### Still Modular Arithmetic

Usual arithmetic deals with natural numbers \(1\), \(2\), \(3\), \(4\), \(5\), ... and integers ..., \(-4, -3, -2, -1, 0, 1, 2, 3, 4, \)... What about Modular Arithmetic? Which numbers are considered in this arithmetic?

Take the example of \[20=8\,(\mbox{mod }12).\]

In this case number \(20\) is identified with number \(8\), that is, numbers \(20\) and \(8\) é are equivalent in Modular Arithmetic modulus \(12\). There is an infinity of other numbers equivalent to \(8\): \(32\), \(44\), \(56\), ... The set \[\{8,20,32,44,56,68,...\}\] is called the equivalence class of \(8\) modulus \(12\) and it is represented by its smallest member, that is, \(8\). Similarly, there are \(11\) more equivalence classes represented by numbers \(0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11\).

These are the numbers in modulus 12 arithmetic: \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \) and \(11\).

Generalizing to any number \(n\), the numbers in modulus \(n\) arithmetic are \(0, 1, 2, ..., n-2\) and \(n-1\).

Since this type of arithmetic deals only with a finite number of "numbers", Modular arithmetic is alsom said to be a finite arithmetic.

Now that we have our numbers, the next step is to study the operations that can be carried out with these numbers, in particular, addition and multiplication.