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
Division by zero is undefined in a very different way from the square root of -1.
For the square root of -1 there is no real value it could be, so if you invent a value for it to be that’s the issue solved.
For dividing by zero there are too many values it could be. 0/0 could be literally any value from +infinity to -infinity (or even an imaginary or complex value). Creating a new value for it to be just makes the problem worse.
Mathematics doesn’t really have any global rules. Operations are defined in some contexts and not in others, depending on what is interesting or useful. For example, if you’re modelling a population of animals, you might well forbid negative or fractional numbers of animals, let alone complex numbers.
There are systems in which division by zero is defined, like the extended real numbers, but they aren’t really all that interesting, and it’s almost always more convenient to leave it undefined. The complex numbers have a very interesting structure that is convenient to use in many contexts.
Because i can imagine a number that isnt real, but if i divide something among 0 people, then no one gets anything, everyone involved gets 0. There simply isnt anywhere to put it. How much was being divided- who can say? no one was on the other side of the divisor to receive it.
In short: Division by zero is not useful.
When we work with complex numbers, they behave in a predictable and consistent manner. They might be as “imaginary” as division by zero is, but the difference is that if you accept imaginary numbers, then you have a functional framework you can use to model all kinds of things. When we use complex numbers to do, say, signal processing, the math works out intuitively and the end result corresponds to what we see in reality.
If you define division by zero as something, then that framework falls apart. Since multiplication is the inverse of division, shoehorning in zero in the wrong spot means that you have to start explaining how x*0 could be equal to something other than zero.
I’m sure other people will provide a dozen and a half explanations for why division by zero results in weird things, but the only thing keeping us from just accepting the weirdness just like we accept the concept of “negative area” with complex numbers is that going down that rabbit hole doesn’t yield anything really useful to us.
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
It’s not necessarily that it’s prohibited. You *could* define an imaginary number to the result of dividing by zero. It’s just that it’s not useful so what is the point?
There is a widely-used mathematical domain where division by zero is allowed, including 0/0.
It’s called [IEEE 754](https://en.wikipedia.org/wiki/IEEE_754).
In this domain you have a finite set of rationals, ±∞, ±0, and NaN.
Any non-zero divided by zero gives one of the infinities, and zero divided by zero gives NaN. Any operation involving NaN also gives NaN, and NaN does not equal itself.
This can result in some very non-intuitive behaviour, but is the system underpinning vast amounts of computing.
If you and I share a pie, how do we divide it? 1 / 2 = 1/2, or .5
If you have a pie to yourself, how should you divide it? 1/1=1, you don’t have to divide it
If nobody has a pie, how should it be divided? Or 1/0=… Divide amongst whom? Why? You can see it’s a nonsense question. It can’t be answered in a meaningful way for most purposes
Imaginary numbers answer meaningful questions with useful results in predictable ways. When you pose division by zero as a word problem it becomes evident that you’re not really asking anything applicable to normal situations. Eg: it’s nonsense. Which we more charitably call undefined
Think of division the other way around:
If you do 12 ÷ 4, you’re asking “how many 4’s do I need to add together to get 12?”. There is only one answer.
But if you ask “12 ÷ 0”, you’re asking “how many 0’s do I need to add together to get 12?”
Well, if you add one 0, you get 0. If you add two, you get 0 still. If you add one million, you still get 0. So there is NO number of 0’s that you can add together to get 12. There is no answer. Hence 12÷0 is “impossible”.
Even if you “0 ÷ 0”. How many 0’s do I need to add together to get 0?
Well, one. Or two. Or zero. Or a million. Any number at all, whatsoever in fact. So there’s no one answer that’s right. Literally every answer is right.
Imaginary numbers are really just numbers “in another dimension”, if you think that way, which we deal with all the time – imaginary numbers crop up in nature all the time – physics, AC electrics, all kinds of unexpected places. They are logical, consistent, you can bring them back into the domain of so-called “real” numbers, and so on. As such, mathematicians like them.
But division by zero gives you either NO ANSWER AT ALL or EVERY ANSWER AT ONCE. It’s practically useless. Hence we just don’t define it.
ELI5: think about your question.
What’s 5/5? Five things divided into 5 groups. How many items per group? That’s 1.
What’s 5/1? Five things divided into 1 group. How many items in that group? That’s 5.
What’s 5/0? Five things divided into 0 groups. Or, five things that aren’t counted in a group at all. It’s not zero, the group isn’t a thing.
It’s not that we need to invent a thing, your question is designed to be unanswerable because of how it’s constructed.
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