# Reading 2022-04-05

## Metadata

- Ref:: Better Explained
- Title:: Math and Analogies
- Author:: Kalid Azad
- Year of publication:: 2019
- Category:: Blog
- Topic::
- Related::

## Notes from reading

Using the symbol `0`

to represent nothingness is a really weird concept. And when you've tied your concept of a number to physical items, concepts like zero become difficult. How about this negative number? What's less than nothing? What's a negative rock?

It took a few thousand years to accept this new feature — negative numbers were still controversial in the 1700s! Negatives aren’t something we can touch or hold, but they describe certain relationships well (like debt). It was a useful fiction.

An idea: numbers are 2D directions

The imaginary dimension lets us get to negatives in two steps. Saying *"i squared is negative one"* is really saying: I'm starting at 1. We multiply by i, multiply by i again, and get to -1. So i represents a 90-degree rotation, and rotating twice points you backwards. That's what *"i squared equals -1"* means.

what multiplication of two complex numbers does:

- Regular multiplication (
*"times 2"*) scales up a number (makes it larger or smaller) - Imaginary multiplication (
*"times i"*) rotates you by 90 degrees

Multiplying by `(2 + i)`

means *double your number and add in a perpendicular rotation*

What about visualizing the multiplication of two complex numbers ("triangles"), like $(3+4i).(2+3i)$. I see this as: *Make a scaled version of our original triangle (times 2) and add a scaled/rotated triangle (times 3i)*

Let’s take a look. Suppose I’m on a boat, with a heading of 3 units East for every 4 units North. I want to change my heading 45 degrees counter-clockwise. What’s the new heading?

An approach by complex number:

- We’re on a heading: 3 units East, 4 units North = $3 + 4i$
- we want to rotate counter-clockwise by 45 degrees = multiply by $1 + i$
- We noticed that 45 degrees is $1 + i$ (perfect diagonal), so we can multiply by that amount

By multiplying together: $(3+4i).(1+i)=3+3i+4i+4i^2=-1+7i$

So our new orientation is 1 unit West (-1 East), and 7 units North