Complex conjugate

Let $$z = a+bi$$ be a complex number where $$a,b\in \mathbb{R}$$. The complex conjugate of $$z$$, denoted by $$\overline{z}$$, is given by $$a - bi$$. In other words, to obtain the complex conjugate of $$z$$, one simply flips the sign of its imaginary part.

Examples

1. $$\overline{4} = 4$$ because the imaginary part of $$4$$ is $$0$$.

2. $$\overline{1+2i} = 1 - 2i$$.

3. $$\overline{3i} = - 3i$$.

One can avoid memorizing the formula for the multiplicative inverse by making use of the complex conjugate. For example, to turn $$\displaystyle\frac{1}{1+i}$$ into the form $$p+qi$$ where $$p,q \in \mathbb{R}$$, multiply the numerator and denominator by the complex conjugate of the denominator. Namely, $\begin{eqnarray} \frac{1}{1+i} & = & \frac{1}{1+i} \frac{\overline{1+i}}{\overline{1+i}} \\ & = & \frac{1}{1+i} \frac{1-i}{1-i} \\ & = & \frac{(1-i)}{(1+i)(1-i)} \\ & = & \frac{1-i}{1-i^2} \\ & = & \frac{1-i}{2} = \frac{1}{2}-\frac{1}{2}i \end{eqnarray}$

Hence, dividing the complex number $$z$$ by the complex number $$w$$ can be performed as $$\displaystyle\frac{z}{w} = \frac{z}{w}\frac{\overline{w}}{\overline{w}}$$. For example, $\frac{3+2i}{2-i} =\frac{3+2i}{2-i}\cdot\frac{2+i}{2+i} =\frac{4+7i}{2^2-i^2} =\frac{4+7i}{4+1} = \frac{4}{5} + \frac{7}{5}i.$

Exercises

1. Give the complex conjugate of each of the following:

1. $$2i$$

2. $$\displaystyle -2+\frac{3}{5}i$$

3. $$\pi$$

2. Let $$z$$ and $$w$$ be complex numbers. Show that

1. $$\overline{\overline{z}} = z$$.

2. $$\overline{z + w} = \overline{z} + \overline{w}$$.

3. $$\overline{z w} = \overline{z}~\overline{w}$$.

4. $$\displaystyle\overline{z^{-1}} = (\overline{z})^{-1}$$.