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.
In modulo arithmetic, numbers are represented by sets of equivalent numbers. These sets are called residue classes or congruence classes. In the context of modulo 11, there are 11 residue classes: [0], [1], [2], ..., [10].
To find the residue class of a number, you take the number and add or subtract multiples of 11 until you obtain a number between 0 and 10. For example, the residue class of 15 modulo 11 is [4] because 15 - 11 = 4. Similarly, the residue class of -30 modulo 11 is [4] because -30 + 3 * 11 = 4.
In modulo arithmetic, addition and multiplication are defined based on the ordinary arithmetic of the numbers within each residue class. For example, to add two residue classes, you add the corresponding numbers from each residue class. For instance, [3] + [8] = [11] = [0] since 3 + 8 = 11. Similarly, to multiply two residue classes, you multiply the corresponding numbers from each residue class. For instance, [2]x[9] = [18] = [7] since 2 * 9 = 18 and 18 - 11 = 7.
It is common to omit the square brackets and simply write the number itself to represent the residue class. So, when we say -2 = 9 modulo 11, we mean that the residue class of -2 is equal to the residue class of 9.