We can break a complex number into it’s 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.
We have that a complex number is equivalent to it’s 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}. z = w if and only if \text{Re}(z) = \text{Re}(w) and \text{Im}(z) = \text{Im}(w).
(\Leftarrow) 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.
(\Rightarrow) Let z = w. Then (x,y) = (u,v). We know that ordered pairs are only equivalent if x = u and y = v. Since \text{Re}(z) = x and \text{Re(w) = u}, we have \text{Re}(z) = \text{Re}(w). A similar argument applies to the imaginary part.