Zero and Infinity

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

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, 
etc, in radians.

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/