Error: Division by Zero
Date: 02/12/2001 at 17:45:34 From: R S Casey Subject: Windows calculator says a number divided by 0 is infinity I've been trying to help my third grader understand that a number divided by zero is undefined but am not getting much help. The calculator they use in school gives the answer 0/E (the teacher told them to write that on their homework paper, but seems not to understand what it means - I'm guessing it stands for "error"). But worse, the calculator on the Windows Accessory program gives the answer "positive infinity" when you divide a number by 0. How are we supposed to swim upstream against the teacher, the school calculators and the computer?
Date: 02/13/2001 at 15:27:59 From: Doctor Twe Subject: Re: Windows calculator says a number divided by 0 is infinity Hi - thanks for writing to Dr. Math. Yes, it's pretty hard to swim upstream - even when you're right. Here's how I'd explain why division by zero is not allowed: Division simply "undoes" multiplication. For example, 12 / 3 = 4 because 4 * 3 = 12. Think of taking 12 oranges and splitting them into piles of three oranges each. We'll make our first pile of 3: O Remaining oranges: O O O O O O O O O O O Pile1 Now we'll make another: O O Remaining oranges: O O O O O O O O O O Pile1 Pile2 And another: O O O Remaining oranges: O O O O O O O O O Pile1 Pile2 Pile3 And finally another: O O O O Remaining oranges: O O O O O O O O Pile1 Pile2 Pile3 Pile4 We end up with four piles of oranges. We can see that this is so because four piles of three oranges each make a total of twelve oranges (i.e. 4 * 3 = 12). Now let's try to divide by zero by creating the "reverse" multiplication problem. Suppose we want to divide 12 / 0. How many piles will we have to make? To parallel the previous problem, where our piles had 3 oranges each, our piles now need 0 oranges each: Remaining oranges: O O O O O O O O O O O O Pile1 Now we'll make another: Remaining oranges: O O O O O O O O O O O O Pile1 Pile2 And another: Remaining oranges: O O O O O O O O O O O O Pile1 Pile2 Pile3 We can see that we're getting nowhere. No matter how many piles we make, we still have 12 oranges remaining. The argument above may seem like a case in favor of the Windows Accessory calculator program's answer of "positive infinity" (by the way, when I try it on my Windows 98 machine, I get the result "Error: Positive Infinity" which at least acknowledges that it is an "error"). To explain why it is not simply positive infinity requires a little more abstraction. Suppose we divide 1 by successively smaller values (I'll use 1, 0.1, 0.01, etc.) Our results are: 1 / 1 = 1 1 / 0.1 = 10 1 / 0.01 = 100 1 / 0.001 = 1,000 1 / 0.0001 = 10,000 1 / 0.00001 = 100,000 We can see that as the denominator gets closer to zero, the quotient increases without bound. (You can introduce the concept of limits and Lim[x->0+, 1/x] = +oo.) But suppose we divide -1 by successively smaller values. What happens then? Our results are: -1 / 1 = -1 -1 / 0.1 = -10 -1 / 0.01 = -100 -1 / 0.001 = -1,000 -1 / 0.0001 = -10,000 -1 / 0.00001 = -100,000 Now our quotient is approaching negative infinity. So we have to make two rules: one if the numerator is positive and one if it is negative. But what if I choose a sequence of successively smaller negative values; for example -1, -0.1, -0.01, etc. What happens then? 1 / -1 = -1 1 / -0.1 = -10 1 / -0.01 = -100 1 / -0.001 = -1,000 1 / -0.0001 = -10,000 1 / -0.00001 = -100,000 Now, even though the numerator is positive, the quotient approaches negative infinity. If I have the problem 1/0, how am I supposed to know whether to use a sequence of positive or negative numbers to approach zero? In fact, I can't know. That's why we say it is undefined instead of positive infinity. As we APPROACH zero from the right or from the left, our quotient approaches positive or negative infinity, but when the denominator IS zero, the quotient is neither - it's undefined. This concept may be too abstract for a third grader. You may have to tell him to take it "on faith" for now, and that later on, when he gets older, he'll understand the reason why. When my son was 3, I didn't tell him why it was a good idea to buckle up in the car - he just had to do it. (I didn't want to scare him with the idea of having an accident.) When he was a little older (about 5, I think), I explained to him WHY it was a good idea. Now he always checks to make sure I'm buckled up, too! One final comment on the school's calculators: I think the "0/E" notation is the manufacturer's cryptic way of expressing "divide by zero error." Personally, I would have made it "/0 E," more literally "divide by zero, error." I hope this helps. If you have any more questions, write back. - Doctor TWE, The Math Forum http://mathforum.org/dr.math/
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