Suppose somebody types 3 ÷ 0 into a calculator, hits the = button, and gets an error message. If the person asks you what is going on, what do you say?
I think in these situations we mostly tend to say, "You can't divide by zero." That isn't wrong, but I suddenly find it odd, because it sounds more like physics than mathematics. You can't travel faster than light, you can't build a perpetual motion machine, you can't unscramble an egg, and you can't divide by zero. One of these things is not like the others, yet the formulation doesn't reflect that difference. A scientist could doubt whether it is really impossible to travel faster than light—all scientific knowledge is provisional—but we can say dogmatically that nobody will ever build a calculator that gives a numerical answer for 3 ÷ 0.
Instead of saying "You can't," in a case like 3 ÷ 0, I think it would be better to say to the student,
"There's no such number."It isn't so much that "you can't divide 3 by 0" as that there's no such number as 3 ÷ 0.
Why is there no such number as 3 ÷ 0? Because remember what 3 ÷ 0 actually means: it means the number that, when multiplied by 0, gives 3.
Think about it! There is no such number!
Any number multiplied by 0 always gives 0, never 3.
"You can't do that" also sounds like an arbitrary rule to remember...just another of the endless rules adults are always hitting kids with. "There's no such number," on the other hand, is a statement of mathematical fact.
Saying "You can't" invites the reply, "Why can't I?" A good answer to which would be: "There's no such number." One might as well skip a step.
It certainly seems absurd to share 3 cookies among no people, or to ask how many servings of size zero are in 3 cups of ice cream...but what is really absurd about the first situation for example is not the vaguely puzzling idea of sharing among no people; it's the concretely outrageous idea that the result of sharing 3 cookies among no people would be shares of half a cookie! Or seven cookies, or any other number of cookies. Likewise, there could not be eleven 0-cup servings in 3 cups of ice cream. No number could be the value of 3 ÷ 0.
So tell me again, why does 3 ÷ 0 give Err on the calculator? Could it be that the school bought cheap calculators to save money? Would a more expensive calculator have told us the number? No. Even at Chichi Academy they get an error message. There is no such number, so the calculator has to display something else.
"Undefined" is the word that careful math educators tend to use for division by zero. And that word is accurate...but saying that "Division by zero is undefined" strikes me as little better or even no better than saying that "You can't divide by zero." After all, what is the force of "undefined," in a discourse where precious little ever gets defined in the first place? I think it means nothing to a student and it gets internally stored in student memory as: "You can't." Anyway, in a case like 3 ÷ 0, "undefined" is shorthand for "there is no such number"! Instead of using shorthand, one could make a statement that has content.
"There's no such number" describes a case like 3 ÷ 0 where the dividend is nonzero. What about the doubly bizarre case of 0 ÷ 0? This, recall, would have to mean the number that, when multiplied by 0, equals 0. But every number, when multiplied by 0, equals 0. So there is a case to be made that any number could be the value of 0 ÷ 0. In a sense, this is the opposite situation to 3 ÷ 0, where there was no such number.
Could there be fifty 0-cup servings of ice cream in 0 cups of ice cream? Sure, why not! Heck, let it be a thousand 0-cup servings. You can feed any number of people with no ice cream, as long as everybody would be satisfied with getting none.
If 0 cookies are shared equally among 0 people, could each person get a dozen cookies? It wouldn't violate any laws of physics!
My own calculator, Mathematica, is pretty chichi, and unsurprisingly it gives two different results for 3 ÷ 0 and 0 ÷ 0, along with some error messages:
Mathematica uses the non-numeric quantity "ComplexInfinity" for various purposes in the analysis of functions. If you ask Mathematica for the value of 0 times ComplexInfinity, you won't get a numeric answer—you'll get "Indeterminate," another non-numeric quantity. So in Mathematica, 0 × (3 ÷ 0) ≠ 3.
I'll close this post with two other cases where "There's no such number" seems like a better answer than "You can't do that":
- Instead of saying "You can't take the logarithm of a negative number," you could say, "There's no such number as Log(−8)." Why? Because remember what Log(−8) means: it means the number that, when you raise 10 to that power, you get −8. But try a few cases and you'll see that there is no such number...10 raised to any real power is always a positive number, never −8. There's no such number as Log(−8).
- Instead of saying "You can't take the square root of a negative number," you could say, "There's no such number as Sqrt(−1)." Why? Because remember what Sqrt(x) means: it means the nonnegative number that, when you square it, you get x. But try a few cases and you'll see that there is no nonnegative number that squares to give you −1. There's no such number as Sqrt(−1).
In these two examples, "number" means "real number." High school students learn that there are two complex numbers whose square is −1, and college students in math or physics learn that there are infinitely many complex numbers that, when you raise 10 to that power, you get −8.
Can complex numbers also help us divide by zero? No. Just like the real numbers, the complex numbers are algebraically a field, which implies that even in the complex numbers, there's no such number as 3 ÷ 0.