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### Isn't 1/0 the Biggest Number?

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Date: 02/28/98 at 10:21:40
From: Garrett Mason
Subject: Isn't  1/0  the biggest number?

Hi,
Isn't 1/0 the biggest number? I mean it makes sense that if you add or
multiply a number to the biggest number you will get the same biggest
number, right? Okay, I'm going to add 9 to 1/0:

1     9     1     9*0     1     0     1
-  +  -  =  -  +  ---  =  -  +  -  =  -
0     1     0     1*0     0     0     0

Okay, now I'm going to multiply 1/0 by 5:

1 * 5     5   (which I'm told is the same as 1/0)
-   -  =  -
0 * 5     0

Is this correct?
Thanks,
Garrett
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Date: 02/28/98 at 15:50:47
From: Doctor Sam
Subject: Re: Isn't  1/0  the biggest number?

Garrett,

Your reasoning is excellent, but it is incorrect.  I really like the
way you reason that the biggest number can't be made larger. That is
exactly the test for "biggest."

The problem is not with your reasoning, it is with the way that you
find a common denominator.  By multiplying 9/1 by 0/0 to get 0/0 you
are claiming that 9/1 = 0/0.

But I could add 1/0 and 5/1 the same way to get:

1/0 + 5/1 = 1/0 + 5*0/1*0 = 1/0 + 0/0 = 1/0

But obviously 0/0 can't be equal to both 5 and 9!

What you have actually discovered is that the ordinary rules of
fractions don't work for 1/0 and 0/0 and for this reason
mathematicians say that these symbols, although they look like
ordinary fractions, don't really stand for anything.  They aren't
numbers.

By the way, your second example doesn't really do what you want to do
because 5/5 = 1.  So you are really multiplying 1/0 by 1, so it isn't
surprising that you get 1/0 as an answer.

By the way, if you aren't convinced yet try repeating your first
example by SUBTRACTING 9 (or anything else) from 1/0.  Your method
again gets 1/0 . . . but what it is true that the largest number can't

-Doctor Sam,  The Math Forum
Check out our web site http://mathforum.org/dr.math/
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