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

You can allow for division by zero. On the extended complex plane often described by the [https://en.wikipedia.org/wiki/Riemann_sphere](https://en.wikipedia.org/wiki/Riemann_sphere) it is alowed.

The rule is
z/0 = ∞ and z/∞ = 0 for all no zero complex numbers.

∞/0 = ∞ and 0/∞ = ∞ but 0/0 and ∞/∞ are still not allowed.

That 0/0 is not allowed to fix the problem that
is commonly used to show division y zero is not allowed. For example, https://en.wikipedia.org/wiki/Mathematical_fallacy#Division_by_zero steps 4 to 5 do 0/0 is still not allowed.

Do not use them if you do not know what mathematical properties you lose and other changes they result in.

An example of what you lose with complex numbers is the absolute order of numbers. If we have the numbers -1, 0, 1 and 5 that is the the order in size. If x is an integer and larger the 0 but smaller than 2 it has to be 1

But how do you order -1, 1, i, -i in order of size? The answer is you can’t do that because with the complex number the best you can do is the norm, that is the distance from 0. All of these numbers have a distance to zero of 1, they are all on a circle with a radius 1. So complex numbers do not have an absolute order -1, 1, i, -i all have the same norm,

The https://en.wikipedia.org/wiki/Complex_logarithm is another large difference with multiple branches. Because hos exponential function relate to trigonometrical and power function is also applies to https://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers

For some maths like calculating residual and complex maths with zeros and poles it is something very practical to do, if you know the limitations.

So it is possible to define maths that allows division by zero except for 0/0 but do not try to use it before you learn enough of the potential