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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'

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.

starwind 5y ago
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
ThrustVectoring 5y ago
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.
Epa095 5y ago
With this intuition, why is it i*i=-1 and not i+i (90+90=180 after all)?
- rthomas6 5y ago
  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!

starwind 5y ago

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

ThrustVectoring 5y ago

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.

Epa095 5y ago

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

rthomas6 5y ago
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!