Why can’t dividing by 0 be done in a theoretical field?

Question

As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?

So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?

In: 27

Answer 1

You can always find a way to define what divide by 0 looks like.

Mathematicians don’t like being told they cannot do something; if they find something that doesn’t have a rule for it they try to come up with one, and *i^2 = -1* is a great example of this. There is no (existing) number that squares to negative one, so we will define one (or two, depending on how you look at it) and see what happens.

But for these definitions and rules to stick they really need to be consistent and, if possible, useful. The *i^2 = -1* does this. You define this new concept, but it still follows all the existing rules of algebra and number theory, and has some useful results.

Dividing by 0 doesn’t work that way. So far no one has been able to come up with a rule for dividing by 0 that is consistent with all our other rules for algebra and numbers (outside the special case of 0/0, which we tackle with limits if we are careful), never mind one that is useful.

The closest we get is the concept of infinity, but that doesn’t work as a number, it doesn’t work with algebra or geometry(ish – there are some exceptions – and this is probably where infinity exists in the most divide-by-zero way), doesn’t even work with infinitesimals and limits. Analysis uses infinity a bit, although not in the divide-by-zero sense, but it isn’t really until we get into set theory that infinity starts to become interesting and useful. But again, not really in the divide-by-zero sense.

You could always start a paper by stating “we will define 1/0 = banana”, but that won’t help you with anything.

There isn’t a way to divide by 0 in a way that is consistent with the rest of maths, or useful.