Liouville number

In number theory, a Liouville number is a real number x with the property that, for any positive integer n, there exist integers p and q with q > 1 and such that

Related Topics:
Number theory - Real number - Integer

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:0 < |x − p/q| < 1/qn.

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A Liouville number can then be approximated "quite closely" by a sequence of rational numbers. An equivalent definition is that for any positive integer n, there exists an infinite number of pairs of integers (p,q) obeying the above inequality.

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It is relatively easily proven that if x is a Liouville number, x is irrational. Assume otherwise; then there exists integers c, d with x = c/d. Let n be a positive integer such that 2n−1 > d. Then if p and q are integers such that q>1 and p/q ≠ c/d, then

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:|x − p/q| = |c/d − p/q| ≥ 1/dq > 1/(2n−1 q) ≥ 1/qn

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which contradicts the above definition.

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In 1844, Joseph Liouville showed that numbers with this property are not just irrational, but are always transcendental (see proof below). He used this result to provide the first example of a provably transcendental number,

Related Topics:
1844 - Joseph Liouville - Transcendental

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c = sum_{j=1}^infty 10^{-j!} = 0.110001000000000000000001000....

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known as Liouville's constant. Liouville's constant is a Liouville number; if we define pn and qn as follows:

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:p_n = sum_{j=1}^n 10^{(n! - j!)}; quad q_n = 10^{n!}

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then for all positive integers n, we have that

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:|c - p_n/q_n| = sum_{j=n+1}^infty 10^{-j!} = 10^{-(n+1)!} + 10^{-(n+2)!} + cdots < 10^{-(n!n)} = 1/{q_n}^n

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This approach provides a useful tool for proving a given number is transcendental. Unfortunately, although every Liouville number is transcendental, not every transcendental number is a Liouville number. It has been proven that π is transcendental, but not a Liouville number.

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More generally, the irrationality measure of a real number x is a measure of how "closely" a number can be approximated by rationals. Instead of allowing any n in the power of q, we find the least upper bound of the set of real numbers μ such that

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:0 < |x − p/q| < 1/qμ

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is satisfied by an infinite number of integer pairs (p, q) with q > 0. For any value μ less than this upper bound, the set of all rationals p/q satisfying the above inequality is an approximation of x; conversely, if μ is greater than the upper bound, then there are no such sequences which get arbitrarily close to x.

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The Liouville numbers are precisely those numbers having infinite irrationality measure.

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