Finite vs. Rational

Date: 9/10/96 at 3:17:20
From: Anonymous
Subject: Finite vs. Rational

We know from the Pythagorean theorem that a right triangle with sides 
1 and 2 has a hypotenuse = the square root of 5. Sqrt (5) is 
irrational - it goes on to infinity without recurring - but the side 
length of a triangle must be finite! I'm confused by this!

Date: 9/10/96 at 20:15:57
From: Doctor Tom
Subject: Re: Finite vs. Rational

By "carries on to infinity," you just mean that the decimal expansion
goes on forever without recurring.

The square root of 5 is:

2.23606797749978969641.....

You can write this as:

2 + 2/10 + 3/100 + 6/1000 + 0/10000 + 6/100000 + 7/1000000 + ...

Forever.

Just because you add an infinite list of numbers doesn't mean that
the sum is infinitely big.  For example, suppose you start with a
chunk of cheese that's 1 cubic inch.

Cut it in half.  Then take one of the halves, and cut it in half.
Take one of those, and cut it in half, and so on, forever.  Pretty
soon you'll get pieces that are too tiny to see, but in principle,
you can go forever.   After you've "gone forever," let's see what
you have:

1 chunk of size 1/2
1 chunk of size 1/4
1 chunk of size 1/8
1 chunk of size 1/16
.....

So the whole cheese (1 cubic inch) is made up of all the chunks:

1 = 1/2 + 1/4 + 1/8 + 1/16 + ...

Forever.

Notice that the chunks in the square root of 5 expansion get smaller 
much faster, so it "converges" even more rapidly.

This is a tough concept, but well worth thinking about. You may
not understand it this time, but keep trying. It's a very important
mathematical idea.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/