Proof that e is irrational
The number e was introduced by Jacob Bernoulli in 1683. More than half a century later, Euler, who had been a student of Jacob's younger brother Johann, proved that e is irrational; that is, that it cannot be expressed as the quotient of two integers. Euler's proof Euler wrote the first proof of the fact that e is irrational in 1737 (but the text was only published seven years later). He computed the representation of e as a simple continued fraction, which is Since this continued fraction is infinite and every rational number has a terminating continued fraction, e is irrational. A short proof of the previous equality is known. Since the simple continued fraction of e is not periodic, this also proves that e is not a root of a quadratic polynomial with rational coefficients; in particular, e2 is irrational. Fourier's proof The most well-known proof is Joseph Fourier's proof by contradiction, which is based upon the equality Initially e is assumed to be a rational number of the form . The idea is to then analyze the scaled-up difference (here denoted x) between the series representation of e and its strictly smaller partial sum, which approximates the limiting