Ask A Scientist Mathematics Archive

name         Steve S.
status       other
age          40s

Question -   My son, Spencer, wanted to know the answer to the question "INFINITY MINUS 
INFINITY=????"
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     The short answer is --Anything you wish. The mathematical concept of
infinity needs to be defined carefully in order to stay out of all sorts of
dilemmas, contradictions, and confusion. First, it has slightly different
definitions depending upon the area of math it is being used -- some
examples:
Numbers like square root of 2, or pi are irrational which means that they
cannot be expressed as a ratio of integers. That is fairly easy to prove. It
means that when the expression for, say pi, is written
pi=3.14159..... the sequence of digits is infinite. There is no pattern.
However, pi - pi = 0. One can also show that 2*pi is infinite in the same
sense. But, 2*pi - pi = pi (not zero).
    When used in the context of "infinite series" it means that there is a
formula adding and infinite number of numbers together according to a
formula, or set of directions. For example:
SUM[n = 1 to infinity] 1/n means 1/1 + 1/2 + 1/3 + ... + 1/1000 + 1/1001
+...+ and so on. If you add up any finite number of terms in this series you
get a number. So:
1/1 + 1/2 + 1/3 = 6/6 + 3/6 + 2/6 = 11/6.
And this is a finite answer. However, the INFINITE
SUM is infinite, that is it just keeps getting larger and larger. In
contrast if you perform the SUM[n = 0 to infinity] (1/2)^n the first few
terms of the series look like
(1/2)^0 + (1/2)^1 + (1/2)^2 + (1/2)^3 + ... = 1 + 1/2 + 1/4 + 1/8 = 8/8 + 4/8 + 2/8 + 1/8 = 
15/8 = 1.875.

However, one can prove that as more and more terms are included in the sum the
SUM ----> 2 (exactly).
    In the first example you see that the term "infinite" refers to the
"answer" you get by doing the SUM, but in the second example it refers, not
to the "answer", but to the number of terms included in the sum.
    It also turns out that some collections containing an infinite number of
numbers ARE LARGER than some other collections containing an infinite number
of numbers!!
    The bottom line is that the concept of "infinity" is one that needs to
be carefully handled in mathematics.

Vince Calder
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Steve S.,
There is no single answer to this, as infinity is not a number in the strictest sense.  
Infinity is a "limit".  It is something you can get closer and closer to, but you never 
quite reach it.  What matters is how fast you approach the first infinity (call it 
infinity01) versus how fast you approach the second infinity (call it infinity02).  If 
you approach both at exactly the same speed, always the same distance from each, then 
infinity01-infinity02=0.  If you are always a little closer to infinity01, then 
infinity01-infinity02 is a positive number.  If you move toward infinity01 twice as 
fast as how you approach infinity02, then infinity01-infinity02=infinity.  The behavior 
of limits is not quite the same as the behavior of numbers.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
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