### Modular addition

(Modular Arithmetic)

In standard arithmetic, \(5 + 10\) is equal to \(15\), but modulus \(12\), in Modular Arithmetic, it is equal to \(3\), since this is the remainder of the divison of \(15\) by \(12\). This is usually expressed has \[5+10=3\,(\mbox{mod }12).\]

What about modulus \(9\) instead of modulus \(12\)? Proceed similarly: since \(15 = 1 \times 9 + 6\), we say that \[5+10=6\,(\mbox{mod }9).\]

Check this *applet* for other examples.

Modular addition has some nice properties like, for instance:

- It is commutative: (\(a + b\) is equal to \(b + a\) for every \(a\) and \(b\));
- It has an identity element (namely, zero) that satisfies \(a+0=a\), for every \(a\))
- Every number \(a\) has an inverse (symmetric) element: there is a \(b\) - possibly equal to \(a\) in some cases - such that \(b+a=0\).

Check these and other properties of modular addition in the following "coloured"
* Addition Table*.

Modular addition may be also seen as the sum of two line segments ("that may be broken") or as the sum of two circular arcs.