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.
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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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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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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It's because it is both positive and negative infinity at the same time, because it doesn't know what side you are taking the limit from. $\frac{1}{0}$ on its own could be either $\lim{x\to 0^-} ~\frac 1 x$ or $\lim{x\to 0^+} ~\frac 1 x$ i.e. it is not a directed infinity. Because of this, it uses the only mathematical symbol available to represent being able to be either positive or negative infinity, $\tilde{\infty}$, which is mathematically correct as a real number can be modelled as a complex number with only a real part. It could have used $\pm\infty$ in the extended real number system but it's not as nice as the concept being unified into a superset concept denoted by a single symbol. The reason it is called a directed infinity intuitively in the complex plane is because there is a complex argument and therefore an angle and a bearing if the sign of infinity is known because it is either moving at 0 degrees or 180 degrees but without that, it has no direction and therefore no defined angle. On the Riemann Sphere however, all infinities have a single point, $\infty$. The Riemann Sphere is a one-dimensional complex manifold which represents the extended complex plane $\overline{\mathbb{C}}$. It's best to imagine that the complex plane has been pressed over the top of a sphere like a sheet around a ball with the origin of the sheet at the pole. All directional infinities converge downwards round the ball to one infinity at the other pole (all complex numbers that have infinite magnitudes), and when the direction is unknown, the fact that all directions lead to that infinity leads to the conclusion that it also must be that infinity, but the state of an infinite complex magnitude having more than one direction to the one point infinity is called complex infinity. $i\times\infty$ comes out as $i\infty$ and not complex infinity because it has a complex argument of $\frac{\pi}{2}$ radians.
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.