Musical instrument been used to play four notes of a diatonic scale. Researchers estimate the flute's age at between 43,400 and 67,000 years, making it the oldest known musical instrument and the only musical instrument associated with the Neanderthal culture. However, some archaeologists and ethnomusicologists dispute the flute's status as a musical instrument. German archaeologists have found mammoth bone and swan bone flutes dating back to 30,000 to 37,000 years old in the Swabian Alps. The flutes were made in the Upper Paleolithic age, and are more commonly accepted as being the oldest known musical instruments. Archaeological evidence of musical instruments was
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.