Tag Archives: irrational

Pi Day Countdown #11 – Pi and Irrationality

There is a bit of misconception that the Greeks believe that Pi was a rational number (a number that can be expressed as a fraction of two whole numbers). Many believe that it was because the Greeks did not believe in the existence of irrational numbers, yet it was the Greeks themselves who discovered and accepted the existence of Irrational numbers.

JHLambert
Johann Heinrich Lambert, 26 August 1728 – 25 September 1777, was one of the first to prove that Pi is irrational

The truth is, the Greeks approximated \pi between two fractions because that is what they knew how to do. Proof that \pi is an irrational number did not exist until the 17th century where it was proven irrational by the Swiss mathematician Johann Heinrich Lambert.

He did this by first showing that his continued fractal expansion of tan(x) holds, then he shows that if tan(x) is rational then tan(x) will have to be irrational. Since tan(\pi /4) = 1 and 1 is obviously rational, \pi/4 must but irrational and ultimately so is \pi.

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