# The Set of Complex Numbers as an Algebraic Field Examples 1

Recall from The Set of Complex Numbers as an Algebraic Field page that the set of complex numbers $\mathbb{C}$ with the operations of addition $+$ and multiplication $\cdot$ is indeed an algebraic field. We will now prove some basic results of this field using the field axioms.

## Example 1

**Prove that for every $z \in \mathbb{C}$ with $z \neq 0$ that the multiplicative inverse $z^{-1}$ of $z$ is unique.**

Suppose that $z^{-1}$ and $z^{-1*}$ are both multiplicative inverses to $z \in \mathbb{C}$, $z \neq 0$. Then:

(1)Multiplying on the left by $z^{-1}$ gives us that $z^{-1} = z^{-1*}$, so the multiplicative inverse of $z$ is unique.

## Example 2

**Using the field axioms, prove that for all $z, w \in \mathbb{C}$ with $z, w \neq 0$ that $\displaystyle{\frac{1}{z \cdot w} = \frac{1}{z} \cdot \frac{1}{w}}$.**

We equivalently want to prove that:

(2)Note that:

(3)So $(z \cdot w)$ is the multiplicative inverse of $(z^{-1} \cdot w^{-1})$ and conversely, $z^{-1} \cdot w^{-1}$ is the multiplicative inverse of $z \cdot w$. But $(z \cdot w)^{-1}$ is also the multiplicative inverse of $z \cdot w$ and so by the uniqueness of multiplicative inverses we see that:

(4)## Example 3

**Suppose that $\displaystyle{\frac{x - yi}{x + yi} = a + bi}$. Prove that then $a^2 + b^2 = 1$.**

Let $z = x + yi$. Then $\overline{z} = x - yi$ and so:

(5)Take the modulus of both sides of the equation above to get:

(6)We know that $\mid z \mid = \mid \overline{z} \mid$ for all $z \in \mathbb{C}$, and so $\displaystyle{\frac{\mid \overline{z} \mid}{\mid z \mid} = 1}$ (provided the denominator is nonzero, and so:

(7)