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Comment by rthomas6 | Hacker News Reader

What finally made imaginary numbers intuitively make sense to me is realizing that the number i just represents a 90 degree rotation on the complex plane. That's why i*i = -1 (it's rotating 180 degrees), why imaginary numbers are orthogonal to real numbers, why e^ix = cos(x) + i*sin(x) makes sense, everything.

When you are dealing with 2 dimensions, complex numbers are a kind of hack for representing both dimensions without any kind of vectors or pairs or anything, just numbers.

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More relevant and less hacky, IMO, is that the real numbers aren't algebraically closed - there are polynomials with real coefficients that you cannot factor into getting real zeros.

This is a big problem when you have something like a spring-mass-damper system. The springiness, damping, and mass of the system generate coefficients of a degree-two polynomial, and the zeros of that polynomial correspond to how the system moves over time.

I majored in math and took a course on waves through the physics department. Everyone struggled with why imaginary numbers would show up and I was just like "we took the square root of this negative number, and the reals aren't closed when we take square roots, therefore imaginary."

I missed out on a lot of physical intuition and had basically no concept of what the imaginary numbers actually meant

With this intuition, why is it i*i=-1 and not i+i (90+90=180 after all)?

Because with complex numbers you are implicitly defining a point on the complex plane. So how on the Cartesian plane you can have 2x + 3y, and that defines the point (2,3), you can have 2 + 3i on the complex plane defining the same point.

If you want to rotate the Cartesian coordinate -90 degrees,

    x' = x*cos(π) - y*sin(π) = -y
    y' = y*cos(π) + x*sin(π) = x
    2x + 3y => -3x + 2y

So check it out:

    (2 + 3i)*i = 2i - 3

It's the same as rotation!

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