Imaginary numbers

Question

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

In: 16

Answer 1

Imaginary can be thought of as:

– Rotation and scaling by 90 degree. Multiplication correspond to combining 2 rotations.

– Rotation and scaling by 180 degree, taking into account the difference between long rotation versus short rotation. Multiplication correspond to combining 2 rotations.

– Signed area. If you remember that determinant of a 2×2 matrix give you a signed area of the parallelogram spanned by the 2 vectors, then imaginary number represent signed area itself. Multiplication correspond to cancel out the height in one direction to obtain the base of a parallelogram.

However these method of thinking are not useful 99% the time when you use complex numbers. It’s there to assure you that complex number directly represent a quantity, and it can be used to generalize complex number into more general settings.

But that’s not *why* we use imaginary number. Imaginary number, or actually, complex number as a whole, makes the math a lot, a lot more convenient.

The conceptual way of thinking of complex numbers is that it’s the *completion* of natural numbers at both the algebraic and analytic level. In simple terms, *anything that is not logically impossible to exist already existed*. More accurately, we have 2 claims. Algebra: if you have a system of polynomial equations in complex numbers, then either: (a) you can prove the system is inconsistent by manipulating it; (b) you can find a root of the system. Analysis: if you have a sequence of complex numbers, then either: (a) there is a positive distance such that the number can’t come closer than this distance and remain there; or (b) there is a complex number that these numbers approach arbitrarily close.

It’s very good for math to run into a situation where a logical possibility is always actualized: you either can disprove something exist that satisfy some conditions, or there is something actually exist that satisfy that condition. Without such object existing, mathematicians would have to abstractly talk about a logical possibility and that’s a lot more inconvenient.