The standard arithmetic and geometry of the real line and real plane are incomplete. There are ways that we work around this incompleteness, but these workarounds can be abandoned for a more unified treatment of these things and I think that these give a more satisfying way to think about them. That's what I'll try to discuss here.

One of the things that I learned during my math training was that it was not possible to divide by zero. Even in high school, I was unsatisfied with this; surely people were just not trying hard enough. And, sure enough, when you start working with more advanced math, I learned that there are ways to do this and that it is, in fact, quite common. The only sticking point is that there is often some kind of strict formalism around how to do this, formalism which makes it inaccessible to most people resulting in workaround that let us divide by zero without actually doing it. But this formalism is not really necessary outside rigorous contexts, so I'll present a way to carefully deal with dividing by zero that will be functional, even if it is not super rigorous.

The trick is to just do it. I want to divide 1 by 0, but the traditional ways of thinking about division, as the inverse of multiplication or just the number of times one number goes into another, quickly lead to dead-ends, guesswork, and pseudo-mathematical pondering.

If I want 1/0 to work, then I will just impose that it does. It should be clear that 1/0 "shouldn't" be a real number, so it must be something else. I will just call it "∞", which should be thought of as just a symbol representing 1/0 without any deep signification tied to it. We need a picture for something that isn't a real number and ∞ works just fine.

So if I just say 1/0 = ∞, what do I get from this from an arithmetic standpoint? The only rules that I want to impose are arithmetic ones such as the addition and multiplication formulas for fractions. I effectively creating a new thing "∞" that is 1/0 and works like 1/0 "should" as a fraction from the standpoint of arithmetic. There first property that I will look at is one of the most important. Let A be any non-zero real number. I can then compute A*∞ directly:

A * ∞  =  A  *  1/0
       =  1/(1/A) * (1/0)
       =  1 / ( (1/A) * 0)
       =  1 / 0
       =  ∞ 

And so ∞ has the property that for any nonzero real number A*∞ = ∞. There is one other number that also has this property: A*0 = 0. The duality between ∞ and 0 will not be incidental. Can I divide by ∞? When I try to do so I get

1/∞  =  1/(1/0)
     =  0/1
     = 0

and so I find that 1/∞ = 0. Next, if A is any real number, then I can compute A ∞ as follows:

A   ∞  =  (A/1)   (1/0)
       =  (A*0   1*1)/(1*0)
       =  1/0
       =  ∞

Therefore A ∞ = ∞ for any real number A. There are a couple of exceptions in the addition and multiplication rules. The first of which is what happens when we try 0*∞? In this case when we try to do the same manipulations as before, we end up with 0*∞ = 0/0 which is not something that I have assigned as existing yet. Moreover, if I try to do ∞ ∞, then I end end up with 0/0 as well. I could try to give 0/0 its own symbol and workout how it works, but this ends in one of two ways. I either end up proving that all numbers are the same and that there is only one number, which is really boring, OR I'm really careful and end up creating Wheel Theory which, while interesting, does not nicely contextualize the things I'm interested in. The issue is that 0/0 looks like it "should" be different things in different contexts. For instance if f(x)=x/x and g(x)=3x/x, then f(0)=1 and g(0)=3, which suggests 0/0=1 and 0/0=3 respectively. The three options I have of resolving this are 1.) There is only one number (boring) 2.) Wheel Theory (not useful or 3.) Leave 0/0 undefined. I will take the third option. So, even if I can allow 1/0 just fine, if I allow 0/0 I break what I'm trying to do and so I won't allow it. Maybe, someday, someone will come up with a fourth option.

So the rules for dividing by zero are 1.) Define ∞:=1/0, 2.) A*∞ = ∞ for nonzero A, 3.) 1/∞ = 0, 4.) A ∞=∞ for non-∞ A and 5.) Any expression that results in 0/0 remains undefined. I will now share some important consequences of this arithmetic extension.

• The first thing to note is that a consequence of A*∞=∞ for nonzero A is that we have -∞=∞. Those who have taken Calculus might have been itching to proclaim that 1/x has no limit at x=0, and so I shouldn't be able to work with 1/0. They are correct, in a way, because as x gets closer to zero, 1/x blows up and it can do so in a negative or positive manner and so it looks like 1/x goes in completely opposite directions depending on what direction you approach zero. The fix to this is to wrap the number line into a circle. Effectively, you take ∞ living at one end of the real line and -∞ living on the other, pick them up, and glue them together making a circle. This is where the duality between 0 and ∞ really shines: Distinguishing between ∞ and -∞ is like trying to distinguish between 0 and -0 and the only way to do so would be to chop the number line in half at 0 and cap each end by either 0 or -0. But I don't want to do that, and so the arithmetic of 0 and the arithmetic of fractions for my hand to say that ∞=-∞

  This new object, the number line wrapped in a circle, has a name: The Real Projective Line. As mentioned, it is a circle, and it has 0 at the bottom of it and ∞ at the top. I like to put 1 and -1 on the equatorial sides of this line, because in this way when I do the operation 1/x to the circle, it merely rotates it about the horizontal axis. When you do Calculus, you actually do effectively work in an extended real line with infinities. But this space does distinguish between ∞ and -∞, and is the real line that has been capped off at both ends. This space is the Extended Real Line. It doesn't work out super nicely arithmetically, but can be better for applications o Calculus on these two things will be similar, but some results will bet different. The difference between these two ways of extending, and the resulting ambiguity in a class known for fundamentally "true" results, could be a reason why we don't see division by zero in high school and college. In Calculus, we use workarounds to deal with the extended real line, but stop short of defining ∞ and working with 1/0.

• The next thing to note is that of indeterminate forms. If you look at a list of all the indeterminate forms, keeping the fact that ∞=1/0, you'll note that they are all expressions that lead to 0/0. For instance, 0*∞ directly leads to 0/0. ∞/∞ leads to 0/0. Even ∞-∞ = 1/0 - 1/0 = (1*0 -1*0)/0 = 0/0. And so the indeterminate forms arise because 0/0 cannot be defined in a nice way like 1/0 can and how the workarounds allow calculus students to use it without actually having to define it. Note one more thing: On the Extended Real Line ∞ ∞ = ∞ and is not indeterminate. But if we work on the Projective Real Line, then fraction arithmetic tells us that ∞ ∞=0/0 and so is indeterminate. This is because on the Extended Real Line, x ∞ pushes things to the right all the way to ∞ which has no issues, but on the Projective Real Line, it is possible to push this out past ∞ into the opposite side of the number line, and so is undefined.

• Finally, you might have seen some faulty proofs that 2=1. These effectively all take advantage of something like 2*0 = 1*0 and, after dividing by zero, we get 2=1. The explanation for why these proofs do not work is usually because we divided by zero in the proof and therefore it is invalid. This is actually not the case. If you carefully note, it's not that division by zero is used, its that we're dividing 0 by 0 to cancel out multiplication by zero. So the real issue is that these proofs divide 0 by 0, meaning they define 0/0 to be a thing. The sad thing is that this explanation is usually used to justify why dividing by zero can't work (for example gives multiple reasons why 1/0 shouldn't exist, reasons that I've countered with this discussion), but, as I've hopefully shown, dividing by zero is totally fine, the issue is dividing 0/0 or, equivalently, using division by zero to cancel out multiplication by zero. This is the difference between thinking about 1/0 as just a fraction and thinking about 1/0 in the context of other ways of understanding division (as the inverse of multiplication, for example). These kinds of proofs are, in fact, how defining 0/0 can lead to all numbers begin equal (eg 2=1).

This is just the beginning on what division by zero looks like. Next time, I want to show how division by zero can be used to actually do some work. They, particularly, work very nicely when working with polynomials and rational functions, and figure out their graphs.