# Complex conjugate properties

Here are some complex conjugate properties and identities that are useful to know for complex numbers $$z$$ and $$w$$.

Complex conjugation is distributive over addition, subtraction, multiplication and division. \begin{align*} \overline{z+w} &= \overline{z} + \overline{w} \\ \overline{z-w} &= \overline{z}-\overline{w} \\ \overline{zw} &= \overline{z} \, \overline{w} \\ \overline{\left( \frac{z}{w} \right)} &= \frac{\overline{z}}{\overline{w}}, \text{ if w\ne0.} \end{align*}

We can relate conjugation with the real part and the imaginary part. \begin{align*} z + \overline{z} &= 2\, \text{Re}(z) \\ z-\overline{z} &= 2i \,\text{Im}(z). \end{align*}

We have an identity for powers of $$z$$. $$\overline{z^n} = (\overline{z})^n \text{, for integer }n.$$

Lastly, the conjugate of a conjugate of a complex number $$z$$ is $$z$$. $$\overline{\overline{z}} = z.$$

Let’s prove a few of these.

Theorem. For complex numbers $$z$$ and $$w$$, we have $$z + \overline{z} = 2\, \text{Re}(z)$$.
Proof. Let $$z = x+yi$$, where $$x$$ and $$y$$ are real numbrs. Then $$z + \overline{z} = x+yi + x-yi = 2x = 2\text{Re}(z).$$

The next one is requires a bit more working, but relies on multiplying the top and the bottom of the fraction by the conjugate of the denominator.

Theorem. For complex numbers $$z$$ and $$w$$, we have $$\overline{\left( \frac{z}{w} \right)} = \frac{\overline{z}}{\overline{w}}.$$
Proof. Let $$z = a + bi$$ and $$w = c+di$$. Then \begin{align*} \overline{\left( \frac{z}{w} \right)} &= \overline{\left( \frac{a+bi}{c+di} \right)} \\ &= \overline{\frac{a+bi}{c+di} \times \frac{c-di}{c-di}} \\ &= \overline{\left( \frac{ac + bd + (bc-ad)i}{c^2 + d^2} \right)} \\ &= \frac{ac + bd-(bc-ad)i}{c^2 + d^2}. \end{align*}

Theorem. For complex number $$z$$, we have $$\overline{z^n} = (\overline{z})^n$$.
Proof. Continually apply the identity $$\overline{zw} = \overline{z} \, \overline{w}$$. That is, we have \begin{align*} \overline{z^n} &= \overline{z^{n-1} \times z} \\ &= \overline{z^{n-1}} \times \overline{z} \\ &= \overline{z^{n-2} \times z} \times \overline{z} \\ &= \overline{z^{n-2}} \times \overline{z} \times \overline{z} \\ &= \dots = \overline{z}^n.\end{align*}

## Problems

1. Prove the following for any complex $$z$$ and $$w$$.
1. $$\overline{z+w} = \overline{z} + \overline{w}$$
2. $$\overline{z-w} = \overline{z}-\overline{w}$$
3. $$\overline{zw} = \overline{z} \, \overline{w}$$
4. $$z-\overline{z} = 2i \,\text{Im}(z)$$
5. $$\overline{\overline{z}} = z$$