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Why Isn't Infinity a Number?

Date: 15 Feb 1995 15:08:54 -0500
From: Steven Bahrij
Subject: infinity

Dear Dr. Math,

I realize that you cannot divide by zero; however, the limit of 1/x as 
x approaches 0 is either positive or negative infinity.  The concept of 
infinity is not considered a number.  Why?

Frances Shefl                 
Grade 11
St. Mary's high School
O'Neill Nebraska                            
math instructor:

Date: 15 Feb 1995 20:25:22 GMT
From: Dr. Math
Subject: Re: infinity

Hello there!

One of the main reasons we don't call infinity a number is that it doesn't
act like one.  Let's say we call infinity the number A (some people
actually do use the hebrew letter Aleph to name one kind of infinity). 
Then I can show you that the equation A + A = A.

Consider the number of terms in the sequence 1,3,5,7,9,11,13,... , and
consider the number of terms in the sequence 2,4,6,8,10,12,14,....  They
have the same number of terms, because I can match them up in a one-to-one
way, right?  Match them up like this:

1 3 5 7  9 11 13 15 ...
2 4 6 8 10 12 14 16 ...

So the number of terms in these sequence is the same, and we'll say that
they each have A terms.  Now what happens if I take both of these
sequences together?  I get the sequence 1,2,3,4,5,6,7,8,....  And here's
the funky part:  this sequence, which looks like it should have twice as
many terms as either of the first two sequences, actually turns out to
have the same number of terms as each of the first two sequences, as I 
can demonstrate by matching them up like this:

1 3 5 7  9 11 13 15 ...
2 4 6 8 10 12 14 16 ...
1 2 3 4  5  6  7  8 ...

So if you believe this stuff, we've just shown that infinity is a number
that you can add to itself and get itself.  That's a pretty bizarre
characteristic to have in a number, unless it's zero (and infinity
certainly isn't zero!).  So this is one of the properties of numbers that
infinity wouldn't stand up to.  

I hope you keep thinking of good questions like these!

-Ken "Dr." Math
Associated Topics:
High School Number Theory

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