Complex infinity ($1/0$)

Question

I've learned that $$1/0$$ is postive and negative infinity, but if I ask wolfram mathematica to calculate $$1/0$$ it gives me: 'complex infinity' but how can we proof that that is true?

Answer 1

To say that $1/0$ is "positive and negative infinity" is at best misleading without further context. One may say that $$ \lim_{x\downarrow 0} \frac 1 x = +\infty\quad\text{and}\quad\lim_{x\uparrow\infty} \frac 1 x = -\infty. $$ And that was probably what was meant if you read that $1/0$ is "positive and negative infinity".

However, there is also an "$\infty$" that is neither $+\infty$ nor $-\infty$ that is approached by the reciprocal of anything that approaches $0$ in the complex plane. It need not even approach from any particular direction; for example, it could spiral infinitely many times around $0$ while its distance from $0$ approaches $0$. That is "complex infinity". And it can be regarded as the value of a rational function at any point at which it has a vertical asymptote or "pole", and likewise with things like the tangent function. That makes rational functions continuous at every point in $\mathbb C\cup\{\infty\}$ and things like tangent and secant continuous at every point in $\mathbb C$.

Answer 2

This comes from thinking of the one-point compactification of the complex plane, called the Riemann sphere, which is basically $\mathbb{C}$ with a point called "$\infty$" added to it, and a topology so that sequences $(z_n)$ such that $|z_n|$ eventually becomes larger than any positive number you choose (i.e. the usual definition of limit) are said to "tend to $\infty$". Then the function $$ T(z) = \frac{1}{z} $$ is a bijection from the set of nonzero complex numbers $\mathbb{C}\setminus \{0\}$ to itself. You can then say that $1/0 = \infty$ by noticing that, for example, the sequence $ 1,1/2,1/3,\dotsc $ maps to the sequence $1,2,3,\dotsc$, which satisfies the property of "tending to $\infty$" that we defined above. Similarly, $1/\infty = 0$ in this context. This is what Mathematica means. (You will find that it also says 1/ComplexInfinity evaluates to $0$.)

Important: This is one way of defining limits at infinity. Real analysis has a different version, in which there are two points added to $\mathbb{R}$, $\infty$ and $-\infty$, basically to preserve the ordering of $\mathbb{R}$. This does not tell you how to evaluate $0/0$.

Further, there are other sorts of infinity. Mathematica has something called DirectedInfinity, which is a generalisation of the $\pm\infty$ used in real analysis. Mathematica can't deal with the types of infinity you find when you look to generalise ordered sets of integers (which leads to ordinals), or generalise the size of sets to infinite sets (which leads to cardinals).