6

I tutor some high school students in Algebra 2 and Pre-Calculus. This is a questions from one of them. Here's the context:

Often, when students are first introduced to the concepts of domain and range, they are told that the domain is "all of the allowed inputs," where an input is allowed if it (a) is a real number and (b) produces a real number as an output.

In general, they are taught that there are two situations that break these rules by accepting a real number as an input but then giving back something that is ill-defined. The two situations are zeroes in the denominators of rational functions and negative numbers under even radicals.

Later, complex numbers will be introduced, and it will be explained that they solve various problems including the ill-defined-ness of the roots of negative numbers. However, the issue of dividing by zero is never resolved.

Are there any number systems which resolve the issue of dividing by zero similar to how complex numbers resolve taking the square root of negative numbers?

While it is not necessary, preferably this number system would be a natural extension of the real or complex numbers that avoids introducing new singularities. It seems to me that the answer to this question might depend on whether or not there are any fields where the additive identity has a multiplicative inverse.

Geoffrey
  • 2,382
  • 1
    Short answer. There's no way to extend the reals to include something like $1/0$ and still preserve the usual rules of arithmetic (field axioms). There are many answers on this site that make that point in one way or another. (Someone will flag your question as a duplicate.) Search. Try https://math.stackexchange.com/questions/26445/division-by-0 – Ethan Bolker Jan 02 '18 at 00:28
  • 1
    Sorry, @EthanBolker, but I think you are unduly pessimistic here. The Question narrows the "1/0" chestnut by asking for a number system in which singularities over rational functions can be eliminated, and mentions the complex numbers as a resolution for taking square roots. The complex numbers with infinity form a number system over which rational functions having singularities are tamed by having poles of integer orders. – hardmath Jan 02 '18 at 00:48
  • 1
    @hardmath Fair enough. But that might not really help the OP help high school students. And he does ask explicitly about a field with $1/0$. – Ethan Bolker Jan 02 '18 at 00:59
  • @EthanBolker I think that this is a valuable observation. I'm not an expert on fields and I am not really conversant in the subtleties of algebra with infinite elements, so I wanted to leave open the possibility that there might be a way to include multiple infinities and/or infinitesimals that sort out the issue. I also thought there could be a chance that some sort of p-adic-like construction might allow you to trade zero for infinity or to eliminate infinity without losing zero. – Geoffrey Jan 02 '18 at 02:06
  • @hardmath Obviously, the question is a bit naive - as evidenced by the fact that it ignores subtleties like branch cuts. The extended complex plane is possibly a good answer to this question. I don't fully understand how to handle $\infty$ as an algebraic element, so it is the type of answer I could probably accept if it were accompanied by an explanation for how it provides some amount of closure to the traditional problems of rational functions. For instance, whether it would be allowed as the root of a polynomial or the multiplicity of a root, or what constructions are still undefined, etc. – Geoffrey Jan 02 '18 at 02:11
  • There are fields with infinitesimals and numbers larger than all the integers. But still no $1/0$. Look up "nonstandard analysis" on this site and elsewhere. – Ethan Bolker Jan 02 '18 at 13:06

1 Answers1

2

If there was it would be very different from the real number system. We can show that the existence of $x$ such that $x=\frac{1}{0}$ leads to a contradicion. We know that the equation $0*1=0*2$ is true. If we multiply both sides by $x$ we obtain $\frac{0}{0}*1=\frac{0}{0}*2$. The $\frac{0}{0}$ terms can be cancelled from both sides to give $1=2$ which is a contradicition.

The "number system" I think you are looking for is the extended complex numbers which can be visualized with the Riemann Sphere.

caden-parajuli
  • 169
  • 10