Okay, so we have the natural numbers, and we've defined an ordering for them based on inequality. We've also defined addition, multiplication, and exponentiation, which means we can do things with them. This leads to a followup question: what *can't* we do with them?

Looking at addition first, we have that given any two natural numbers a and b, a+b=n, where n is also always a natural number. But if a+n=b, there is not always an n which makes that statement true. Let me provide a concrete example: if a=2 and b=1, then a+b=3, and 3 is a natural number. But there is no natural number n such that 2+n=1.

This leads to the definition of the integers as an extension of the natural numbers. Given any natural numbers a and b, there is an integer n such that a+n=b. Note that this extension preserves all of the properties of the natural numbers. The integers are ordered. a<b if a+n=b and n is an integer greater than 0. (There's the possibility of circular definition here. n>0 can't simply mean that n=0+n, because that's always true. Saying that an integer n>0 if n is also a natural number avoids this problem.)

Addition is defined on the integers, and has closure and is commutative and associative. In addition, we can now define 0 as the additive identity, giving us another property for addition. Also, it is now true that for any integer a, there exists a number b such that a+b=0. This also means that we can define subtraction as negative addition.

Multiplication keeps all the same properties and is unchanged. Exponentiation is unchanged as long as the exponent is non-negative. a^{0} is defined as 1 for all integers except 0. 0^{0} is undefined. Basically, depending on how you get there, sometimes 0^{0} should equal 1, and sometimes it should equal 0. If you define it one way or the other, it will end up breaking math. Instead, math says 0^{0} has no meaning, so if you've done a bunch of math and end up writing that down, you should start over with an approach that doesn't end up using it.

We can't define negative exponents yet, but we will soon. In fact, that's a good place to start next time.