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Name: Steve S.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: 5/4/2004


Question:
My son, Spencer, wanted to know the answer to the question "INFINITY MINUS INFINITY=????"



Replies:
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


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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