Irrational Numbers

Date: 08/13/97 at 22:38:51
From: Julian Potvin-Bernal
Subject: Irrational Numbers

It's Julian's Dad typing this in, but it's truly Julian's question.  
First, I'll tell you how we got to the question... I had my two kids 
working out 1/3 of 100.  Julian thought of using long division, and 
they ended up with 33.333...  which they recognized as an irrational 
number.  (We've been  having fun with that stuff before - I've 
described irrational numbers as being like peanut butter stuck in 
between all the rational numbers.) 

So Julian asks: "How can three irrational numbers added together give 
you a rational number?" Hmmm... I was stuck for an explanation. My 
thinking is that an irrational number can never be exactly defined. 
(That's where I'm probably imprecise in my thinking...?) So how can 
two or more inexactly defined numbers be combined logically to give an 
exact number?

Julian's sleeping in bed now, but he asked me to send this question to 
you.  Thanks for your great site!  (Curiosity question...Are "you" 
many generous people out there, or one amazing individual who answers 
peoples' questions?  Thanks either way.)

Date: 08/14/97 at 08:47:37
From: Doctor Jerry
Subject: Re: Irrational Numbers

Hi Julian, 

It has been proved that pi is not rational and is, by definition, 
irrational.  It has been proved (by Euclid, I think) that sqrt(2) is 
also irrational.  

The numbers pi+sqrt(2), 3*sqrt(2)-pi, and 10-4*sqrt(2) are all 
irrational (this can be proved but at this point you may wish to take 
my word for it).  Note that their sum is 10, a nice rational number.

It's true that pi can't be written out exactly, using a terminating 
decimal, but most of us agree that pi is well-defined.  If you can 
agree, for example, that the circumference of a circle with radius 
1 meter is a well-defined number, then you have accepted the existence 
of 2*pi.

We are many people, not one.  One person would have to be truly 
amazing to handle the numbers of questions we get.

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