I teach math to kids, usually 11- to 15-year olds. When we talk about integers, I remind them that when they were little, teachers told them that you can't subtract eight from five. Now, of course, they know that you can; it's just that the answer is a negative number, and teachers didn't want to go there.
My Algebra students have a quiz today on different categories of numbers:
- Counting numbers: 1, 2, 3. . . (the way you teach a baby to count)
- Whole numbers: 0, 1, 2, 3. . .(same as above plus the doughnut hole)
- Integers : . . . -3, -2, -1, 0, +1, +2, +3 . . . (whole numbers and their opposites)
- Rational numbers: All of the above plus fractions and decimals that terminate or repeat. (Technically, rational numbers are any numbers that can be written in the form a/b, where a and b are both integers and b is not zero Eg: .5 can be written 1/2, and is therefore a rational number.)
- Irrational numbers: Funky numbers with decimal parts that do not terminate or repeat, like pi, or the square root of 2, or 5.12122122212222. . .(which has a discernable pattern, but not a block of numbers that repeat.)
- Real numbers*: All of the above.
So the brighter kids in the class want to know, aren't all numbers real numbers? What other kind of number is there? I tell them about the existence of imaginary numbers, and tell them that they'll learn more later. You would be amazed at what a little bit of suspense does. They're dying for me to explain imaginary numbers to them.
Dividing by zero is the same sort of thing. Sadly, I was one of those people, even in freshman Calculus, who couldn't remember which was zero and which was undefined,
--- or ---
This is how I explain it to my students:
You can have $0 and share with any number of people; everyone gets nothing. I could have NO COOKIES right now, and I would be kind enough to share, and everyone would get the same amount: NO COOKIES. However, if I had, say, five cookies, I couldn't put them into zero piles. One pile, sure (I could eat them all mysellf and they would be in a pile in my stomach; five piles, yes; even ten piles would be possible. But not zero piles.
(A visual image helps.)
Once they have digested that bit of information, I tell them that, later on in mathematics, they will be shown another, more sophisticated way to think about dividing by zero. It still won't be possible, but there are other ways to visualize it (when they learn about limits.)
Then again, I'm thinking of just telling them that division by zero makes the world blow up.
Another hint for the non-mathematically inclined: when the zero is UNDERneath, the answer is UNDEfined.
*The textbook calls rational and irrational numbers together the family of real numbers. I have my kids picture a family picnic, with the rational 'people' behaving normally and the crazy relatives, THAT side of the family, the irrationals, howling while crouched under the picnic tables. What do you say when you take your leave of this sort of gathering? "It's been real."
These days, you can just ask Siri, although she gets a little snarky: “Imagine that you have zero cookies,” Siri’s response begins, “and you split them evenly among zero friends. How many cookies does each person get? See? It doesn’t make sense.
“And Cookie Monster is sad that there are no cookies, and you are sad that you have no friends.”
Ask Siri ‘what is zero divided by zero?’ and she will send you to the burn ward