# Mike's Notes

Assorted notes

November 2nd 2018

# Complex Numbers

## Better Explained

### A Visual, Intuitive Guide to Imaginary Numbers

This was a nice intro to imaginary numbers. I already had a bit of an understanding of them, which probably helped.

Thinking of imaginary numbers as a rotation was a new concept to me. It is a neat way to understand imaginary numbers.

Complex numbers take the form $a + bi$

### Intuitive Arithmetic With Complex Numbers

This was a bit harder to follow than the previous article. I got a bit lost in the explanation of complex division.

The magnitude of a complex number is straight forward Pythagoras;

Multiplication also follows the same rules as vector multiplication. You multiply all the components with all the other components. Remeber that $i^{2} = -1$

I struggled to follow the description of division. What does it mean to divide a complex number? In the end I read some other articles to try and get a clearer understanding.

The key to division is that you want to multiply out the denominator, so that you can get rid of the division by i. You use the conjugate of the denominator to achieve this. The conjugate simply means flipping the sign of the imaginary part, so that the i axis is mirrored.

Using the relationships we already know for magnitude, $\lvert{z}\rvert = \sqrt{a^{2} + b^{2}}$, we can express $z \cdot \bar{z}$ as $\lvert{z}\rvert^{2}$. The magnitude is a real number, so we just need to perform the multiply of the top part of the fraction with $\bar{z}$, and then divide by a real number.

## Understanding Why Complex Multiplication Works

Complex multiplication adds the angles, but why? I’m not convinced that this article left me fully enlightened; my notes are not particularly coherent. Conceptually, I seem to understand what is going on, but I need to work on explaining in a clearer way.

1. Change the mulitply to polar coordinates.
2. Expand out, and group the parts by real and imaginery.
3. We now have something resembling the sin and cosine addition formulae.

Multiplication scales a number. $i$ is a rotation, so scaling of $i$ is a rotation.

• Scale - multiply by a real number
• Rotate - multiply by i
• Scale & Rotate - multiply by a complex number

Can visualise complex rotation as either a path, or 2 steps.

## Intuitive Guide to Angles, Degrees and Radians

A circle is $2\pi$ radians.