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### Dividing by Zero

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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
nothing?

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
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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
http://mathforum.org/dr.math/faq/faq.divideby0.html

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
this:

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
http://mathforum.org/dr.math/
```
Associated Topics:
Elementary Division
Elementary Infinity
Middle School Division

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