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### Zero and Infinity

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Date: 04/24/97 at 07:43:09
From: Anonymous
Subject: 1/0 and 0/0

I am a junior in high school. What are the algebraic proofs for the
values of 1/0 and 0/0? Does 1/0 equal infinity? What does it mean to
be indeterminate?

Thanks,
Anonymous
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```
Date: 04/24/97 at 08:11:06
From: Doctor Jerry
Subject: Re: 1/0 and 0/0

Hi there,

The set R of real numbers does not include an object called infinity,
although when mathematicians work with the sets of cardinal or ordinal
numbers there are objects that correspond to infinity in one of
several senses.

The expressions 1/0 and 0/0 are not defined. That doesn't mean they
are infinite. The standard definition for the set Q of all rational
numbers is that Q is the set of all real numbers p/q, where p and q
are integers and q is not zero.  That leaves 1/0 and 0/0 not included
in Q since they don't fit the definition.

Some people say that 1/0 is infinity as a kind of short hand for what
happens to 1/x as x approaches 0. Note that if x approaches 0 from
the right, 1/x becomes larger and larger; often we say that 1/x
approaches infinity. Note also that 1/x becomes smaller and smaller
as x approaches 0 from the left, so that 1/x approaches negative
infinity.

0/0 is often used as shorthand for an indeterminate form in which
numerator and denominator approach 0.  The ratio is not determined in
the sense that it can approach almost anything.

sin(x)/x is a 0/0 indeterminate form; as x approaches 0, both x and
sin(x) approach 0; it is known that sin(x)/x approaches 1 as x
approaches 0.  Try calculating sin(x)/x for x = 0.01, 0.001, 0.0001,

sqrt(|x|)/x is also a 0/0 indeterminate form; as x approaches 0, both
sqrt(|x|)  and x approach 0; it is known that the ratio becomes
unbounded as x approaches 0.  It is misleading to say that it
approaches infinity since depending on whether x approaches 0 from the
right or left, the ratio becomes very large or very small.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Number Theory

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