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Why We Never Divide by Zero… Why we never divide by zero… Zero hit the USS Yorktown like a torpedo. On September 21, 1997, while cruising off the coast of Virginia, the billion-dollar missile cruiser shuddered to a halt. Yorktown was dead in the water. Warships are designed to withstand the strike of a torpedo or the blast of a mine. Though it was armored against weapons, nobody had thought to defend the Yorktown from zero. It was a grave mistake. The Yorktown computers had just received new software that was controlling the engines. Unfortunately, nobody had spotted the time bomb lurking in the code, a zero that engineers were supposed to remove while installing the software. But for one reason or another, the zero was overlooked, and it stayed hidden in the code. Hidden, that is, until the software called it into memory — and choked. When the Yorktown’s computer system tried to divide by zero, 80,000 horsepower instantly became worthless. It took nearly three hours to attach emergency controls to the engines, and the Yorktown then limped into port. Engineers spent two days getting rid of the zero, repairing the engines, and putting the Yorktown back into fighting trim. Charles Seife: ZERO — The Biography of a Dangerous Idea, Penguin, New York 2000, p. 1-2. You may have seen this little proof that 2=1: a = b [true for some a’s and b’s] a+a = a+b [add a to both sides] 2a = a+b [a+a = 2a] 2a–2b = a+b-2b [subtract 2b from both sides] 2(a–b) = a+b–2b [factor: 2a–2b = 2(a–b)] 2(a–b) = a–b [b–2b = –b] 2 = 1 [divide both sides by a–b] You may doubt that 2=1. So, where is the mistake? Think about it. http://www.jimloy.com/algebra/two.htm Division by zero error Division by zero is the operation of taking the quotient of any number and 0, i.e., . The uniqueness of division breaks down when dividing by zero, since the product is the same for any, so cannot be recovered by inverting the process of multiplication. 0 is the only number with this property and, as a result, division by zero is undefined for real numbers and can produce a fatal condition called a “division by zero error” in computer programs. http://mathworld.wolfram.com/DivisionbyZero.html “Proof” Assume you can divide by zero. Then we start with 2x0=1x0, which is true. Divide by zero, and you get 2=1 which is false. We have our contradiction. It doesn’t work. http://www.jimloy.com/algebra/two.htm Division by Zero “Don’t do it! It’s bad! Very bad! Never divide by zero!” -- Your Teacher Dramatic, isn’t it? You’ve probably heard something like that from your math teacher before, and they were right. As soon as you divide by zero, or by a more complex expression that is equal to zero, you have broken the rules of mathematics and your answer is now suspect. Let’s think about it from a simplistic point of view. What does it mean to multiply? If I have 3 times 2, that could mean three groups of two, for a total of six. Multiplication is repeated addition in that sense. You can multiply by zero because that means you have zero groups, and thus a total of zero. But what about dividing by zero? If you represents the slices of a pizza into a fraction, then 4/8 means you have four of the eight pieces. What would 4/0 mean? That you have 4 of the zero pieces? Imagine dividing up 6 dol- lars among 3 people. Each person would have 6/3, or 2 dollars. But now divide 6 dollars by 0 people. How much does each person get? It doesn’t make sense, because there aren’t any people to divide the money among! That’s why division by zero is undefined. When you divide by zero the answer isn’t zero, or infinity, or negative infinity. It’s undefined. As in... it doesn’t make sense. The folks at mathforum.org actually put together a nice list of some questions and answers about dividing by zero, and some nice ways of visualizing it in your mind. Check it out to convince yourself that division by zero is unde- fined. http://www.freemathhelp.com/division-by-zero.html.
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