Eli5 Why do we need complex numbers in real life?
In: 11
Mathematical concepts correspond to human ideas.
Multiplication, as its taught in school, corresponds to areas of squares.
Negative numbers correspond to direction; you’re going in the backwards direction as opposed to forward.
In this sense, imaginary numbers represent rotation of shapes, among many other things.
If you multiply a number by the square root of -1, it corresponds to rotating that point by 90 degrees in a certain context.
Multiplying a number by just -1 rotates it by 180 degrees, you’re going backwards to where you came from.
This extends to any angle you want; if you multiply a number by the square root of the square root of -1, you get rotation of 45 degrees. Multiplying by the cube root of -1 gives rotation of 60 degrees. Multiplication by the cube root of the square root of -1 gives rotation of 30 degrees, and so on.
But imaginary numbers have many more uses than just this, just like multiplication is not just used to find the area of a square; this is just one of the first properties that crops up.
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