Midy's theorem explained

In mathematics, Midy's theorem, named after French mathematician E. Midy,^[1] is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period . If the period of the decimal representation of a/p is 2n, so that

(a)/(p) =0 .\overline{a _{1a} _{2a} _{3...} a _{na} _{n+1} ... a _{2n} }

then the digits in the second half of the repeating decimal period are the 9s complement of the corresponding digits in its first half. In other words,

a _{i+a} _{i+n} =9 a _{1...} a _{n+a} _{n+1} ... a _{2n} =10 ^{n-1.}

For example,

(1)/(13) =0 .\overline{ 076923 } and 076+923=999 . (1)/(17) =0 .\overline{ 0588235294117647 } and 05882352+94117647=99999999 .

Extended Midy's theorem

If k is any divisor of h (where h is the number of digits of the period of the decimal expansion of a/p (where p is again a prime)), then Midy's theorem can be generalised as follows. The extended Midy's theorem^[2] states that if the repeating portion of the decimal expansion of a/p is divided into k-digit numbers, then their sum is a multiple of 10^k - 1.

For example,

(1)/(19) =0 .\overline{ 052631578947368421 } has a period of 18. Dividing the repeating portion into 6-digit numbers and summing them gives 052631+578947+368421=999999 . Similarly, dividing the repeating portion into 3-digit numbers and summing them gives 052+631+578+947+368+421=2997=3 x 999 .

Midy's theorem in other bases

Midy's theorem and its extension do not depend on special properties of the decimal expansion, but work equally well in any base b, provided we replace 10^k - 1 with b^k - 1 and carry out addition in base b.

For example, in octal

\begin{align} & (1)/(19) =0 .\overline{ 032745 } _{8} \\[ 8 pt] & 032 _{8+745} _{8=777} _{8} \\[ 8 pt] & 03 _{8+27} _{8+45} _{8=77} _{8. \end{align}}

In duodecimal (using inverted two and three for ten and eleven, respectively)

\begin{align} & (1)/(19) =0 .\overline{ 076 l{E}45} _{12} \\[ 8 pt] & 076 _{12} + l{E}45 _{12} = l{EEE} _{12} \\[ 8 pt] & 07 _{12} +6 l{E} _{12} +45 _{12} = l{EE} _{12} \end{align}

Proof of Midy's theorem

Short proofs of Midy's theorem can be given using results from group theory. However, it is also possible to prove Midy's theorem using elementary algebra and modular arithmetic:

Let p be a prime and a/p be a fraction between 0 and 1. Suppose the expansion of a/p in base b has a period of ℓ, so

\begin{align} & (a)/(p) = [ 0 .\overline{a _{1a} _{2...} a _{\ell}]} _{b} \\[ 6 pt] & ⇒ (a)/(p) b ^{\ell} = [a _{1a} _{2...} a _{\ell.\overline{a} _{1a} _{2...} a _{\ell}]} _{b} \\[ 6 pt] & ⇒ (a)/(p) b ^{\ell} = N + [ 0 .\overline{a _{1a} _{2...} a _{\ell}]} ^{b = N + (a)/(p)} \\[ 6 pt] & ⇒ (a)/(p) = (N)/(b ^{\ell-1}) \end{align}

where N is the integer whose expansion in base b is the string a₁a₂...a_ℓ.

Note that b ^ℓ - 1 is a multiple of p because (b ^ℓ - 1)a/p is an integer. Also bⁿ-1 is not a multiple of p for any value of n less than ℓ, because otherwise the repeating period of a/p in base b would be less than ℓ.

Now suppose that ℓ = hk. Then b ^ℓ - 1 is a multiple of b^k - 1. (To see this, substitute x for b^k; then b^ℓ = x^h and x - 1 is a factor of x^h - 1.) Say b ^ℓ - 1 = m(b^k - 1), so

(a)/(= p) (N)/(m(b ^{k-1)}) .

But b ^ℓ - 1 is a multiple of p; b^k - 1 is not a multiple of p (because k is less than ℓ ); and p is a prime; so m must be a multiple of p and

(am)/(= p) (N)/(b ^{k-1})

is an integer. In other words,

N\equiv 0 \pmod{b ^{k-1}.}

Now split the string a₁a₂...a_ℓ into h equal parts of length k, and let these represent the integers N₀...N_h - 1 in base b, so that

\begin{align} N _{h-1} & = [a _{1...} a _{k]} _{b} \\ N _{h-2} & = [a _{k+1} ... a _{2k} ] _{b} \\ & {} \vdots \\ N _{0} & = [a _{l-k+1} ... a _{l]} _{b \end{align}}

To prove Midy's extended theorem in base b we must show that the sum of the h integers N_i is a multiple of b^k - 1.

Since b^k is congruent to 1 modulo b^k - 1, any power of b^k will also be congruent to 1 modulo b^k - 1. So

N = \sum ^{h -1} _{i =0} N ^{ik} _{ib} = \sum ^{h -1} _{i =0} N ^{k} _{i(b} ) ^{i} ⇒ N \equiv \sum ^{h -1} _{i =0} N _{i} \pmod{b ^{k-1}} ⇒ \sum ^{h -1} _{i =0} N _{i} \equiv 0 \pmod{b ^{k-1}}

which proves Midy's extended theorem in base b.

To prove the original Midy's theorem, take the special case where h = 2. Note that N₀ and N₁ are both represented by strings of k digits in base b so both satisfy

0 \leq N _{i} \leq b ^{k-1.}

N₀ and N₁ cannot both equal 0 (otherwise a/p = 0) and cannot both equal b^k - 1 (otherwise a/p = 1), so

0 < N _{0+N} _{1} < 2 (b ^{k-1)}

and since N₀ + N₁ is a multiple of b^k - 1, it follows that

N _{0+N} _{1} = b ^{k-1.}

Corollary

From the above,

(am)/(p) is an integer

Thus

m \equiv 0 \pmod p

And thus for

k = (\ell)/(2) b ^{\ell/2} + 1 \equiv 0 \pmod p

For

k = (\ell)/(3) and is an integerb ^{2\ell/3} + b ^{\ell/3} + 1 \equiv 0 \pmod p

and so on.

Notes

1. Leavitt. William G.. A Theorem on Repeating Decimals. The American Mathematical Monthly. June 1967. 74. 6. 669–673. Mathematical Association of America. 10.2307/2314251. 2314251. 0211949.
2. Bassam Abdul-Baki, Extended Midy's Theorem, 2005.

References

• Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 158–160, 1957.
• E. Midy, "De Quelques Propriétés des Nombres et des Fractions Décimales Périodiques". College of Nantes, France: 1836.
• Ross, Kenneth A. "Repeating decimals: a period piece". Math. Mag. 83 (2010), no. 1, 33–45.