It is my understanding that a group must satisfy the requirements of
- Closure
- Associativity
- Invertibility
- Identity.
so….
- If you multiply two complex numbers, you get a complex number back.
- The order in which you do so does not matter.
- $z^{-1} = z^*/(zz^*)$
- But what is the identity element in the multiplicative group of complex numbers? Doesn't this mean that some complex number should exist, whereby, multiplication by it returns the same number? Of course the number $1$ does this, but $1$ is a real number. Or do we just consider $1 + 0i$ to be the identity element?
Best Answer
The identity of $(\mathbb C\setminus\{0\},\times)$ is $1$. Yes, $1$ is a real number, but all real numbers are complex numbers too.