Why can’t you divide by zero?

Part 1: Dividing By Smaller and Smaller Numbers by a High School Math Teacher Suppose you’ve got a pizza. A nice charcoal-cooked one from New Haven, or an oven-hot Chicago deep-dish, or even one of those organic San Francisco artisan pies that somehow make artichoke hearts seem like they belong on a pizza. And, generous soul that you are, you’ve decided to share. How many people can you feed if everybody gets half of a pizza (a hearty helping)? Well, it’s 1 pizza ÷ ½ pizzas per person = 2 people. And how many can you feed if everybody gets 1/10 of a pizza (a cheesy snack)? 1 pizza ÷ 0.1 pizzas per person = 10 people. And how many if everybody gets 1/100 of a pizza (a bite-sized morsel)? 1 pizza ÷ 0.01 pizzas per person = 100 people. And how many can you feed if each gets 1/1000 of a pizza (a crumb with a dab of sauce)? 1 pizza ÷ 0.001 pizzas per person = 1000 people. The smaller the slice you give to each person, the more people you can feed. Or, more abstractly: the smaller the number you divide by, the bigger the result. Now, take it one step further: What if each person gets 0% of a pizza? 1 pizza ÷ 0 pizzas per person = ??? How many people can you feed? Well, there’s no limit, because you’re not actually feeding them anything. If the earth’s seven billion people all show up at your door, asking for their share of pizza, you can say “No problem!” because “their share of pizza” amounts to nothing at all. Add another seven billion, and you’d say the same thing. How many people can you feed? There’s no answer. When you divide a number by 0, there’s no single answer. To divide is to break something into piles of a certain size. And breaking something into piles of size zero just doesn’t make sense. Part 2: “Inverse of Multiplication” by a Math PhD Candidate While she was washing dishes, I asked my fiancee why you can’t divide by zero. Her off-the-top-of-her-head answer was more concise than mine. (In my defense, I get the dishes cleaner than she does.) When you divide by a number – let’s say 4 – you’re asking, “How many times can 4 go into the number?” So: But when you divide by 0, you’re asking, “How many times can 0 go into the number?” And no matter how many zeroes you add, 0 + 0 + 0 + 0 … will never equal 12. So 12 ÷ 0 is undefined. Part 3: “Inverse of Multiplication” Redux by an Elementary-Level Math Specialist I then ran both of these explanations by my sister Jenna, a K-8 math specialist. She liked Taryn’s answer, and gave her own even more concise version. Division is the inverse of multiplication. So when you divide 12 by 4, you’re saying: “What times 4 gives you 12?” Therefore, dividing by zero is like asking, “What times 0 gives you 12?” There’s obviously no answer, since any multiple of 0 will be 0. Part 4: Tying it All Together by a Professor (my dad) At dinner with my father James (a professor of Operations Research), I asked him to explain why you can’t divide by zero. He gave an explanation pretty similar to mine, and then summarized the relative merits of the two approaches quite nicely. The Taryn/Jenna explanation, he said, cuts to the chase, and will satisfy a wider (and younger) audience. It starts by saying, “Well, here’s what division is,” and then showing that the concept makes no sense when applied to zero. The Ben/James explanation, meanwhile, is valuable because it doesn’t cut to the chase. It connects the question “Can you divide by zero?” to other ideas (limits and asymptotic behavior), and gets more to the conceptual heart of the problem. Anyway, there you have it. Four math professionals, two basic explanations, and one more blog adding its voice to the din of answers on this subject.