Why is it mathematically consistent to allow imaginary numbers but prohibit division by zero?

Question

Couldn’t the result of division by zero be “defined”, just like the square root of -1?

Edit: Wow, thanks for all the great answers! This thread was really interesting and I learned a lot from you all. While there were many excellent answers, the ones that mentioned Riemann Sphere were exactly what I was looking for:

https://en.wikipedia.org/wiki/Riemann\_sphere

TIL: There are many excellent mathematicians on Reddit!

In: 1691

Answer 1

>Couldn’t the result of division by zero be “defined”, just like the square root of -1?

No. The very cornerstone of the numbers we use is that a number cannot equal its own successor: that 0 does not equal 1, 1 does not equal 2, 2 does not equal 3, etc forever. Two very important properties of number spring up from this idea: the Zero Property, and the Multiplicative Identity.

The Zero Property says zero times anything is zero. The Multiplicative Identity says for all numbers that are not zero, there is a number you can multiply with it to get one. These properties are mutually exclusive. We can show these properties are true in proofs *using only the above concept of successors* (numbers not equaling the number that comes after them).

The Multiplicative Identity is how we define division. That for any number that is multiplied with its prime version, so that their product is one, that prime version is a divisor. Since this exists for every number except zero, division is defined for every number except zero. We could say, for example, 2 times the multiplicative inverse of 3, is 2 divided by 3.

If we were to define a number that multiplies with zero to get one, to create a situation where zero is a valid divisor, then we would have to say that number times 0, ***equals both zero and one***, so zero equals one. That’s a violation of the premise of successors that our number system is based on.

That’s why division by zero cannot just be defined.

For comparison, the square root of one is an operation defined on all inputs. You can put any number into it and get an output with only the catch of the even root of -1 being not yet defined, but being consistent in value. Meaning, we can have an input of negative one without undermining any other rules of numbers, and as such we can create and define an output for it.