“Because the universe will blow up,” was the usual answer I got when my teachers tried to explain why we couldn’t divide by zero. From a young age, I was a sort of anti-Pythagorean in that I believed people created numbers, not that the universe was ruled by them. So why then did we create the divide-by-zero bomb?

The best way I’ve found to describe why dividing by zero will destroy everything is to go back to translating fractions. What does “1/2” really mean? “1/2” translates to “1 out of 2” or “I have one piece of candy out of the two pieces on the table, so I have half of what is on the table. My sister is a good sharer.”

Now try this with “0/2”. This translates to “zero out of 2” or “I have zero pieces of the two that are on the table. My sister’s cheap!”

Both of these situations are real. You can have one piece of candy out of two. You can have none of the pieces of candy. Even if the fraction is an improper fraction, like “3/2”, certainly you can’t have three out of two pieces of candy; this makes no sense at all. But then we remember that improper fractions can be written into mixed fractions, so “3/2” becomes “1 and ½”, and we sure can have 1 and a half of the pieces of candy on the table [leaving our cheap sister with just ½! Haha!]!

So then comes “2/0”, which would translate to “2 out of zero” or “I have two pieces of candy out of the zero that are on the table.” HUH?? This obviously doesn’t make sense! Despite what Little Orphan Annie and Jay-Z may lead us to believe, you can’t make something out of nothing. It’s just basic physics.

Once a student begins learning about slope and functions, the impossibility of “2/0” becomes even more obvious. Let’s think of a graph that measures your height against your age. “2/0” represents a rise (y-value or “height”) of 2 and a run (x-value or “time”) of 0. This is to say that, for example, at time 0 you are 2 feet tall. Ok, so maybe you were born 2 feet tall. That’s possible. Now let’s move up from coordinate (0, 2). The slope of “2/0” tells us to move up 2 and over 0. We move up two spaces to 4 feet tall and over to… over to nothing! We stay at zero! So a slope of “2/0” says that you can be 2 feet and 4 feet tall at the same point in time. This is impossible!

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I rather like the word problem interpretations: 2/0: If I have 2 cookies, and I put 0 cookies on each plate, how many plates do I need for all of the cookies? OR If I have 2 cookies, and I put them on 0 plates, how many cookies are on each plate? [The answer, in my opinion, to both questions is HUH?! That doesn’t make any sense!]. A lot of my students like the fact-family explanation 2/0=__ is in the same fact family as 0x__=2–so, what can you multiply 0 by to get 2? [Well, obviously there isn’t such a number]

0/0 is much more mysterious. Indeed, I sometimes explain differential calculus as a way of figuring out what 0/0 is (the answer, interpreted in the differential calculus way, is–it depends on what sorts of 0’s you have: a delightfully enticing answer, I think–but that’s because I like calculus, I suppose).

With that, “if I have 0 cookies and put 2 on each plate” would also get a “HUH?” but 0/2 is defined. The “something out of nothing” explanation always works best for me.

I should probably mention that 2/0 is probably best seen as infinity… for those at high school, look at a 1/x hyperbola. Look at what happens as x approaches 0 from the right side… 1/x gets really big as x gets really small. It’s fairly easy to see that 2/0 (or 3/0, 4/0 etc) is infinity in that light.

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