3.2 Whole numbers
If we append 0 to N as a new smallest value we get the whole numbers, denoted W. The whole numbers are not used a lot by themselves in mathematics but let’s see what we get with this additional value.
We are still closed under addition and multiplication and not closed under division. We do now have to watch out for division by zero. Expressions like 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, which is a new concept for us to consider. I’ve put it in bold to show how special it is. This 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 + 0 = 0 + 14 = 14. Also, 0 × w = w × 0 = 0.
For the whole numbers W we have a collection of values
...
