What is infinity in complex plane and what are operation with infinity extended to complex numbers?

Question

For a real number $a$, $$\infty + a = \infty,$$ and if $a$ is positive, $$\infty \cdot a = \infty$$

What is $\infty + a$ and $\infty*a$ if $a$ is non-zero complex number, where $\infty$ is real positive infinity?

Answer 1

I agree with Daniel Fischer's comment

For complex numbers, one doesn't consider "real positive infinity". One considers only "complex infinity", and there's only one of that, you have $\infty +a=\infty$ for all $a\in\mathbb C$, and $\infty \cdot a = \infty$ for all $a\in\mathbb C\setminus \{0\}$.

but will remark that it's sometimes convenient to describe half-lines in $\mathbb C$ borrowing notation from real axis: for example, $\mathbb C\setminus [0,i\,\infty)$ is the plane slit along the positive part of the imaginary axis. More exotic examples are possible: $[1+i,(3-2i)\,\infty)$ and so on. In this context, we are not really performing arithmetic operations with the object $\infty$, but rather use it as a shorthand for $\{1+i + (3-2i)t : 0 \le t<\infty\}$.

To some extent, $+\infty$ and $-\infty$ also play this role on the real axis: they are not a destination, they are road signs that tell us to go in a certain direction and never stop.