TITLE: What is the real and imaginary part of complex infinity?
QUESTION [5 upvotes]: I know that complex infinity is a pole on the Riemann-sphere, but what is actually the real and imaginary part of it? Is it $\pm\infty \pm \infty i$? And how is it different from simple non-complex infinity?

REPLY [5 votes]: According to the definition from Wolfram MathWorld "Complex infinity is an infinite number in the complex plane whose complex argument is unknown or undefined".
Basically, it's a number in the complex plane whose magnitude is infinite, but whose phase is unknown. Unlike with a 1-dimensional number where we have only one direction to go to positive or negative infinity, we can approach complex infinity from an infinite number of directions depending on the phase.