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:

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.