the zero of imaginary numbers.

Question

So generally 2 the power of 0 equals 1. same goes for 3 and 4 and so on. You know what I mean. So what is zero in the imaginary dimesion? i is the first of the imaginry numbers, like 1 is in the natural numbers. so what  is the zero of imginary numbers that when its a power will produce i. like 3i to the power of this imaginary zero will give i. i would really like to know (if its even defined) thx

Answer 1

Recall that you can express any complex number z in the form

 

z = re^iφ

 

r = modulus, a real number

φ = phase, a real number

 

Then note that

 

z⁰ = (re^iφ)⁰ = r⁰e^(0)iφ =  r⁰e⁰ = 1, as Ravi V. pointed out.

Answer 2

Complex number to the power of 0 is also 1. Any complex number times 1 will give you the original complex number. Don't know if this is what you are looking for.