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 e is irrational numbers. Still another proof can be obtained from the fact that Define formula_11 as follows: Then: which implies: for any integer formula_15 Note that formula_16 is always an integer. Assume formula_17 is rational, so, formula_18 where formula_19 are co-prime and formula_20 It's possible to appropriately choose formula_21 so that formula_22 is an integer i.e. formula_23 Hence, for this choice, the difference between formula_22 and formula_16 would be an integer. But from the above inequality, that's impossible.So, formula_17 is irrational. This means that formula_27 is irrational. In 1840, Liouville published a proof of the fact that "e" is irrational followed by