If it doesn’t exist, why and how do we use it?
In: Mathematics
> how do we use it?
Complex numbers (numbers of the form a+bi for real numbers a and b, where i^(2) = -1) are very useful because they can encode rotation.
When we multiply by a positive real number, we “scale” by that factor. Multiplying by 2 scales up by 2; multiplying by 1/2 scales down by 1/2. Multiplying by a negative real number “scales” but also “reflects” the number from positive to negative. If you think about a number line, multiplying by a negative will “reflect” your number about 0.
Complex numbers extend this geometric intuition. Instead of representing complex numbers on a line, we represent them on a plane, where the x-axis is the real number line. Now, multiplying by a complex number both scales your original number and *rotates* it by a certain amount about the origin.
Multiplying by i, for instance, rotates a number 90 degrees counterclockwise. Multiplying by i^(2) rotates by 90 degrees, twice—so it rotates by 180 degrees. If you imagine starting with a positive real number and multiplying by i twice, then you end up with a negative real number. And this is exactly what you expect, since i^(2) = -1, so multiplying by i twice is the same as multiplying by -1, and we already know -1 flips positive reals to be negative reals.
The idea though is that complex numbers let us rotate by any amount. This makes them very useful for compactly and concisely representing lots of more complex phenomena, since instead of having to write multiple equations to capture some behavior, we can just use complex numbers to capture all sorts of scalings and rotations in single equations.
Another reason i is so useful is because of the formula e^(ix) = cos(x) + i*sin(x). This formula lets us encode all sorts of waves concisely in terms of complex numbers.
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