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 π is irrational In 1761, Lambert proved that is irrational by first showing that this continued fraction expansion holds: Then Lambert proved that if "x" is non-zero and rational then this expression must be irrational. Since tan(/4) = 1, it follows that /4 is irrational and therefore that is irrational. A simplification of Lambert's proof is given below. This proof uses the characterization of as the smallest positive number whose half is a zero of the cosine function and it actually proves that is irrational. As in many proofs of irrationality, the argument proceeds by reductio ad absurdum. Consider the sequences ("A") and