Irrational PiDate: 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 fractions? 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 fraction. 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 http://mathforum.org/dr.math/ |
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