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Irrational Pi

Date: 12/22/2001 at 15:12:07
From: Zach
Subject: Pi

Why must pi be irrational?

If you have a circle with a diameter of 1 unit, the circumference 
(pi d) must be equal to 1 times pi or pi, but the circumference has 
two endpoints, the starting and ending point, which are the same point 
on the exterior of the circle. If pi is irrational then the circle 
would never actually be drawn and would never be completed because the 
definition of an irrational number is that it never repeats or 
terminates. How can this be?

If you have a circle with a radius of 1 unit, then the area of the 
circle (pi r squared) is pi and therefore cannot be measured or 
counted. But the circle has a given area, which it can not exceed, so 
how can pi be irrational?

Date: 12/22/2001 at 18:55:50
From: Doctor Jordi
Subject: Re: Pi

Hello, Zach. Thanks for writing.

Consider the following question: 

Suppose I draw a rectangle of any length and any width. I want to 
measure this rectangle to find some unit that will tell me how wide by 
how long it is. However, I'm picky. I don't like fractions. I want my 
measurements to be whole numbers and nothing but whole numbers. If the 
units I choose to measure are "flicks," the only measurements I will 
accept are two flicks, five flicks, or a million and one flicks; 1/2 
of a flick is not an acceptable measurement. If I get a measurement of 
1/2 flicks, well, that's easy to fix: I'll just choose a smaller unit 
to measure, the "fleck."  One fleck is half a flick. So the side that 
I measured as 4 and 1/2 flicks is actually 9 flecks.

Perhaps using flecks I get fractions again. Maybe I measure 2/3 of a 
fleck. No problem. I'll use an even smaller unit, the "flack." One 
flack is a third of a fleck. So 2/3 of a fleck is 2 flacks. Here's the 
picture so far:

    3 flacks make a fleck
    2 flecks make a flick
   (therefore, 6 flacks make a flick)

So here's the problem: if you give me any rectangle, can I always find 
units to measure it, no matter how small, so that I don't have to use 

I'll give you the answer right away: No, I can't do it! I can't do it 
because the sides of my rectangle might be irrational. This is what 
irrationality means. Rational values are those that can be measured 
with fractions.

I suppose you have read the proof that the square root of two is 
irrational (the proof that pi is irrational is much harder). What this 
proves is that if you take a square and measure its sides in some unit 
you cannot find any other unit, no matter how small, to measure the 
square's diagonal without using fractions. This remarkable fact was so 
shocking to the the ancient Pythagoreans, a group of Greek 
mathematicians, that they actually drowned one of their members when 
he spread the secret.

What a dilemma. It looks as if we will have to use fractions after all 
to measure a square's diagonal (or a circle's circumference). But even 
so, the situation is not so pretty. If we could use fractions to 
measure everything, we could just as easily get whole numbers by using 
a smaller measuring unit. Some quantities (diagonals and 
circumferences) can only be *approximated* by fractions. The "real 
value" is somewhere around the fractional value, but not exactly this 

Take a square exactly one inch on each side (enough about those silly 
flicks, flecks, and flacks).  If you use a ruler to measure its 
diagonal, you'll see that it's about 1.4 or 1.5 inches (terminating 
decimals are just fractions, remember). That's just an approximation, 
mind you. If you had a better ruler, with more markings, you might be 
able to measure even better, and see that it's actually about 1.41 or 
1.42 inches. If you had the best ruler in the universe (some sort of 
laser measurement) you would find that the diagonal is about 
1.4142135623730950 inches, and that is *still* an approximation. The 
exact value is the square root of 2, a value that we cannot express as 
a terminating decimal.

That's the present state of affairs: there are certain exact values 
out there that we cannot express as nicely as we would like. It's like 
trying to translate an English poem into another language. The other 
language just doesn't have the correct words for saying exactly what 
the poem says in English, but if you pick the right words, you can 
make the poem sound almost the same. Similarly, the correct, true, and 
precise language for saying how long the diagonal of a unit square is 
"The square root of two," and we can give a rough translation of what 
this means by saying "about 1.4142135623730950488016887242097"

Does this clear any doubts?  Please feel free to write back if you 
would like to talk more about this.

- Doctor Jordi, The Math Forum   
Associated Topics:
High School Number Theory

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