 With the set of integers in hand, we can define the four basic arithmetic operations of addition, subtraction, multiplication, and division. We can define addition as an exercise in counting. A plus B means start with the number specified by the first term, A, and count the number specified by the second term, B. In this example, we start with A at 11 and count seven more for B. Zero is the additive identity in that adding zero to any number gives you the same number you started with. Subtraction would then be start with the first term and count backwards the number specified by the second term. Here we are counting seven times back from 11. This addition and subtraction are tied directly to counting and counting has been shown to accurately represent anything you can count. We can also define multiplication in terms of addition, which is based on counting. A times B says add the number A to itself, B times. Here we are starting with three and adding it to itself five times. Counting it one time leaves it unchanged. Thus the number one is the multiplicative identity, like zero was the additive identity. But what does it mean to add a number to itself zero times? To deal with this, we define a number being added to itself zero times to be the number zero. We define division as the inverse of multiplication. In the division of one number, say A, by another number, say B, we are asking how many times can we subtract B from A? Or more generally, what number, when multiplied by B, gives us A? For example, we would ask how many times can we subtract three from fifteen and the answer is five.