The Neighborhood of Infinity

Did you know that:

1) Although the more commonly used geometric representation of complex numbers is the complex plane (developed by Gauss (1799) and Argand (1806)), one can also use the sphere (Riemann (19th century)) to obtain what is known as a stereographic projection (see figures below). Here points, P, of the sphere are projected from the North Pole, N, onto the tangent plane at the South Pole as follows: The line passing through N and P intersects the tangent plane at the point P'. This P' represents the sphere point P in the plane. Observe that with the exception of the North Pole itself, each point of the sphere corresponds to exactly one point in the plane. (Wait...there's coolness coming...keep reading...)

2. While we need two symbols '+infinity' and '-infinity' when we extend the real line R to infinity (the extended real line is called R*), we need only one symbol 'infinity' when extending the complex plane C to C*. This results form the fact that there is no ordering relation '<' in complex numbers.

3. Definition: Each set in C of the form {z : |z| > r >= 0} is called a neighborhood of 'infinity', or a ball with center at 'infinity'.

4. Now consider the stereographic projection described above. If we remove the apparent exception at the North Pole by regarding it as the geometric representation of 'infinity', we get a one-to-one correspondence between the extended complex plane C* and the total surface of the sphere. If the South Pole is the origin of the complex plane, the exterior of a "large" circle in the plane (|z| > r) will correspond to a "small" spherical cap about the North Pole on the sphere (neighborhood of 'infinity').

How cool is that for a geometric illustration of why we have defined a neighborhood of 'infinity' by an inequality of the form |z| > r !

Enjoy 😉!

Sources: Mathematical Analysis by Tom M. Apostol and the Internet.






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