Editing Division by zero (section)

== Alternative number systems ==

===Extended real line===
The [[affinely extended real numbers]] are obtained from the [[real number]]s <math>\R</math> by adding two new numbers <math>+\infty</math> and <math>-\infty,</math> read as "positive infinity" and "negative infinity" respectively, and representing [[point at infinity|points at infinity]]. With the addition of <math>\pm \infty,</math> the concept of a "limit at infinity" can be made to work like a finite limit. When dealing with both positive and negative extended real numbers, the expression <math>1/0</math> is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define <math>1/0 = +\infty</math>.

===Projectively extended real line===
The set <math>\mathbb{R}\cup\{\infty\}</math> is the [[projectively extended real line]], which is a [[one-point compactification]] of the real line. Here <math>\infty</math> means an unsigned infinity or [[point at infinity]], an infinite quantity that is neither positive nor negative. This quantity satisfies <math>-\infty = \infty</math>, which is necessary in this context. In this structure, <math>\frac{a}{0} = \infty</math> can be defined for nonzero {{math|''a''}}, and <math>\frac{a}{\infty} = 0</math> when {{math|''a''}} is not <math>\infty</math>. It is the natural way to view the range of the [[tangent function]] and cotangent functions of [[trigonometry]]: {{math|tan(''x'')}} approaches the single point at infinity as {{math|''x''}} approaches either {{math|+{{sfrac|π|2}}}} or {{math|−{{sfrac|π|2}}}} from either direction.

This definition leads to many interesting results. However, the resulting algebraic structure is not a [[Field (mathematics)|field]], and should not be expected to behave like one. For example, <math>\infty+\infty</math> is undefined in this extension of the real line.

===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.