Chapter 11: Modular Arithmetic: Situation 5 From the MACMTL–CPTM Situations Project

A group of high school Mathematics Club members was examining the concept of modular arithmetic. They were working in mod 5, and as they were becoming familiar with mod 5, a student asked whether it is possible to write fractions in mod 5. She wondered about the meaning of expressions such as

mod 5 .

The symbol

⁠, and more generally,

$ab$

such that a and b are integers and b ≠ 0, can be interpreted in a variety of ways: as a single rational number (commonly called fraction ), as a ratio of two numbers, or as a quotient of two numbers. However, these interpretations may cause confusion when dealing with operations within integer rings ℤⁿ (where n is a positive integer). Thus, it is important to move beyond the previously mentioned common interpretations of the symbol

$ab$

and regard it as only a symbol.