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/