the point at infinity
In ordinary arithmetic, 1/z gets larger and larger as z approaches 0, and there is no number it 'arrives at' — division by zero is undefined and the value runs off to infinity. The bold and tidy fix of complex analysis is to add a single brand-new point, called the point at infinity and written with the symbol for infinity, and declare that as |z| grows without bound, z heads toward this one point. The plane plus this extra point is the extended complex plane, often written C-hat.
Crucially there is just ONE point at infinity, not a different infinity for each direction — unlike the real line with its separate plus and minus infinity. No matter which way you travel outward in the plane (north, southwest, spiralling), you approach the same single point. With it, formerly-forbidden expressions get clean meanings: 1/0 = infinity and 1/infinity = 0, which makes the map z -> 1/z a perfect, glitch-free swap of 0 and infinity over the whole extended plane.
Why go to the trouble? Because adding infinity makes the whole theory rounder and more symmetric. Mobius transformations become genuine one-to-one maps of the extended plane onto itself; rational functions become well-defined everywhere; and 'behaviour near infinity' becomes something you can study exactly like behaviour near any ordinary point (by substituting w = 1/z and looking near w = 0). The natural home for this one-point completion is a sphere — the Riemann sphere.
The map z -> 1/z sends 0 to infinity and infinity to 0, and is a clean one-to-one map of the whole extended plane to itself. Without the point at infinity it would be undefined at z = 0; adding the single point repairs that gap.
One extra point makes 1/z a perfect swap of 0 and infinity.
Infinity is a single added point, not a number you can do free arithmetic with: infinity + infinity and infinity times 0 stay undefined. It earns clean rules only for the specific operations (like 1/0 = infinity) where the limit genuinely exists.