The Extended Complex Plane

Complex Analysis · Axiom Academy

LESSON The Extended Complex Plane Adding a single point at infinity to close the plane into a sphere. 1. One Point at Infinity — Not Two On the real line you can run off in only two directions, so you need two ends, and . The complex plane is different: "going to infinity" just means the distance |z| grows without bound, and you can leave the origin along any ray. Every one of those escape directions is identified as the same single point . Once is a genuine point, we extend the four operations so that division by 0 no longer "breaks." For every finite (and where noted): — any finite shift leaves you at infinity. — scaling by a nonzero finite number stays at infinity. — a finite amount, spread over an infinite one, vanishes. — the operation that was undefined in now has a home. The reason one point at infinity is exactly right: it makes w = 1/z a bijection of onto itself . Filling the two holes the reciprocal had on takes just the two new rules — and they trade 0 and for each other. 0 maps to , and maps back to 0 Push z toward 0 : its reciprocal 1/z races off to — and it does so no matter which direction you approach 0 from. Send z out to : its reciprocal 1/z settles down onto 0 . The map folds the inside out and the outside in, fixing only the unit circle |z| = 1 . The closer z sits to the origin, the larger |1/z| grows — without bound. From every direction of escape, 1/z lands at the single point 0 .

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