Eli5: in complex numbers, what is the meaning of adding a “+b*i” part? It looks to me similar to ‘normal’ coordinates (+b*y), but with another name. I understand that the meaning of i as the square root of a negative makes it different but could never understand how so.

Question

We were told about complex numbers, and did the algebra by solving problems with the general form of “a*n + b*i”. It does not really behave differently than standard algebra, it seems. What am I missing in the intention or meaning of the “i” part?

In: Mathematics

Answer 1

This is a great starting point! It’s not that different from having two different spatial dimentions: (almost) all of the basic algebra still works. I believe this is often the starting point for teaching complex numbers to avoid confusion, ease students in: you start with what you are familiar with, go over the rules that still work as before.

But the biggest difference is when defining products. The product for (ax + by) *(cx+dy) is not really defined, if we consider x and y separate spatial dimensions. But (an + bi) *(cn+di) is defined. If you consider that n=1, the real unit, this simplifies to (a + bi) *(c+di) = ac +(ad+bc)i + bd*i^2, and since we have defined i as the square root of -1, then also i^2 = -1, so

= ac + (ad+bc)i -bd = ac-bd + (ad +bc)i

This is pretty much the only major difference from just having a separate spatial dimension. But this very basic rule of i^2 = -1 and products being defined adds a ton of useful tools to algebra. This is quite typical in mathemathics: you add a simple rule, which might seem almost non-consequential, but it interacts with the pre-existing system in various, interesting and useful ways.