Question

Original qustion posted here:

http://www.jiskha.com/display.cgi?id=1239379999


If you do computations Modulo some number, say Modulo 11, then you identify numbers from the ordinary number system that differ by a mulitple of 11.

A rigorous mathematical definition of these new numbers is as follows. For every ordinary number n, you consider the set [n], defined as containing all the numbers that differ from n by a multiple of 11. n itself is included in this set. So, we have:

[n] = {n, n + 11, n - 11, n + 2*11, n-2*11,....}

There are then only 11 of these sets:

[0], [1], [2],...,[10].

If you consider [11], then you see that this is the same as [0]. The numbers in the number system Modulo 11 are these 11 objects [0], [1],...,[10].

Addition is defined in terms of addition of ordinary numbers by:

[A] + [B] = [A + B]

Multiplication by:

[A]x[B] = [AxB]

It is conventional to omit the square brackets. So, when we say that -2 = 9 when working Modulo 11, the -2 and the 9 refer to the [-2] and the [9] defined above, not the -2 and the 9 of the ordinary number system.