Recorder (musical instrument) Recorder (musical instrument) The recorder is a woodwind musical instrument in the group known as "internal duct flutes"—flutes with a whistle mouthpiece. A recorder can be distinguished from other duct flutes by the presence of a thumb-hole for the upper hand and seven finger-holes: three for the upper hand and four for the lower. It is the most prominent duct flute in the western classical tradition. Recorders are made in different sizes with names and compasses roughly corresponding to different vocal ranges. The sizes most commonly in use today are the soprano (aka "descant", lowest note C), alto (aka "treble",
Proof that e is irrational 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 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