3.2 Whole numbers

If we append 0 to N as a new smallest value, we get the whole numbers, denoted W. They are both infinite sets of numbers, but N is a subset of W. We do not use the whole numbers a lot in mathematics, but let’s see what we get with this additional value. whole number number$whole W`bold

We are still closed under addition and multiplication and not closed under division. We do now have to watch out for division by 0. Expressions such as 3 – 3 or n – n, in general, are in W, so that’s a little better for subtraction, but this does not give us closure.

So far, there’s not much that we’ve gained, it seems. Or have we?

0 is an identity element for addition, a new concept for us to consider. I’ve put it in bold to show how special it is. This element is a unique (meaning there is one and only one) number such that for any whole number w, we have w + 0 = 0 + w = w.

Thus, 14...