Problems and Proofs in Numbers and Algebra by Richard S. Millman Peter J. Shiue & Eric Brendan Kahn

Author:Richard S. Millman, Peter J. Shiue & Eric Brendan Kahn
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

This result allows us to picture the collection of equivalence classes (mod 12) as a clock. (We could write instead of as they are equal sets but it is better to use as indicated by the division algorithm, and soon we’ll see why.)

Fig. 3.1Usual 12-clock

We are thus led to make the following (Fig. 3.1).

Definition.

The set of equivalence classes of with respect to the “equivalence relation modulo n” is called the integers modulo n and is written .

Notation.

has n elements: equivalence classes of 0, 1, …, n − 1. To simplify the notation, we shall write to represent the equivalence class , under , of the integer k since n is fixed. Of course, since . We continue to write for .

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