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Even simpler, complex numbers are really 2D vectors with addition and multiplication defined: a field. There's nothing "imaginary" about that second dimension,

Even simpler, complex numbers are really 2D vectors with addition and multiplication defined: a field. There's nothing "imaginary" about that second dimension, very frustrating to see them defined that way because it makes people think of it as an "escape hatch" out of real numbers. When you're working with complex numbers, you are working with a different system: `5 + 0i` is still a complex number because it's really `(5, 0)`.

rrauenza 1y ago
Today I learned that complex numbers can be represented by matrices... thanks! https://www.youtube.com/watch?v=HbUewIIpl6I https://www.youtube.com/watch?v=HbUewIIpl6I
AnotherGoodName 1y ago
Yeah https://xkcd.com/2028/ https://xkcd.com/2028/ hit the nail on the head on this.
woopsn 1y ago
But working with complex numbers I hardly if ever write (a, b) for a+ib, while I use the "escape hatches" all the time. They solve equations that have no real solution, they give me paths from x=-1 to x=1 that don't cross the origin, etc. There's only so much to learn about C as a vector space, while the theory tying it to R (and even N) is very deep.
- phailhaus 1y ago
  Thing is, there's no such thing as an escape hatch. Either you are working in the reals, or you are working in the complex plane. They don't "solve equations that have no real solution", that equation is either a real number equation or a complex number equation, not both. If you work in the complex plane, that is a different equation describing a different space! It just looks the same in standard notation. If you don't realize this, then you can draw conclusions that don't make sense in the space you're working with. Take a simple equation like y = -x^2 - 5, representing a thrown ball's trajectory. It never crosses zero, there are no solutions. You can't "pop into the complex numbers and find a solution" because the thing it represents is confined to the reals. So if you find yourself reaching for complex numbers, you have to understand that the thing you are working with is no longer one-dimensional, even if that second dimension collapses back to 0 at the end.

rrauenza 1y ago

Today I learned that complex numbers can be represented by matrices... thanks! https://www.youtube.com/watch?v=HbUewIIpl6I https://www.youtube.com/watch?v=HbUewIIpl6I

AnotherGoodName 1y ago

Yeah https://xkcd.com/2028/ https://xkcd.com/2028/ hit the nail on the head on this.

woopsn 1y ago

But working with complex numbers I hardly if ever write (a, b) for a+ib, while I use the "escape hatches" all the time. They solve equations that have no real solution, they give me paths from x=-1 to x=1 that don't cross the origin, etc. There's only so much to learn about C as a vector space, while the theory tying it to R (and even N) is very deep.

phailhaus 1y ago
Thing is, there's no such thing as an escape hatch. Either you are working in the reals, or you are working in the complex plane. They don't "solve equations that have no real solution", that equation is either a real number equation or a complex number equation, not both. If you work in the complex plane, that is a different equation describing a different space! It just looks the same in standard notation. If you don't realize this, then you can draw conclusions that don't make sense in the space you're working with. Take a simple equation like y = -x^2 - 5, representing a thrown ball's trajectory. It never crosses zero, there are no solutions. You can't "pop into the complex numbers and find a solution" because the thing it represents is confined to the reals. So if you find yourself reaching for complex numbers, you have to understand that the thing you are working with is no longer one-dimensional, even if that second dimension collapses back to 0 at the end.