(This is Part II of a multi-part series. See Part I)
Consider the following statements about integer arithmetic.
even + even = even
even + odd = odd
odd + even = odd
odd + odd = even
even * even = even
even * odd = even
odd * even = even
odd * odd = odd
Speaking loosely, we can say that we have an Algebra consisting of the elements “even” and “odd.” We call this Algebra the Integers modulo 2. We write 0 as shorthand for even and 1 as shorthand for odd, and use the following notation to distinguish this modular arithmetic from our standard arithmetic.
0 + 0 = 0 (mod 2)
0 + 1 = 1 (mod 2)
1 + 0 = 1 (mod 2)
1 + 1 = 0 (mod 2)
0 * 0 = 0 (mod 2)
0 * 1 = 0 (mod 2)
1 * 0 = 0 (mod 2)
1 * 1 = 1 (mod 2)
A note on notation: think of the (mod 2) as describing the statement as a whole, not the element on the right-hand-side of the equals sign. What we are really saying is that in the Algebra of the Integers modulo 2, we have 1 + 1 = 0.
The following statements are a bit more of a mouthful, but are analogous to the example above.
[A Multiple of 3] + [A Multiple of 3] = [A Multiple of 3]
[A Multiple of 3] + [A number that is one more than a multiple of 3] = [A number that is one more than a multiple of 3]
Some simple notation is necessary. We use (mod 3) rather than (mod 2) here to indicate that we are thinking about multiples of 3 rather than multiples of 2. We use 0 to mean [A Multiple of 3], 1 to mean [A number one more than a multiple of 3], and 2 to mean [A number that is two more than a multiple of 3].
Then, we can write the statements above as
0 + 0 = 0 (mod 3)
0 + 1 = 1 (mod 3)
We leave it to you to write out the remaining 7 statements of addition and the 9 statements of multiplication. These statements describe the Algebra of the Integers Modulo 3.
So far, we have seen the Integers Modulo 2 and the Integers Modulo 3. It’s not hard to imagine the notation or interpretation for the Integers Modulo 4 or the Integers Modulo 37.
So if you are asked to find 2+2, you better know what Algebra you are working in! In the Algebra of the Reals or the Integers, 2+2=4. But in the Integers Modulo 3, we have 2+2=1(mod 3).
Given that our motivation is to understand what “An Algebra” is rather than to understand the intricacies of any particular Algebra, we’ll close our discussion of this family of Algebras here. The curious reader may want to explore whether division makes sense in this Algebra.