I am going into Honors Algebra II and while I am fine at using imaginary numbers in a formulaic sense I never understood them conceptually. I can’t tell if they exist just to make math work better or because there is an actual logical way to understand them
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They’re good for solving differential equations, especially in quantum mechanics, and there are lots of theorems that are easier to prove if you use complex numbers.
In everyday math, complex numbers aren’t really necessary, and answers involving complex numbers would be unphysical or nonsensical. For example, if you’re given the path of two objects and asked where they intersect, it would be far better to simply say “they don’t intersect” (assuming that was the right answer) instead of giving a weird answer involving imaginary numbers. Any variable that can be measured in the real world will not have an imaginary component.
I tend to look at them as another batch of numbers, orthogonal to the standard set of real numbers we have. So to map one to the other we multiply by i. Multiply by i again and we map them back to the real numbers (negated, because we’re essentially rotating 90 degrees each time)
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