We think pi is probably a normal number, but we cannot yet prove it merely by looking at the frequency of the digits and strings of digits that appear in its decimal expansion. You might think that similarly, although we think pi is irrational since it doesn't happen to repeat through trillions of digits, maybe at some point it will start?

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It turns out, however, that that's not the case, and we've known for several hundred years now. Johann Lambert first proved that pi is irrational in the mid-18th century. And although the details are a little tough to work out in a 60-second short, the gist of it is that Lambert showed a particular infinite continuing fraction *had* to produce an irrational output if its input was rational (and non-zero). Contrapositively, if the output of that fraction was rational, that had to mean the input was irrational.

Conveniently, that infinite continuing fraction is equivalent to the tangent of the input. And since we know that various fractions of pi have tangent values of 1, we know it would produce a rational output in the infinite continuing fraction form. Therefore, that fraction of pi (π/4) must be irrational. And since we know it's not the 4 making it irrational, it must mean that pi is irrational.

#PiMysteries #IrrationalNumbers #MathHistory

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