Pi, Irrational Numbers

Date: 29 Dec 1994 19:48:56 -0500
From: Megin Charner
Subject: For Sydney, from Sam

Dear Sydney,

        Please tell me: What is Pi and how does it work?

What are rational numbers and what are irrational numbers? How do they work?

A joke: Why was the algebra book so sad?

Thanks Partner,
                 Sam

Answer: Because it had a lot of problems......

he he he he he hehe he eheheheheheheehehehheehheeh

Date: 30 Dec 1994 13:20:34 -0500
From: Dr. Sydney
Subject: Re: For Sydney, from Sam

Dear Sam, 

        Hey there!  How's it going?  Sorry it took me a while to write
back.  I've been pretty busy here with Christmas activities.  Anyway, 
I'm glad you wrote back!

        Pi is a number approximately equal to 3.14; that is, by definition,
the ratio of a circle's circumference to its diameter.  The circumference 
of a circle is the distance around the edge of the circle.  The diameter of 
the circle is the length of the line that starts at one point on the circle,
then goes through the center of the circle, and then goes through the 
point directly opposite the original point on the circle.  I wish I could 
show you with a picture, but it is hard to draw on the computer, so maybe 
your mom, dad, or sister could draw you a picture.  

         Anyway, early on, mathematicians realized that no matter how big 
or small a circle is, if you divide the circumference by the diameter, you 
always get the same number.  

         What is so interesting about this number is that it is an infinite 
decimal.  That means it has infinitely many numbers behind it's decimal 
point.  Unlike numbers like 3, 9.876, and 4.5, which have finitely many 
nonzero numbers to the right of the decimal place, pi has infinitely many 
numbers to the right of the decimal point.  Computers have calculated pi 
to lots and lots of decimal places (I'm not sure of an exact figure, sorry!).  

        Another interesting thing about pi is that if you write it down in
its decimal form (which we know is an infinite decimal), the numbers 
to the left of the 0 follow no pattern whatsoever.  You see, some infinite 
decimals have patterns.  For instance the infinite decimal .3333333... 
which has all 3's to the right of the decimal point, has a definite pattern -- 
every number is 3. Likewise, the number .123456789123456789123456789... 
also has a pattern -- the sequence 123456789 is repeated.  Pi, on the other 
hand, has no such patterns.  Many mathematicians have tried without 
success to find patterns.  There are many other neat aspects to pi, but I 
don't want to overload you with stuff now. Perhaps one of the other math 
doctors will jump in with something about pi.

         On to your other question...before I can answer this, I must make 
sure you understand what an INTEGER is.  An integer is a positive or negative 
whole number.  So, 2, -28, and 0 are integers, but 2.5, pi, and -9.90 are not.
A RATIONAL number is a number that can be expressed as a fraction 
where the numerator (the top number of the fraction) and the denominator 
(the bottom number in the fraction) are both INTEGERS.  Numbers like 
3, 2.5, and even -.333... are rational numbers because they can be 
represented as fractions with integer numerators and denominators.  
3 = 3/1 and 2.5 = 5/2, and -.333... = -1/3.  Showing this last one is a little 
tricky.  You can test it out on a calculator, by dividing 1 by 3.  You should 
get .333...  

         Some numbers cannot be represented as a fraction with integer 
numerators and denominators.  These numbers are called IRRATIONAL 
numbers.  It can be proven that numbers with square roots, like the square 
root of 2, are irrational.  That means the square root of 2 cannot be written 
as a fraction where the numerator and denominator are integers. 

As it turns out, there are a lot more irrational numbers than there are
rational numbers.  There are an infinite number of both kinds of numbers,
but there are many many more irrational numbers than rational numbers.
That's kind of fun to think about, yes?

Well, I hope this helps. Write back if you have any more questions.  

Happy New Year!

--Sydney, Dr. "math rocks" Foster