Infinity a Concept, Not a Number

Date: 03/16/2003 at 01:08:54
From: Kaiser
Subject: 1/infinity

Hi everybody!

1/infinity = 0

In words, if 1 chocolate bar is divided among an infinite number of 
people, no one gets anything! Where did the chocolate bar go?  
Doesn't it imply that 1/infinity = infinitesimally small?

Date: 03/16/2003 at 02:06:09
From: Doctor Wallace
Subject: Re: 1/infinity

Hello Kaiser,

I think you have the basic idea down, but you may have also fallen 
prey to a common misconception that I'd like to clear up.

Where did you get the idea that 1/infinity = 0?

The very sentence "1/infinity = 0" has no meaning. Why? Because 
"infinity" is a concept, NOT a number. It is a concept that means 
"limitlessness." As such, it cannot be used with any mathematical 
operators. The symbols of +, -, x, and / are arithmetic operators, and 
we can only use them for numbers.

To write 1/infinity and mean "1 divided by infinity" doesn't make any 
sense. 1 cannot be divided by a concept. It can only be divided by 
a number. Similarly, "infinity + 1" or "2 times infinity" are also 
meaningless.

As another example, what does this mean:  "1 / justice = 5"?

That's right! It is as meaningless as "1 / infinity = 0" because 
justice is a concept, not a number.

In math, when you hear people say things like "1 over infinity is 
zero" what they are usually referring to is something called a limit. 
They are just using a kind of shorthand, however. They do NOT mean 
that 1 can actually be divided by infinity. Instead, they mean that, 
if you divide 1 by successively higher numbers, the result becomes 
closer and closer to 0. If I divide 1 by a very large number, like a 
billion, then I get one-billionth, which is a VERY small number, but 
it isn't 0. Since there is no largest number, I can always divide 1 by 
a bigger number. But that will just produce an even smaller number, 
right? It will NEVER produce 0, no matter how high I go. But since the 
answer to the division is getting closer to and closer to 0, we say 
that "the limit of the expression is zero." But we have still not 
divided anything by infinity, since that isn't a number.

To go back to your chocolate bar, what if you divide it among every 
person living on earth? Each person would get roughly 1 six-billionth 
of a chocolate bar. That's a very, very small amount, and you'd 
probably need a microscope to see your piece, but it wouldn't be zero, 
right? Ah, but you asked about dividing it up amongst an infinite 
number of people. Well, we can't. Why? Because infinity isn't a 
number, so you can't show me an infinite number of people. If you try 
to, I will just add one more person, and then we'd realize that the 
number you thought was "infinity" actually wasn't.

So, to finish up, you are perfectly correct in saying that "1/infinity 
= infinitesimally small." But only if you realize that you REALLY mean 
"1 divided by a REALLY big number is a REALLY small number."

Thanks for writing to Dr. Math. Don't hesitate to write again if you 
need further help with this or another question.

- Doctor Wallace, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/17/2003 at 03:32:34
From: Kaiser
Subject: Thank you (1/infinity)

Thanks a lot, Dr Wallace. I knew that a number cannot be divided by a 
concept but I did not remember that 1/infinity is a limit that tends 
to 0.