eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?
In: 277
The elementary branch of mathematics dealing with infinities would likely be Set Theory. Putting a symbol doesn’t mean anything until there are properties and rules defined how to use it and the kind of problems that it helps solve or areas of exploration it helps to illuminate.
At this point x/0 is just not useful. There are already established approaches to infinities and x/0 is a fairly narrow, non useful construct.
Let’s try it. Let’s define a symbol (let’s call it, I dunno, Q) and say that x / 0 = Q for all real numbers x.
Now, when we write a / b = c, we mean a = bc. For example, 12 / 4 = 3 because 3*4 = 12. So since, say, 3 / 0 = Q, we would need 3 = 0*Q. And since 5 / 0 = Q, we would need 5 = 0*Q. So 0*Q equals both 3 and 5, and in fact every other real number. We can, therefore, prove that all real numbers are equal.
Needless to say, this is not particularly useful math.
It turns out that adding *i* does not have this sort of consequence. Nothing about complex numbers disrupts, in any way, the arithmetic of real numbers. The same goes for real-but-not-rational numbers like pi: the reals extend the operations on the rationals without disrupting how rational numbers behave in and of themselves.
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EDIT since some people are objecting and saying you should just define 1/0 = Q and then, say, 3/0 = 3Q. Turns out that doesn’t work either. Consider the expression 2 * 0 * Q. Multiplication is associative, so we can either write:
(2 * 0) * Q = 0 * Q = 1
Or
2 * (0 * Q) = 2 * 1 = 2
So 1 and 2 are both equal to 2 * 0 * Q and therefore to one another. We can make 1 equal any real number x by doing 1 = 0 * Q = (x * 0) * Q (since 0 = x * 0 for all x) = x * (0 * Q) = x * 1 = x.
More generally, the properties of 0 as a number prevent you from defining division by 0 – at least not if you want to be able to *multiply* anything by 0. The fundamental problem is the expression 0 * Q, which immediately generates this kind of contradiction. While there are constructions that effectively define 1 / 0 = Q (usually using the symbol for infinity for this new value), those constructions don’t allow expressions like 0 * Q or Q/Q, so they’re at least as complicated as just leaving division by 0 undefined.
They have. “∞” specifically in the context of [the projectively extended real line](https://en.wikipedia.org/wiki/Projectively_extended_real_line) is one example. You just hear about it less because there’s not as much interesting stuff to do with it.
you mean, sort of like the [riemann sphere](https://en.m.wikipedia.org/wiki/Riemann_sphere)?
They have done this, the symbol for it is ∞. But there are a few caveats.
Firstly, you have to be careful when doing arithmetic operations with it. Simple things like 1+∞=∞ and 2*∞=∞ work out nicely, but there are exceptions that are not allowed. For instance, you *can’t* do 0*∞, if you *could* then you could cancel out the zeros in 2*0=1*0 by multiplying through by ∞ to get 2=1 ^# . The key thing is that while x/0 is fine for *x not equal to 0*, the fraction 0/0 is the really bad thing. There are other less obvious exceptions that you can’t do because they pass through a 0/0 at some point, such as ∞+∞ and ∞-∞ which cannot be defined.
Secondly, this ∞ has another interesting quirk in that +∞=-∞. This is because when you multiply -1 by 1/0, the negative sign is eaten by the 0. The fact that +∞=-∞ should be seen as equivalent to +0=-0. And, just as 0 glues together the positive numbers to the negative numbers *because* of this, ∞ actually glues together the *other ends* of the positive numbers to the negative numbers. Just as you can go from positive to negative by passing through zero, you can also go from positive to negative by passing through ∞ but you do it “at infinity”. This means that this ∞ turns the number *line* into a *circle*, which mathematicians call the [Projective Real Line](https://en.wikipedia.org/wiki/Projectively_extended_real_line).
The circle you get from this is actually different from how ∞ is used in Calculus. In Calculus, and other applications, +∞ and -∞ are *different*, so you don’t actually get a zero. This would be like having +0 and -0 being different, which could result in cutting the real line in half, when +∞ and -∞ are different it cuts the circle into a line with two caps at +∞ and -∞. This actually allows ∞+∞=∞ to be a pretty okay rule, but it also means that the limit of 1/x as x goes to 0 is undefined because 1/x wants to go to both +∞ and -∞ at the same time which is no issue when +∞=-∞, but it becomes a problem when +∞ and -∞ are different ^# . This extension of the number line to include infinity is called the [Extended Real Line](https://en.wikipedia.org/wiki/Extended_real_number_line). Generally, the Extended Real Line has more applications and is less abstract than the Projective Real Line which is why we typically stick to the Extended Real Line and just say that division by zero can’t be done, but it is actually very common to see the Projective Real Line used in math.
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^# These two points are the two reasons most often cited as why you cannot divide by zero – you’ll likely see them in responses to this post. But this is only a problem when you have a commitment to the Extended Real Line, and *both* of them are non-issues when you work with the Projective real line. So, though he’s a great teacher, when [Eddie Woo says that division by zero is “undefineable”](https://www.youtube.com/watch?v=J2z5uzqxJNU), he’s not exactly correct: Division by zero is definable, as long as you’re careful and okay with the number line turning into a circle. A more commonly accepted version of the Projective Real Line is the [Riemann Sphere](https://en.wikipedia.org/wiki/Riemann_sphere), which is like the complex number version of the Projective Real Line and turns the complex plane into a sphere. I have no idea why the Riemann Sphere is a common thing in pop-math content but the Projective Real Line isn’t. Maybe we should call it the Riemann Circle.
And here I thought all those symbols on the chalkboards in the movies were spontaneous. Is there a book with identifications and definitions?
Because it serves no useful purpose.
Please don’t think, like everyone does, in “imaginary” numbers. They are *complex* numbers which contain a component in another dimension, we call it imaginary because you can’t really see it, but it’s DEFINITELY there and the maths works all the way through it and back to the “real” axis all the time for almost everything – AC electrics, astrophysics, and even your MP3, JPG and MPG files (using Fourier transforms). There’s nothing imaginary about them. They are useful, they exist in nature, they crop up all the time, and they are rigorous and predictable.
Same as pi – it exists in nature, crops up all the time, all over the place (not just circles), and its application is rigorous and predictable.
However, division by zero is meaningless.
Think of it like this:
If you were dividing 30 by 5, say, then that’s the same as asking “how many 5’s would I need to add together to get 30?”. 5. 10. 15. 20. 25. 30.
If you were dividing 30 by 3, it’s “how many 3’s would I need to add together to get 30?”. 3, 6, 9, etc.
If you are dividing by 0, it’s “how many 0’s would I need to add together to get 30?”
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. …..
There is no answer. There is NO number of zeroes that you can add together to get 30. Ever. No matter how long you go for or how hard you try. The question literally has no answer.
Alternatively, if you choose, say, 0 divided by 0:
“How many 0’s would I need to add together to get 0?”
0. One zero suffices. But hold on.
0 + 0. So does that. Two zeroes also answers the question.
0 + 0 + 0. So does that.
In fact, every possible number of zeroes will add up to 0.
So dividing by zero only ever gives you “no answer whatsoever” or, in a single special case, “every possible number in existence”.
As such it’s useless, and technically it’s undefined – there is no universally correct answer, way of determining it, etc. Having a symbol, constant or formula for it wouldn’t help at all. And it has almost no real world relevance because – in precisely the same way that there is no possible answer – it doesn’t crop up in nature. Nature avoids ever doing it too. No useful use of division by zero has been found.
Which is totally different to complex numbers (formed by two dimensional components, called “real” and “imaginary” but both equally valid), and pi.
It’s like asking why there isn’t a symbol for “and then the purple crocodile eats this number”. What areas of crocodilatry have you invented that requires such a symbol? Is it a shorthand for a longer Alligator formula that gets tedious to write out all the time? Does it allow you to analogise the crocodile maths to areas of spotted-chicken maths utilising the “and the hen sits on this number” operator, and thus make a breakthrough connection between them? Does it open up new areas of Crocodithmetic which we need to start teaching kids?
For all intents and purposes, division by zero is undefined and the answer doesn’t actually exist.
Isn’t this what i is for?
Pi and i are the results of use “solving” polynomial equations which don’t have solutions by inventing one. We also do this to get fractions from the counting numbers.
For example, lets create 5/3 from the counting numbers. We could ask “what is x if 5x=3?” Since no counting number satisfies this, we define a new number 3/5 which satisfies this.
We can similarly create i by asking “what is x if x^2=-1?” Constructing the reals (including pi) is more complicated.
The problem with creating a number 5/0 is any solution to “what is x if 0x=5” breaks a fundamental (very mild lie) algebraic rule: 0y = 0 for all y. If 5 =/= 0, then we have a big problem! The other numbers we invented earlier don’t break any rules, so those are okay.
Complex numbers do exist – we just didn’t understand them and didn’t have a way to describe them. It isn’t right to say that imaginary numbers can’t exist without i. Rather, i is the letter we use to describe those numbers.
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