Midi 's theorem - a theorem in mathematics, named after the French mathematician Midi (ME Midy), states that if in decimal notation fractions (Where Is a prime number ) the length of the fraction period record consists of digits, that is:
then
In other words, the sum of the decimal notation in the first half of the period and the corresponding digit in the second half is 9.
For example,
- and
Midi's theorem in systems with a different basis
Midi's theorem does not depend on the base of the number system . For a number system other than decimal , in it you need to replace 10 with the base of the system - k , and 9 with k-1 . So, for example, in the octal number system :
Links
- Weisstein, Eric W. Midy's Theorem at Wolfram MathWorld .