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Dr. Euler's Fabulous Formula by Paul J. Nahin

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Chapter 3

The Irrationality of π2

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3.1 The irrationality of π.

The search for ever more digits of π is many centuries old, but the question of its irrationality seems to date only from the time of Euler. It wasn’t until the 1761 proof by the Swiss mathematician Johann Lambert (1728–1777) that π was finally shown to be, in fact, irrational. Lambert’s proof is based on the fact that tan(x) is irrational if x ≠ 0 is rational. Since tan(π/4) = 1 is not irrational, then π/4 cannot be rational, i.e., π/4 is irrational, and so then π too must be irrational. ...

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