Dear Straight Dope:
My dream was interrupted in my seventh period physics class when I overheard my teacher say that pi is not an exact number. I would like you to prove that I am right in stating that because it is a relationship between a radius and circumference of a circle, it must be exact. Also, as with e (1+1/k)^k, it can be expressed in an equation. Thank you.
There is no such thing as “not an exact number.” A number is a number. A measurement can be inexact (in fact, all measurements are inexact), but a number–such as 1, or 42, or -2.485, or pi–is a uniquely identified point on the real number line.
Not sure what the teacher meant … that’s the problem with your having heard only part of his/her lecture. Stay awake next time, huh? I can think of a few possibilities.
(a) If the teacher meant that we can only approximate pi in the real world, because it is an infinite nonrepeating decimal, well, OK. But that doesn’t mean you can’t identify it exactly. It just means that pi is a little … different. Most people can handle the idea of repeating decimal fractions, e.g., that 1/3 in decimal terms is 0.3333333333… ad infinitum. A few can even tell you that this is strictly a function of our base-10 counting system. Pi is a little more mysterious, since it’s basically a geometric concept and defies conventional numerical representation. That’s one reason it’s called a transcendent number. But it’s still an exact number.
(b) Now, if the teacher meant that mathematics and reality are different, we’re into philosophy. Mathematics says that a line has no thickness, and that each point on the line represents a real number (and each real number is represented by a point on the line). Now, any line that you draw with a pencil, say, has thickness. Furthermore, the line is drawn with graphite, on paper, so if you look at it under an electron microscope, you will see that it is not smooth and continuous, it’s bumpy and rough and irregular.
If the teacher is saying that pi only exists as a math-theoretic construct, that’s true but not very helpful. The solid table-top only exists as a theoretic construct, too, it’s really got lots of little sub-atomic holes in it.
(c) The final thought is that the teacher might mean something like this: pi is the ratio of circumference to diameter of a circle. However if you draw a circle with a pencil on paper, your measurement of its circumference will be only approximate, and your measurement of its diameter will be only approximate, so the ratio between the two will only be an approximation.
Does that cover it? If not, I don’t know what to tell you, brother. Maybe it’s time to break out the peyote.
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