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why is pi an irrational number
Pi's digits never end, but that alone proves nothing. One third never ends either: three, three, three. Irrational means something stricter: no fraction of whole numbers equals it.
People use twenty-two sevenths for pi. That is 3.142857, off by about a thousandth. Even three hundred fifty-five over one hundred thirteen is only close, never exact.
Where does pi come from? Divide any circle's circumference by its diameter and you get pi. So pretend pi is a fraction: a over b, with a and b whole numbers.
Now build a hump: x to the n, times a minus b x to the n, all over n factorial. It is zero at both ends, and a bigger n makes it flatter and smaller.
Now find the area under f times sine of x, from zero to pi. Integrating by parts again and again, every end term is a whole number. So the area is a whole number.
But the hump is positive, and at most pi times a pi to the n over n factorial. Factorials beat powers, so for big n that is below one. No whole number sits strictly between zero and one.
So pi equals no fraction. The chain: suppose a over b, build the hump, get a whole number, trap it between zero and one. Lambert showed it first, published in 1768; Niven found this short proof in 1947.
Pi isn't irrational because its digits roam forever; it is irrational because no fraction can fit, and a clever area proves it.