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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 can not 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 rational numbers can't be written as infinite continued fractions, ''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 second degree polynomial with rational coefficients; in particular, ''e''2 is irrational.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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