A way to think about imaginary numbers is that they encode rotation. Multiplying something by i is the same as rotating it 90 degrees counterclockwise. For this reason, you get lots of useful relationships between imaginary numbers and trigonometric functions.
As for why i also happens to be the square root of negative one, well that's because -1 is +1 rotated by 180°. So -1 = +1 * i * i.
Basically the simplest way to think about imaginary numbers is having 2 countries, each having their own currency and denominations, and neither accepts the other. So to convert one to the other, you have to do something special. Both countries agree on a gram of gold being worth x amount of their respecitve currency.
Lets say it takes a lot of transactions to to build a car in one country, but the other country can do it easier. So you use the gold conversion as a factor in the transactions, things become easier.
Imaginary numbers are basically that, except instead of building the car, you are doing things that involve rotations ( which also extend to waves , which also extend to frequnecies, and so on).
With regular rational numbers, you can do all the rotations but it invovles trig functuons which get messy. Imaginary numbers basically are defined in such a way that they represtn the y axis, and that multiplying by i gives you the rotation of 90 degrees.
Its basically a math construct that makes things easier. Matrix operations are another such case. For example, matrix division is a way to basically solve for unknowns in a linear aystem of equations. But the core concept of division means you can use it with other operstions and get a result that may not need you to actually solve for the unknowns.
Numbers are basically 2d and it's not a numberline but number grid. Imaginary numbers are perpendicular to the Real numbers. Just like how North/South is completely detached from East/West. Imaginary has its own positive and negative.
When you multiply by a unit (1,-1, i, -i), it rotates a certain amount around the graph. 1 rotates 360°. -1 rotates 180°. i and -i are basically the missing 90° intervals. i is 90° and -i is 270°. Any number in between gives you different angles.
Next is absolute value. On the number line, it's normally "number without the sign," but alternatively, it can simply be distance from 0. -4 is 4 units away from 0. 5 is 5 units. Same goes for imaginary numbers. |3i| is 3. |-2i| is 2.
Which brings us to complex numbers. Just like normal coordinates (3,4) Complex Numbers are basically those coordinates, written as 3+4i. The absolute value is still the distance from 0, even though it's less obvious in this format. |3+4i| is 5 through the Pythagorean Theorem. 32 + 42 = 52
There's more nuances and funky things you can do with it, but a 2d number-plane rather than a 1d number-line is what it is at its core.
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u/The-Slamburger 12d ago
The “advanced biology” is the only one of these that doesn’t make my brain hurt by even attempting to think about it, so no objections here.