Midi Theorem

Midi 's theorem - a theorem in mathematics, named after the French mathematician Midi (ME Midy), states that if in decimal notation fractions $a/p$ ${\ displaystyle a / p}$ $a / p$ (Where $p$ ${\ displaystyle p}$ $p$ Is a prime number ) the length of the fraction period record consists of $2n$ ${\ displaystyle 2n}$ $2n$ digits, that is:

{\frac {a}{p}}=0.{\overline {a_{1}a_{2}a_{3}\dots a_{n}a_{n+1}\dots a_{2n}}},

{\ displaystyle {\ frac {a} {p}} = 0. {\ overline {a_ {1} a_ {2} a_ {3} \ dots a_ {n} a_ {n + 1} \ dots a_ {2n} }},}

{\ frac {a} {p}} = 0. \ overline {a_ {1} a_ {2} a_ {3} \ dots a_ {n} a _ {{n + 1}} \ dots a _ {{2n}} },

then

a_{i}+a_{i+n}=9

{\ displaystyle a_ {i} + a_ {i + n} = 9}

{\ displaystyle a_ {i} + a_ {i + n} = 9}

a_{1}\dots a_{n}+a_{n+1}\dots a_{2n}=10^{n}-1.

{\ displaystyle a_ {1} \ dots a_ {n} + a_ {n + 1} \ dots a_ {2n} = 10 ^ {n} -1.}

a_ {1} \ dots a_ {n} + a _ {{n + 1}} \ dots a _ {{2n}} = 10 ^ {n} -1.

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,

{\frac {1}{17}}=0,{\overline {0588235294117647}},

{\ displaystyle {\ frac {1} {17}} = 0, {\ overline {0588235294117647}},}

{\ frac 1 {17}} = 0, \ overline {0588235294117647},

and

05882352+94117647=99999999.

{\ displaystyle 05882352 + 94117647 = 99999999.}

05882352 + 94117647 = 99999999.

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 :

{\frac {1}{19}}=0.{\overline {032745}}_{8}

{\ displaystyle {\ frac {1} {19}} = 0. {\ overline {032745}} _ {8}}

032_{8}+745_{8}=777_{8}

{\ displaystyle 032_ {8} + 745_ {8} = 777_ {8}}

03_{8}+27_{8}+45_{8}=77_{8}.

{\ displaystyle 03_ {8} + 27_ {8} + 45_ {8} = 77_ {8}.}

Links

Weisstein, Eric W. Midy's Theorem at Wolfram MathWorld .

Midi Theorem

Midi's theorem in systems with a different basis

Links

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