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### Division by Zero

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Date: Wed, 2 Nov 94 14:20:49 EST
From: "Terry Strohecker"
Subject: Question!

I am in need of a detailed answer to the following question:

Why can't you divide a number by 0?

Terry
stroheck@student.msu.edu
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Date: Wed, 2 Nov 1994 14:50:44 -0500
From: Melissa D. Binde
Subject: Re: Question!: division by zero

I don't know of the "real" answer to this one, but this is how I always
thought of division.

When you divide a number by another number, you can think of splitting it
up into groups.

For example, 10/5 (ten divided by five): take the ten and divide it up
between five groups so there is nothing left of the ten.  You get two in
each group, so 10/5 = 2.

This works for numbers that don't divide evenly also: 5/2: take the five
and divide it up between two groups so there is nothing left.  You can put
two in one group and two in the second group, but that leaves a remainder
of one.  So you can split the one between the two groups and get 2.5 (2 and
1/2) in each group.  So 5/2 = 2.5.

However, if you try to do this with zero, it doesn't work.  Take any number
-- we'll use 10 to make it simple.  What happens when you try to split ten
up into zero groups with no remainder??  You can't do it!  If there aren't
any groups, then you can't put the ten anywhere.

To make this idea clear, mathematicians call division by zero "undefined"
-- meaning that it doesn't really make any *sense* to try to split a number
into zero groups.

If I'm wrong on this, I'm sure one of the other "doctors" will correct me!

Melissa
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From: Dr. Ken
Subject: Re: Question!
Date: Wed, 2 Nov 1994 16:53:27 -0500 (EST)

Hello Terry!

more, and I think we've come up with something about why you can't divide by
zero.

There are sort of two reasons.  For one thing, when you divide one number by
another, you expect the result to be another number.  So in particular, look
at the sequence of numbers 1/(1/2), 1/(1/3), 1/(1/4), ... .  Notice that the
bottoms of the fractions are 1/2, 1/3, 1/4, ..., and that they're going to
zero.  So if there's a limit to this sequence, we would take that number and
call it 1/0.  So let's see if there is.

Well, the sequence turns out to be 2, 3, 4, ..., and that goes to infinity.
Since infinity isn't a real number, we don't assign any value to 1/0.  We
just say it's undefined.

But let's say we did.  Let's say that infinity is a real number, and 1/0 is
infinity.  Then look at the sequence 1/(-1/2), 1/(-1/3), 1/(-1/4), ..., and
notice again that the denominators -1/2, -1/3, -1/4, ..., are going to zero.
So again, we would want the limit of this sequence to be 1/0.  But looking at
the sequence, it simplifies to -2, -3, -4, ..., and it goes to negative
infinity.  So which would we assign to 1/0?  Negative infinity or positive
infinity?  Instead of just assigning one willy nilly, we say that infinity
isn't a number, and that 1/0 is undefined.

I hope this helps.

-Ken "Dr." Math
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Associated Topics:
Middle School Division

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