Irrational Numbers

Date: 01/07/98 at 03:59:41
From: she jianwei
Subject: Irrational numbers

Are pi and the square root of 2 irrational numbers? Why? What about 
1.33333... and 1.181818... ?

Date: 01/07/98 at 06:08:17
From: Doctor Mitteldorf
Subject: Re: Irrational numbers

Dear Xi,

A rational number is one that can be expressed as A/B where A and B 
are integers.  1.33333...= 4/3 and 1.1818... = 13/11.

Sqrt(2) cannot be expressed as A/B. Here's the way the proof is 
usually written: Start with two numbers A and B so A/B = sqrt(2).  
Then reduce the fraction to lowest terms in the usual way, so there is 
no number that can be divided into both A and B.

Now A/B = sqrt(2), so A^2/B^2 = 2.  This means that A^2 must be even. 
So A must be even. So A^2 must really be divisible by 4. So B^2 must 
be even as well. So B must be even. But then A/B must not be a 
fraction in lowest terms, and I asked you before to put the fraction 
in lowest terms.

   That's the proof - it finds a hidden contradiction in the 
assumption that sqrt(2) can be written as a fraction A/B in lowest 
terms.

It's more difficult to prove that pi is irrational, but it is. In 
fact, pi isn't even the square root or the cube root of anything.  
There's no equation that you can write that has only rational numbers 
in it where pi is the solution to the equation. This kind of number is 
called "transcendental."

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

Date: 01/10/98 at 06:17:01
From: she jianwei
Subject: Irrational numbers

Are 1.33333333 and 1.1818181818 (repeated decimals) rational? Why?

Date: 01/14/98 at 04:55:45
From: Doctor Mitteldorf
Subject: Re: Irrational numbers

Dear Xi,

When you divide one integer by another, and compute the result as a 
decimal, there are two things that can happen. 

First, it could come out even after a while: for example, 1/8 is 
exactly 0.125. If you tried to compute more decimal places, they would 
all be zeros.

The second possibility is a repeating pattern. If you divide 4 by 3, 
you get 1.33333... The 3's go on forever. The definition of a rational 
number is any number that can be written as a fraction. Hence
4/3 is rational, and 1.33333... is 4/3.

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