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Pi = C / D (circumference / diameter) . I have read that if circumference can be expressed as an integer then diameter cannot and vice-versa, so that the ratio can never be expressed as a/b where both a,b are integers & hence Pi is irrational. However as far as i know a ratio of two decimals can always be expressed as a ratio of two integers by adding 0s after the deciaml. eg- 56.89 / 23 can always be written as 5689/2300 . So we should also always be able to express C/D as integers . Why then is Pi irrational ? Can't we always measure the diameter and circumference accurately and express them as integers in the ratio ?

rahul891
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    Irrational numbers, i.e. numbers you can't express as the ratio a/b, a,b integers, are without exception infinite and non-repeating after the decimal point in their decimal representations. So you have $3.14159265...$ and you can't express it as $314159265.../100000000...$. There's no ending in the decimal representation. – user26486 Mar 07 '15 at 15:43
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    You may find the answers here useful. – Unochiii Mar 07 '15 at 15:43
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    The fact that $\pi$ can be approximated as accurately as wanted by rational numbers does not contradict the irrationality of $\pi$. – hardmath Mar 07 '15 at 15:44

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There are two possible answers, depending on which of two possible points is confusing you:

  1. It isn't true that a ratio of any two "decimals" can be expressed as a ratio of two integers, if by "decimal" you really do mean any decimal expansion at all. The problem is basically that some decimal expansions are infinite, so your idea of multiplying by a power of ten to clear the decimal point doesn't always work. It only works for finite decimal expressions.

  2. If you already know that your idea only works for finite decimal expansions, maybe you just don't realize that not only can you never find a circle such that both $C$ and $D$ are integers, you actually can't even find a circle such that both $C$ and $D$ have finite decimal expansions.

Jack M
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A real number is rational, if it is the ratio of two integers. Otherwise it is called irrational.

If $D$ is the diameter of circle, then $C= \pi D$ is it's circumference. If $\pi$ is irrational and $D$ rational, then $C$ is irrational. If it were rational, then $C/D = \pi$ would be rational.

"Turning the argument around": We know, that the circumference of a circle with rational diameter is always irrational. Therefore $C/D = \pi$ is irrational. If it were rational, then $C = \pi D$ would be too.

So regarding your last question:

No! You cannot measure both diameter AND circumference of a circle with arbitrary accuracy. At least one of them is irrational and therefore does not have a representation as the ratio of two integers.

Stefan Perko
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    This does not address OP's misunderstanding that you can just 'remove the decimal point' and divide by a corresponding power of ten to get a fraction. – user26486 Mar 07 '15 at 17:10
  • It should. I defined, what a number being irrational means. – Stefan Perko Mar 07 '15 at 17:18
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    All he said in his misunderstandings is that you can just remove the decimal point and divide the number by a corresponding power of ten. He said this purely because he doesn't know that $\pi$ is infinite and non-repeating after the decimal point, and he wouldn't have said this otherwise since it is obvious that you can't remove the decimal point and divide by a corresponding power of ten to get a fraction made up of integers in a number that has an infinite amount of digits after the decimal point. This should've been the first thing someone addresses when answering this question – user26486 Mar 07 '15 at 17:26
  • Hi @StefanPerko !lets forget about π for a min. Is it theoretically impossible to measure two arbitrary lengths correctly ? – rahul891 Mar 12 '15 at 15:17
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    If it's circumference and diameter of a circle, then yes; it's impossible. – Stefan Perko Mar 12 '15 at 16:11
  • @user3573749 The question is somewhat meaningless - if we're talking about mathematics, then "theoretically" and "measure" are two words you shouldn't even be using in the same sentence! If you're measuring lengths, then you're outside of pure mathematics and into the physical world, where things are a lot messier. Philosophically, the problem is that it's not clear what "correct" means in your question. – Jack M Mar 12 '15 at 16:27
  • Yes, i do realize the point you make . I used to think that it has been experimentally observed that the ratio of C to D is always a constant and is irrational, which led to the confusion regarding "measurements" and "accuracy". As @David K pointed out : in reality we cannot make such accurate measurements and that the proof of irrationality of PI is mathematical (and not observational/experimental) in nature. – rahul891 Mar 12 '15 at 18:03
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The question seems to come down to this: "Can't we always measure the diameter and circumference accurately …?"

Suppose you are given a circle whose circumference is either $C = 10.123456789 12345678 1234567 12345678 123456789 23456789 1$ or $C = 10.123456789 12345678 1234567 12345678 123456789 23456789 2$. How would you go about measuring the circumference to determine which of the two equations is true?

In practice you cannot even measure an object in real life to such accuracy that you can give its size an exact finite decimal expansion with the certainty that it cannot be any other finite decimal. The alternative is to use mathematics to determine what $C$ should be, ideally, for some given $D$. If you start with a finite decimal $D$ and do the mathematics correctly, however, you will never reach the last digit of $C$, so you'll never be able to use the "add zeros after the decimal" trick.

David K
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  • That's exactly what I wanted to know ! "Can't we always measure the diameter and circumference accurately …?" Thanks ! – rahul891 Mar 12 '15 at 16:17
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    This seems to me to be a misleading answer, because our ability to measure accurately has, in fact, nothing to do with the issue. Even if we could measure everything with perfect accuracy, we would still find that the circumference is an irrational multiple of the diameter. – WillO Mar 16 '15 at 13:54
  • @WillO The point is that this hypothetical "perfect" accuracy is tantamount to using the mathematical idealization of $\pi$, which implies the "perfect" measurements cannot both be expressed as finite strings of decimal digits, and you can't reduce $C/D$ to a ratio of integers. Of course this statement relies on the fact that it has already been proved by abstract mathematics (not physical measurements) that $\pi$ is irrational. But I didn't set out here to prove the irrationality of $\pi$; rather, I merely present a counterargument to a particular "squaring the circle" fallacy. – David K Mar 16 '15 at 14:11
  • @DavidK: Point taken, though my guess is that this subtlety will be lost on the OP, who is in danger of thinking that if only we had perfect measuring instruments, $\pi$ would cease to be irrational. – WillO Mar 16 '15 at 14:42
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    @WillO I used to think that it has been experimentally observed that the ratio of C to D is always a constant and is irrational, which led to the confusion regarding "measurements" and "accuracy". But David K clearly pointed out that the proof of irrationality of PI is mathematical (and not observational/experimental) in nature :- "In practice you cannot even measure an object in real life to such accuracy.... The alternative is to use mathematics to determine what C should be, ideally, for some given D, If you start ....", which is why I accepted his answer . – rahul891 Mar 16 '15 at 15:27
  • @user3573749: I worried that this would be unclear to you, but I'm glad to find out that I was wrong. Thanks for posting this --- and thanks to DavidK for the answer. – WillO Mar 16 '15 at 15:34
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The problem is if you have a number of form $C/D$ where any of $C$ or $D$ have infinite decimal expansion you cannot do the trick with adding zero - actually you have too many digits, and you can only approximate $C/D$ by replacing $C$ and $D$ by some values that are close to $$C and $D$ respectively but have finite decimal expansion.

Michal Seweryn
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It should come as no surprise that our /accurate/ approximate measurements of circumference and diameter can give us no more than an /accurate/ approximation of pi.

Jonathan Hebert
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