eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?


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.


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


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.

^# 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.