Ive always thought that infinity isn’t really a number. It is just an idea – a name we attach to something that grows without bound.
So in real analysis, when the terms of a sequence or partial sums of a sequence (series) keep increasing without an upper bound, we say the sequence or the series goes to infinity. Negative infinity is the same idea, but with a minus sign, that is negative terms, which keep decreasing without any lower bound go to −∞. And this “infinity” object is bigger than any number you could possibly think of (because it isn’t a number in itself). All fine.
But attached with this idea of bigger than everything else, is the notion of big or small, i.e. order. However for complex numbers, there is no total order. We just can’t compare any 2 given complex numbers and say which is “bigger”. How, then is infinity thought of in complex analysis? It can’t be an element that is bigger than all other elements, because “bigger” doesn’t make any sense.
And since there is only one infinity in the complex plane, unlike +∞ and −∞ in R, does that mean that the complex infinity is more of a scalar (with only a modulus and no direction or argument) than a vector (like we can associate a vector with all other complex numbers)?
I know I sound very confused. I am. Please shed some light.
∞ is the compactification point of the complex plane. In this context, ∞ is very real: it is the north pole of the Riemann sphere. 0 is the south pole.
In fact, 0 and ∞ are closely related:
they both only have a modulus and no direction or argument.
the function z↦1z exchanges them.
they 0 and ∞ are absorbing elements for multiplication: 0⋅z=0, ∞⋅z=∞, for z≠0,∞.