Editing Division by zero (section)

===Riemann sphere===
The subject of [[complex analysis]] applies the concepts of calculus in the [[complex numbers]]. Of major importance in this subject is the [[extended complex numbers]] <math>\C \cup\{\infty\},</math> the set of complex numbers with a single additional number appended, usually denoted by the [[infinity symbol]] <math>\infty</math> and representing a [[point at infinity]], which is defined to be contained in every [[Domain (mathematical analysis)|exterior domain]], making those its [[topology|topological]] [[neighborhood (topology)|neighborhoods]].

This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point <math>\infty,</math> a [[one-point compactification]], making the extended complex numbers topologically equivalent to a [[sphere]]. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inverse [[stereographic projection]], with the resulting [[spherical distance]] applied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and conversely complex arithmetic can be interpreted in terms of spherical geometry. As a consequence, the set of extended complex numbers is often called the [[Riemann sphere]]. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for example <math>\hat\C = \C \cup\{\infty\}.</math>

In the extended complex numbers, for any nonzero complex number <math>z,</math> ordinary complex arithmetic is extended by the additional rules <math>\tfrac{z}{0}=\infty,</math> <math>\tfrac{z}{\infty} = 0,</math> <math>\infty + 0 = \infty,</math> <math>\infty + z = \infty,</math> <math>\infty \cdot z = \infty.</math> However, <math>\tfrac{0}{0}</math>, <math>\tfrac{\infty}{\infty}</math>, and <math>0\cdot\infty</math> are left undefined.