Theory

Theory 1

The complex numbers $ℂ$ are sums of real and imaginary numbers. Every complex number can be written uniquely in ‘Cartesian’ form:

$$z=a+bi,a,b\in ℝ$$

To add, subtract, scale, and multiply complex numbers, treat ‘$i$’ like a constant.

Simplify the result using $i^2=-1$.

For example:

$$\begin{array}{c}(1+3i)(2-2i)\gg \gg 2-2i+6i-6i^2\\ \\ \gg \gg 2+4i-6(-1)\gg \gg 8+4i\end{array}$$

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Complex conjugate

Every complex number has a complex conjugate:

$$z=a+bi\gg \gg \bar{z}=a-bi$$

For example:

$$\begin{array}{cc}\overline{2+5i}& =2-5i\\ & \\ \overline{2-5i}& =2+5i\end{array}$$

In general, $\overline{\stackrel{‾}{z}}=z$.

Conjugates are useful mainly because they eliminate imaginary parts:

$$(2+5i)(2-5i)\gg \gg 4+25\gg \gg 29$$

In general:

$$(a+bi)(a-bi)\gg \gg a^2-abi+bia-b^2{i}^2\gg \gg a^2+b^2\in ℝ$$

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Complex division

To divide complex numbers, use the conjugate to eliminate the imaginary part in the denominator.

For example, reciprocals:

$$\begin{array}{c}\frac{1}{a+bi}\gg \gg \frac{1}{a+bi}\cdot \frac{a-bi}{a-bi}\\ \\ \gg \gg \frac{a-bi}{a^2+b^2}\gg \gg \left(\frac{a}{a^2+b^2}\right)+\left(\frac{-b}{a^2+b^2}\right)i\end{array}$$

More general fractions:

$$\begin{array}{c}\frac{a+bi}{c+di}\gg \gg \frac{a+bi}{c+di}\cdot \frac{c-di}{c-di}\\ \\ \gg \gg \frac{ac+bd+(bc-ad)i}{c^2+d^2}\gg \gg \frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i\end{array}$$

Multiplication preserves conjugation

For any $z,w\in ℂ$:

$$\overline{zw}=\bar{z}\bar{w}$$

Therefore, one can take products or conjugates in either order.