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The quirky nature of Pi... how is it possible?


jcarlson
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pi is probably the strangest number on the planet.

 

On one hand, Pi is defined as a ratio... that of the Circumference of a circle to its diameter.

 

However, as far as I know, it is a nonrepeating infinite decimal, therefore making it an irrational number, correct?

 

So how is it that you can take a ratio, and get an irrational number out of it?

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[math]\pi = \frac{C}{D}[/math]

 

One of the two (C or D) is an irrational number (if not both). So dividing the two would get you another irrational number: [math]\pi[/math].

 

An example:

 

[math]C = 2\pi[/math], [math]D=2[/math]. [math]\frac{2\pi}{2} = \pi[/math]

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So how can we know that one of the two is irrational, when in order to determine PI, you have to measure the circumference and diameter of a circle, and even with the most advanced instruments, a measurement will never produce an irrational number because anything used to measure has finite precision?

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That's why we can only go up so far. We only know 6.4 digits. The world record for the most digits of [math]\pi[/math] memorized is held by Hiroyuki Goto, who's memorized over 42,000 digits, taking him more than 9 hours to recite!

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Do not confuse mathematics with the things it models. In the "real world" all caclulations are done with rational multiples with respect to some base measurement, but that says nothing about the acutal mathematical properties of numbers, it merely states something about the "real world". Numbers are not things that exsit in any physical sense, but there are things that exist in the physical world that numbers are useful for desribing properties of. God that was an ugly sentence.

 

 

But, basically, pi is a real number, it is irrational, even transcendental, and it is the the ratio of a geometric NOT real world quantities.

 

It can theoretically be evaluated in base ten to arbitrary precision, given enough time by one of the many series formulae for it. pi squared of six is for example the sum of the reciprocals of the squares of the natural numbers.

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