I was working on some basic complex numbers and suddenly have a question about it. As you all know, the complex numbers are created as we cannot find a place in the real number system to fit in the value (-1)^(1/2).
So we use i to represent it, but how do we know that any number to the power of any other number (including fraction, negative and others) are able to be expressed in a+bi? Do we need another number system for like (-1)^(1/4)? Although that is not a very good example, because we are able to show that (1/2 + i/2)^4 = -1 after some working.
After doing some researches on this topic, I think I come to the conclusion that we don't need another complex system but I don't know how to prove it. Can anyone tell me how?
I really appreciate any help on this.
any complex number can be written in the form $$a+ib\to r(\cos\theta+i\sin\theta)=re^{i\theta}$$ if you raise the term to $n$ being any real or complex number. $$(a+ib)^n=r^ne^{in\theta}$$which is once more a complex no.
as per your need i will eloborate for a complex $n=c+id$ $$r^ne^{in\theta}=r^{c+id}e^{i(c+id)\theta}$$ $$ =r^c.r^{id}e^{-d\theta}e^{ic\theta}....(1)$$ now let $$r^{id}=e^{i\alpha}$$ $$i\alpha=id\ln r$$ therefore $$ r^{id}=e^{id\ln r}$$ substituting in $(1)$ $$(r^c.e^{-d\theta})e^{i(d\ln r+c\theta)}=(a+ib)^{c+id}$$ which is $$(r^c.e^{-d\theta})(\cos ( d\ln r+c\theta )+i\sin ( d\ln r+c\theta ))$$ which is finally of the form$$r'(a'+ib')$$