The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Dividing by Zero

Date: 05/18/2001 at 13:18:52
From: Iain MacEachern
Subject: Dividing by zero

I cannot comprehend that a human being is not able to divide a number 
by zero, because by definition 0 is nothing, and if you can multiply 
by nothing, and add and subtract by nothing, why can't you divide by 

Let's say you have 10 apples and you divide them by 0 - don't you 
still have 10 apples? I cannot see why this cannot be done!

Thank you for giving this question notice.

Sincerely confused,
Iain MacEachern

Date: 05/18/2001 at 13:40:45
From: Doctor Peterson
Subject: Re: Dividing by zero

Hi, Iain.

Did you see our FAQ on this subject, which answers your question in 
several ways?

   Dividing by 0   

You don't seem to be thinking closely about what it MEANS to divide by 
zero. Let's watch what happens when we try to divide those apples.

First, let's divide the 10 apples into piles of 2, so we can give 2 to 
each of our friends. (When we run out, we'll be out of friends!) We do 

     oo oo oo oo oo
     -- -- -- -- --

We've managed to make 5 piles; 10 divided by 2 is 5.

Now let's try dividing the apples into piles of ZERO to give to our 
enemies, and see when we run out:

     -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- ...

This is getting hard! No matter how many empty piles I make, I haven't 
used up any of my apples. I guess I can have infinitely many enemies 
to give them to!

This is why we can't divide by zero: we can never finish the job. And 
your mention of human beings is interesting; it's precisely because we 
are human, and therefore finite, that we can't do this.

To put all of this into mathematical terms, dividing by 2 means 
finding a number (5) by which we can multiply 2 to get 10:

     10 / 2 = 5 because

     10 = 2 * 5

If we could divide 10 by 0 (I'll call the answer X), we would be 
saying that:

     10 / 0 = X because

     10 = 0 * X

But zero times anything is 0, so I will never find an X for which this 
is true. That's what happened when I tried dividing the apples.

- Doctor Peterson, The Math Forum   
Associated Topics:
Elementary Division
Elementary Infinity
Elementary Number Sense/About Numbers
Middle School Division
Middle School Number Sense/About Numbers

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.