A short note on the Riemann sphere and some of its properties. 

Bernhard Riemann's great contribution to mathematics led to the discovery of the Riemann sphere, the Riemann sphere is an extraordinary sphere that has a wide range of application in mathematics. 

What is the Riemann sphere?
The Riemann sphere also referred to as the set of extended complex number or extended complex plane can be defined as a plane that contains the complex numbers $C$ and an infinity $\infty$. The set of the extended complex numbers is usually denoted as $C\bigcup\{\infty\}$ or $\mathbb{C}_\infty$. 
With the Riemann model, the point $\infty$ is near to very large numbers, just as the point $0$ is near to very small numbers.
The extended complex plane is also referred to as the closed complex plane.  


Source: Wikipedia


One of the importance of the extended complex numbers is its usefulness in complex analysis because they allow division of numbers by zero in certain circumstances, in a way that makes expressions such as $\frac{1}{0}=\infty$ well-behaved. 
In geometry, the Riemann sphere can be thought of as a Riemann surface while in projective geometry the sphere can be thought of as a complex projective line. 
Addition and multiplication of complex numbers can be extended by defining $z\in{C}$ such that $z+\infty=\infty$ and $z\times\infty=\infty$ but $\infty-\infty$ and $0\times\infty$ are undefined. 
Unlike the complex numbers, the extended complex numbers do form a field since $\infty$ does not have a multiplicative inverse. 
In division, it is good we know that $\frac{z}{0}=\infty$ and $\frac{z}{\infty}=0$ $\forall{z>0}$ but $\frac{0}{0}$ and $\frac{\infty}{\infty}$ are all undefined. 
When discussing the three dimensional real space $\mathbb{R}^3$, the Riemann sphere can be defined as the unit sphere $x^2+y^2+z^2=1$.