Modular arithmetic

This article is about the concept that uses the "a (mod m)" notation. For the binary operation mod(a,m), see Modulo.

In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching or exceeding a certain value, called the modulus. The modern approach to number theory using modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar setting exhibiting modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in 7 + 8 = 15, but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is congruent to 3 modulo 12, and we write 15 ≡ 3 (mod 12), so 7 + 8 ≡ 3 (mod 12).

Similarly, if one waits 8 hours and then 8 more hours (thus 16 hours in total), the clock will show the same time change as if one waited 4 hours. This is reflected by the identity 2 × 8 ≡ 4 (mod 12). After a wait of exactly 12 hours, the hour hand will be right where it started, so 12 acts as 0; one writes 12 ≡ 0 (mod 12).

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