 We've assumed certain properties of complex numbers. To prove them, we need to introduce a formal definition of a complex number. We'll take the approach presented by William Ron Hamilton in 1835 in his theory of conjugate functions or algebraic couples with a preliminary and elementary essay on algebra as a science of pure time. For our purposes, the key part of Hamilton's work is the introduction of what he called number couples, though we'd call them ordered pairs of real numbers. After some preliminary examination of how we'd like these number couples to act, Hamilton introduces the following definitions, which we'll put this way, let A, B, C, D be ordered pairs of real numbers, upon which the following operations are defined. The sum of two number couples will be defined component-wise, and the product will be defined in some rather bizarre way which Hamilton does justify earlier in his paper. So at this point we have an obligatory math joke, well not really, but how many legs does a dog have if you call a tail a leg? And the answer is four. Calling a tail a leg doesn't make it a leg. And how this connects to higher mathematics is the following, it's worth keeping in mind things that act the same are the same. And in particular, in mathematics we don't really care what something is called, what we do care about is what its properties are. So it's easy to show that the number couple A0 corresponds to the real number A. In particular, everything we can say about the real number A is also true about the number couple A0, and conversely. For example we have commutativity of addition A plus B is equal to B plus A for all real numbers A and B. But meanwhile for all number couples A0, B0, we can add them. And remember definitions are the whole of mathematics, all else is commentary. We have a definition for how to add number couples, and if we apply it, but remember A and B themselves are real numbers so we do know we have commutativity of the real numbers, and definitions work both ways. If we have the sum, we can break it apart into the summands. And so A0 plus B0 is B0 plus A0 for all A0, B0. And again these number couples act just like real numbers. Now consider the product 01 by 01. We have this product, again we have our definition for multiplication, so if we apply it, we get negative one zero. And so that tells us the square of 01 is negative one zero. But remember the number couple A0 corresponds to the real number A, so this number couple negative one zero corresponds to the real number negative one. So the square of whatever 01 is is negative one. Consequently, the number couple 01 corresponds to the pure imaginary number I. And because the components of the ordered pair are real numbers, we can invoke all the properties of the real numbers, and consequently we can prove the following. The complex numbers satisfy an additive identity, a multiplicative identity, associativity of addition and multiplication, commutativity of addition and multiplication, and distributivity of multiplication over addition. In other words, all of the arithmetic properties we've been assuming hold true can in fact be proven to hold true. So you should prove all the properties, but just as an example let's prove commutativity. We have the sum of the two number couples, definitions of the whole of mathematics, all else is commentary. Our definition tells us how to add the two, and since everything inside is a real number, commutativity of addition holds so we have. And again, definitions work both ways. We can break this apart into its components, and that proves commutativity.