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How do we know the value of pi?

The question asks: If pi is not equal to 22/7, how do we know its value? (Question details: Didn’t it derive from the circles, didn’t we acquire it by dividing two numbers? I have a hard time understanding how it is not equal to the ratio that gives it a birth initially and how do we calculate it if it is not equal to that ratio? Did we use the ratio from the circles to find other ways for calculating it? There are proofs here which seems like they generate the number “pi”, but I don’t quite understand them completely.)

My answer: (To vote and comment on Quora, visit the link here.)

There are helpful answers here, but I am afraid the actual question asked here is much more fundamental. The asker is wondering why is it that pi is defined as a ratio between two numbers, but we still can’t accurately express it as a ratio between two numbers! Answering the questions one by one:

Didn’t it derive from the circles, didn’t we acquire it by dividing two numbers?

Ans.: Yes, we derived it from the circles, and we acquired it by dividing two numbers, but of those two, at least one was a non-terminating, non-repeating decimal! If the diameter was 7 cm, we found that the circumference was close to but not 22 cm. It was something we couldn’t find with precision:

To measure pi, we first measured the diameter. A simple, common number  like 7 cm. Then, we measured the circumference with a scale with least  count of a millimetre. We recorded it as 22.0 cm.

Was that accurate? We checked by measuring the circumference with a scale  with least count of a tenth of a millimetre. We recorded it as 21.99 cm.

Then least count of 1/100 of a millimetre. And oh, the circumference came  out to be 21.991 cm. Take that a micrometre, and the circumference is  21.9911 cm.

The decimal places didn’t end even when we reduced the least count by a million times. And so, there you have a ratio which is a ratio, but not a ratio between two beautiful, simple numbers like 22 and 7.

How it is not equal to the ratio that gives it a birth initially and how do we calculate it if it is not equal to that ratio?

Ans.: Pi is definitely equal to the ratio that gave birth to it, but out of the  two components of the ratio (circumference and diameter), there exists at least one which is not known to be a terminating or repeating decimal.

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We never calculated pi as 22/7. The calculations are never ending. We can  only approximate it: by using approximate fractions such as 22/7 (for an intuition of approximations, do read Joshua Engel‘s answer), or by terminating the decimal at our  will (such as 3.14 or 3.1415). Such approximations work well in  everyday calculations. Properties of circles and triangles help us evaluate pi as sums of series accurately… but they again won’t give you two simple numbers like 22 and 7 which were divided by each other to  get pi.

So, pi is not the ratio between circumference and diameter which are two simple rational numbers. Pi is the ratio between two values, one of which must be irrational (non-repeating, non-terminating) if the other is rational. And I hope that explains your query.

1 Response
  • Kanha Batra
    Oct 5, 2014

    Another interesting way to deduce pi though again an interesting one but again only as precise as the number of iterations is through computation, wherein say I am hitting a dart board with some arrows randomly and the dart board has a circle precisely inscribed in square such that all four sides of square are tangent to it then if I measure the area of square it’s (a square) then fir circle the area (pi a square by 4) if I divide the arrows in circle by arrows in square (including the ones in circle as well) the ratio gives me (pi by 4)

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