 18th partial quotient, we get
that approximates Champernowne constant with an error of about 9×10-190.
Computation
Champernowne's constant for a given base b can be written as an infinite sum by:This sum may be used as a tool to analyze the constant.
The naïve method of adding the digits one by one may also be slower to perform on a computer than other more sophisticated algorithms.
See also
- Copeland–Erdős constant, a similar normal number, defined using the prime numberPrime numberA prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s - Liouville's constant, another constant defined by its decimal representation
External links
- Sequence in the On-Line Encyclopedia of Integer SequencesOn-Line Encyclopedia of Integer SequencesThe On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
.