Axioms and Construction

In algebra we learned Complex numbers exist in the form z = u + vi, where u,v are real numbers and i = \sqrt{-1}. Complex analysis requires rigorous proof, so we must define exactly what we mean by a complex number.

Axioms

Complex numbers are not usually stated as axioms; they are instead defined as ordered pairs of real numbers or as a field extension of the reals. These definitions lead to construction-dependent properties without giving a list of explicit axioms.

We want to choose a set of axioms which give all the desired properties of complex numbers shown in the last section. I have listed below a slightly modified version of the axioms found from MetaMath, a formal math database.

Axioms

The complex numbers are a set denoted by \mathbb{C}, such that the following properties hold

(C1) \mathbb{C} is a field.

(C2) \mathbb{C} contains a subfield isomorphic to \mathbb{R}. We will call this subfield \mathbb{R}_C

(C3) There exists a number i \in \mathbb{C} such that (i \cdot i) + 1 = 0.

(C4) For every complex number z \in \mathbb{C}, there exist x,y \in \mathbb{R}_C such that z = x + (y \cdot i).

There are some differences between these axioms and the ones stated by MetaMath, such as the former defining \mathbb{R} as a genuine subset of \mathbb{C}. The axioms given above allow us to use our familiar construction of the reals given in real analysis.

Note that while although \mathbb{R}_C and \mathbb{R} are set-theoretically different, they are isomorphic. We will use them interchangly, and occasionally make the false statement

Construction

Now we must show that we can construct a structrure satisfying these axioms. There are a few equivalent ways to do this, the most common being the use of ordered pairs.

Ordered pairs

We will use the most common construction of the complex numbers, defining them as ordered pairs of real numbers.

Definition

The set of complex numbers \mathbb{C} is equivalent to the cartesian product \mathbb{R} \times \mathbb{R}.

We also use the following definitions for addition and multiplication

Definition

The Addition relation +_\mathbb{C} \colon \mathbb{C} \to \mathbb{C} is defined (x_1,y_1) +_\mathbb{C} (x_2,y_2) \to (x_1 + x_2, y_1 + y_2).

Definition

The Multiplication relation \cdot_\mathbb{C} \colon \mathbb{C} \to \mathbb{C} is defined (x_1,y_1) \cdot_\mathbb{C} (x_2,y_2) \to (x_1x_2 - y_1y_2, x_1y_2 + x_2y_1).

We will immeditably abuse notation and denote +_\mathbb{C} as +, and \cdot_\mathbb{C} as \cdot (or simply A \cdot B as AB). Now to show that this definition satisfies our axioms!

Theorem (Axiom C1)

The complex numbers (\mathbb{C}, +, \cdot) form a field. That is

(F1) Addition is closed.

\forall z,w \in \mathbb{C}, z + w \in \mathbb{C}.

(F2) Addition is commutative.

\forall z,w \in \mathbb{C}, z + w = w + z.

(F3) Addition is associative.

\forall z,w,t \in \mathbb{C}, (z + w) + t = z + (y + z).

(F4) There exist an additive identity.

Theorem (Axiom C2)

The complex numbers contain a subfield \mathbb{R}_C isomorphic to \mathbb{R}.

Theorem (Axiom C3)

There exists a number i \in \mathbb{C} such that (i \cdot i) + 1 = 0.

Let i = (0,1). Then from some algebra we obtain

\begin{align*} ((0,1) \cdot (0,1)) + (1,0) &=\\ (0 - 1, 0 + 0) + (1,0) &=\\ (1 - 1,0) = (0,0) &= 0. \end{align*}

Where 1 = (1,0) and 0 = (0,0) as defined from our field.

Theorem (Axiom C4)

For every complex number z \in \mathbb{C}, there exist x,y \in \mathbb{R}_C such that z = x + (y \cdot i).

We know that \mathbb{R}_C = (a,0) where a \in \mathbb{R}. Then after algebra we have for all z \in \mathbb{C}, exists x,y \in \mathbb{R} such that z = (x,y), which is exactly our definition.

And just like that, we’ve shown that ordered pairs are a model for the complex numbers.