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 fraction, "e" is irrational. A short proof of the previous equality is known. Since the simple continued fraction of "e" is not periodic, this also proves that "e" is not a root of second degree polynomial with rational coefficients; in particular, "e" is irrational. The most well-known proof is Joseph Fourier's proof by contradiction, which is based upon the equality Initially "e" is assumed to be a rational number of the form ⁄. Note that "b" could not be equal to 1 as "e" is not an integer. It can be shown using the above equality that "e" is strictly