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

>What am I missing in the intention or meaning of the “i” part?

**You are missing the elegancy, and that’s it.**

Hamilton’s approach was exactly like you thought – 2D reals equipped with special algebra. This is in principle the same as the complex number, in the sense that the special algebraic structure R^2 is homomorphic to C.

Historically speaking, the imaginary unit was introduced to complete closed polynomial roots. In other words, given any polynomial equation, you wanted to find a number system that is closed for all the roots.

For example, the roots of x^2 +1=0 is non-existence in ‘conventional’ number system (ie, the reals.) To workaround this issue, you need to add some auxiliary part. An elegant way to do that is by defining the *i* (this was Euler’s idea).