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 Proof that π is irrational In the 1760's, Johann Heinrich Lambert proved that the number (pi) is irrational: that is, it cannot be expressed as a fraction "a"/"b", where "a" is an integer and "b" is a non-zero integer. In the 19th century, Charles Hermite found a proof that requires no prerequisite knowledge beyond basic calculus. Three simplifications of Hermite's proof are due to Mary Cartwright, Ivan Niven, and Nicolas Bourbaki. Another proof, which is a simplification of Lambert's proof, is due to Miklós Laczkovich. In 1882, Ferdinand von Lindemann proved that is not just irrational, but transcendental as well.