We will start our analysis of Complex numbers by starting with our definition we used to construct them.
Let x,y \in \mathbb{R}. Then z = (x,y) is a complex number.
We commonly denote z with x + yi instead of (x,y).
The set of all complex numbers is denoted \mathbb{C}. Recall our definitions for addition and multiplication
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).
The Multiplication relation \times_\mathbb{C} \colon \mathbb{C} \to \mathbb{C} is defined
(x_1,y_1) \times_\mathbb{C} (x_2,y_2) \to (x_1x_2 - y_1y_2, x_1y_2 + x_2y_1).
We will abbreviate +_\mathbb{C} as +, and \times_\mathbb{C} as either \times, \cdot, or concatenated variables.
Real and imaginary parts
We define the following unary functions
The Real part of a complex number is given by the function \text{Re} \colon \mathbb{C} \to \mathbb{R} such that
\text{Re} \colon (x,y) \to x.
The Imaginary part of a complex number is given by the function \text{Im} \colon \mathbb{C} \to \mathbb{R} such that
\text{Im} \colon (x,y) \to y.
Let z = 2 + 3i. Then by our definition we have {z = (2,3).} This lets us calculate our real and imaginary parts as \text{Re}(z) = 2, and \text{Im}(z) = 3.
Now we can start using these functions for theorems. Our first theorem states that z is equal to its real and imaginary parts.
Let z \in \mathbb{C}. Then z = (\text{Re}(z),\text{Im}(z)).
Let z \in \mathbb{C}. Then z = (x,y). We have from our definitions of real and imaginary parts that \text{Re}(z) = x and \text{Im}(z) = y. Then
(\text{Re}(z),\text{Im}(z)) = (x,y) = z.
Our next theorem states that if two complex numbers have the same real and imaginary parts, they are the same number.
Let z,w \in \mathbb{C}. If \text{Re}(z) = \text{Re}(w), and \text{Im}(z) = \text{Im}(w) then w = z.
Let z = (x,y) and w = (u,v). If \text{Re}(z) = \text{Re}(w) then we have x = u. Similarly, if \text{Im}(z) = \text{Im}(w) then we have y = v. This gives us our equivalence
z = (x,y) = (u,v) = w.