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I typed a number divided by 0 on Wolfram Alpha and thought that it would say "undefined". However, when I pressed enter it told me that the answer is complex infinity.

Complex infinity

I have always been taught that it is undefined and do not understand what complex infinity is. Is complex infinity just another way of saying undefined or does it mean something else?

Help would be appreciated.

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Whether or not things are "undefined" largely depeneds on what framework you are working in.

If we are working in the naturals, we might say that $3-5$ is undefined.

There are many systems where it makes sense to assign $\frac{n}{0}$ some value. In this particular example, it is defined to be complex infinity, which can be thought of as follows: suppose we are looking at the complex plane. Similarly to how the complex number "0" is represented by a zero vector of arbitrary direction, we wish to associate all complex numbers of infinite absolute value (regardless of direction) to a single point.

This is complex infinity, and geometrically, by associating all complex numbers of infinite absolute value to be the same on the plane, we have formed a sphere, one with zero on the bottom, and complex infinity on the top. This is called the Riemann Sphere.

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  • $\begingroup$ Why is 0/0 not complex infinity? Wolfram Alpha says that it is indeterminate. $\endgroup$ – anonymous May 23 '15 at 1:18
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    $\begingroup$ Because we cannot "decide" any value for it. Suppose it is $x$, then we have $x=\frac{0}{0}$, so equivalently, $0x = 0$, but all $x$, where defined, have this property. $\endgroup$ – Jonathan Hebert May 23 '15 at 1:31
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    $\begingroup$ When my mathematical physics professor mentioned the Reimann Sphere, I thought he was mad. $\endgroup$ – Mateen Ulhaq May 15 '17 at 4:58
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Sometimes it is useful in complex analysis to consider the complex numbers plus the "point at infinity". See this wiki article for details: Riemann Sphere

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  • $\begingroup$ So are you saying that the top center point of this sphere is complex infinity? $\endgroup$ – anonymous May 23 '15 at 0:55
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    $\begingroup$ Yes, that is the usual mapping. You can imagine the sphere resting on top of the complex plane with its "bottom" matching the complex number zero. Then you project down stereographically. $\endgroup$ – muaddib May 23 '15 at 0:56
  • $\begingroup$ Then why is 0/0 not complex infinity? $\endgroup$ – anonymous May 23 '15 at 1:16
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    $\begingroup$ Well 0/0 is a pretty bizarre concept, so this is just a guess, but... choose any $z \neq 0$ in the complex numbers and $\epsilon$. Then as $\epsilon$ gets smaller $z/\epsilon$ moves off toward that infinity point. However, if $z = 0$ you could reason that since $0/\epsilon = 0$ that point goes nowhere, hence doesn't converge to infinity. $\endgroup$ – muaddib May 23 '15 at 1:26
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The idea consists of the following: consider the unit sphere $\mathbb{S}^2 \subset \Bbb R^3$, and look at $$\Bbb C \equiv\Bbb R^2 \cong \{(x,y,0)\mid x,y \in \Bbb R \}\subset \Bbb R^3$$

Consider the north pole $N= (0,0,1)$. Consider the stereographic projection from $\Bbb S^2 \setminus N$ to $\Bbb C$. Then you take a object $\infty \not\in \Bbb C$ and define the value of the projection at $N$ to be $\infty$.

Wolfram Alpha means this infinity, and not $+\infty$ or $-\infty$ that you meet in calculus. It is often useful to define in this context thing like $1/0 = \infty$ (just symbology, I am not dividing by zero) and $1/\infty = 0$.

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