Midy's theorem
Midy's theorem states that when the decimal expansion of 1/p has period 2n, its two n-digit repetend halves sum to 10ⁿ − 1.
Long division: An arithmetic algorithm that repeatedly divides, multiplies, and subtracts to produce quotient digits. The decimal digits and successive remainders used in Midy's theorem arise directly from this procedure.
Unit fraction: A fraction whose numerator is 1 and whose denominator is a positive integer. The theorem applies to the decimal expansion of the unit fraction 1/p.
Étienne Midy: A French mathematician associated with the nineteenth-century statement of the decimal repetend result now called Midy's theorem. His name became attached to the theorem about the halves of certain repeating decimals.
Proof by modular arithmetic: A proof method that establishes statements by working with congruences and remainders. This method derives Midy’s half-sum identity from powers of 10 modulo the denominator.
1/7: The rational number one seventh, whose decimal expansion repeats the six-digit block 142857. Splitting its repetend into 142 and 857 gives 999, the case n = 3.
Repeating decimal: A decimal expansion in which a finite block of digits repeats indefinitely. Midy's theorem concerns the repeating block in the decimal expansion of a unit fraction.
Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The denominator p is prime, which structures the modular remainders behind the result.
Journal de mathématiques élémentaires: A French mathematical periodical founded by Émile Lemoine in the nineteenth century. Midy’s result appeared in this periodical in 1836.
Divisibility: The relation in which an integer is a multiple of another integer with no remainder. The theorem’s sum is 10ⁿ − 1, a number divisible by 9 and composed of n nines.
1/17: The rational number one seventeenth, whose decimal expansion repeats a sixteen-digit block. Its two eight-digit halves sum to 99,999,999, illustrating the theorem with n = 8.