The question asks: (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.)
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.
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‘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.