Modular arithmetic

Modular arithmetic, sometimes also called clock arithmetic, is a way of doing arithmetic with integers. Much like hours on a clock, which repeat every twelve hours, once the numbers reach a certain value, called the modulus, they go back to zero.

In general, given a modulus ${\displaystyle n}$, we can do addition, subtraction and multiplication on the set ${\displaystyle \{0,1,\ldots ,n-1\}}$ in a way that "wrap around" ${\displaystyle n}$. This set is sometimes represented by the symbol ${\displaystyle \mathbb {Z} _{n}}$, and called the set of integers modulo ${\displaystyle n}$.^[1][2][3]

People talked about modular arithmetic in many ancient cultures. For instance, the Chinese remainder theorem is many centuries old. The modern notation and exact definition of modular arithmetic were first described by Carl Friedrich Gauss.^[4]

1. ↑ "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-12.
2. ↑ Weisstein, Eric W. "Modular Arithmetic". mathworld.wolfram.com. Retrieved 2020-08-12.
3. ↑ "2.3: Integers Modulo n". Mathematics LibreTexts. 2013-11-16. Retrieved 2020-08-12.
4. ↑ Richard Taylor (2012). "Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity". Institute for Advanced Study. Retrieved 7 March 2013.