Midy's theorem describes a symmetry hidden in the repeating decimal expansion of a fraction a over p, where p is a prime, whenever that expansion has an even period of 2n digits. It states that the digits in the second half of the repeating period are the 9s complement of the corresponding digits in the first half, so that, for example, the repeating decimal for 1/13 splits into 076 and 923, which sum to 999. The result is named for the French mathematician E. Midy, though William G. Leavitt's 1967 paper in The American Mathematical Monthly, titled A Theorem on Repeating Decimals, is the source that documents it in English. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementIf a/p, with p prime, has a repeating decimal period of even length 2n, the digits in the second half of the period are the 9s complement of the digits in the first half. 1 Classification
Statement Form Connections
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In Branch
Source Wolfram MathWorld: Midy's Theorem
Sources
1. Midy's theorem (Wikipedia)
IntroductionQuote, Introduction
then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half
Wolfram MathWorld: Midy's Theorem
Wolfram MathWorldIn Branch: Arithmetic, Subject classificationsQuote, In Branch: Arithmetic, Subject classifications
Number Theory Arithmetic
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