Complex numbers appeared initially as a way to solve equations that don't have real solutions. However, what are we getting from this? We can write that i is the solution to x^2=-1, but what exactly is this result? It is an imaginary number, so it is not something that can represent for example area. The obvious result from this line of thinking is - are complex numbers really needed? What problems do they solve?

Somewhat surprisingly complex numbers solve a lot of different problems or at least make them easier. In this post I want to introduce one particular problem that is solved by using complex numbers. The problem is to define a^b for all values of b and for all values of a except zero. We can partially solve this problem without complex numbers. For example, we can agree that the functions log and exp are defined as usual and then:

As a start, lets try to define exp(z) and log(z):

exp(iy)=cos(y)+isin(y)

exp(x+a)exp(iy+ib)=exp(x)exp(a)[cos(y+b)+isin(y+b)]

cos(y+b)+isin(y+b)=cos(y)cos(b)-sin(y)sin(b)+isin(y)cos(b)+icos(y)sin(b)

cos(y+b)+isin(y+b)=[cos(y)+isin(y)][cos(b)+isin(b)]

exp(z+w)=exp(x)exp(a)exp(iy)exp(ib)

exp(z+w)=exp(x)exp(iy)exp(a)exp(ib)

exp(z+w)=exp(x+iy)exp(a+ib)=exp(z)exp(w)

As you can see it is rather easy to extend exp to the complex numbers. Extending log is more difficult. Lets start with what we want the function to do. We want that for any complex number z:

Now that we do not have these numbers we can define log. We know that any complex number z can be written as z=r*exp(it), where t represents the angle between the line that connects zero and z. Obviously exp(it)=exp(it+2(pi)i). However, since we have some numbers removed we cannot go in a circle around zero. Therefore, for numbers that have a negative real or complex part we will take t to be 2pi minus the angle. This solves our little problem because we now have exactly one value of t assigned to every z. And now we can define log exactly as we tried to do before. Since the negative numbers were not removed we have log defined for negative numbers as well. For example: log(-1)=i*pi.

Now that we have this we can indeed define that for all real a,b such that a is not zero:

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