Division by Zero is Not Illegal, Just Undefined
The grade school method of teaching arithmetic seems to often be a cycle of white lies and revelation: you can’t subtract a larger number from a smaller one, until we define negatives; you can’t divide an odd number by an even one, until we define fractions. We get wise to this game of Charlie Brown football, however, and so when it comes to division by zero our teachers have to stress that, no, for real this time, division by zero is not allowed.
That is, of course, until it is. But more accurately, division by zero is undefined, because something necessarily breaks when you try to define it. Breaking things is the job of a mathematician, however, so that doesn’t mean dividing by zero is off the table forever.
In fact, something broke each time Lucy pulled the football earlier: when we defined negative numbers, we no longer could guarantee that any nonempty set of numbers has a minimum. When we defined fractions, we lost the ability to talk about the “next” number. Defining the imaginary unit i causes us to lose a well-behaved notion of “less than” on numbers. In each of these cases, we just decided that the flexibility of having negatives, rationals, and imaginary numbers was worth the cost.
The most common solution is to introduce what is called the extended real line: put simply, we introduce two new numbers at either end of the real number line: positive infinity and negative infinity. This is also a good time to point out that if someone complains that infinity is not a number, they’re wrong; infinity is not usually a number. Adding numbers to our line means we have to define how they work with our other numbers, which involves the following rules:
- Infinity times any positive number is infinity
- Infinity times any negative number is infinity
- Any positive number divided by zero is infinity
- Any finite number divided by infinity is zero
From these rules you can deduce other rules: ∞ + ∞ = ∞ and -1/0 = -∞, for example. But, some quantities are left undefined. 0⋅∞, ∞-∞, and ∞/∞ do not have defined values. Oh well, we’ll just call these “indeterminate” and leave that issue to the end of Calculus 1.
What does this definition break? We lose the following property relating multiplication and division: it is no longer the case that a/b = c is always equivalent to a = b⋅c. Certainly, for finite numbers, 6/2 equals 3 because 2⋅3 = 6, but our rule 1/0=∞ clashes with the indeterminacy of 0⋅∞. In particular, we have 1/0 = 2/0, which under the previous regime would be a terrible contradiction.
This has a simple fix: we will just have to remember that the equivalence between multiplication and division only holds for finite values.
Sure, that’s reasonably complex, but not entirely out of the question. Another reason why we leave division by zero undefined is that there are multiple ways to define it! To motivate a new definition, let’s take a look at the graph of 1/x:
This function isn’t defined at 0, but that doesn’t deter us — we just defined a bunch of undefined stuff. The problem is that there isn’t a clear value to give the function at zero. Certainly, it can’t be finite, but the right wing looks like it wants positive infinity, while the left wing wants negative infinity.
So, what if instead of adding two infinities, we just added one? We could just combine the positive and negative infinities from before into one grand infinity that rules the ends of the number line. We could even still have the same rules as before for handling this “infinity”.
In essence, we’ve tied the two ends of the number line together and called the knot “∞” — this version is called the projectively extended real line. Our function f(x)=1/x now has a good value for f(0); it can be this singular infinity, being approached from opposite sides just as the function tends to approach all of its other, finite, values from opposite sides.
This, of course, comes at a cost. While our extended real line still had a strong notion of “less than”, by tying the two sides together we have our ∞ being both larger and smaller than all the finite numbers. Alas, this is a sacrifice we have to make. But! If we did the same thing in the complex numbers, where we already sacrificed such comparisons, there would be no problem! Indeed, doing such an extension to the complex numbers produces what is usually called the Riemann sphere.
These aren’t the only ways to define division by zero — just the most popular. What definitions can you come up with? What breaks?
Being able to work with infinity and to divide by zero is tremendously useful, but does require some maturity to remember what you are and are not allowed to do. It makes sense why this secret is usually revealed only in upper-level mathematics courses, after students have practiced the importance of being careful. But this leads to a wider public absolutely terrified of dividing by zero for fear of opening an inter-dimensional rift in the fabric of mathematics. Or worse, that some hidden beauty of the universe is waiting to be discovered if only we could figure out how to divide by zero.
Division by zero is just undefined. You’re free to define it, as long as you’re willing to pay the price.