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### Pi, Irrational Numbers

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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

he he he he he hehe he eheheheheheheehehehheehheeh
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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,

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

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.

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
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