Musical instrument material composition, their size, etc. However, the most common academic method, Hornbostel-Sachs, uses the means by which they produce sound. The academic study of musical instruments is called organology. A musical instrument makes sounds. Once humans moved from making sounds with their bodies—for example, by clapping—to using objects to create music from sounds, musical instruments were born. Primitive instruments were probably designed to emulate natural sounds, and their purpose was ritual rather than entertainment. The concept of melody and the artistic pursuit of musical composition were unknown to early players of musical instruments. A player sounding a flute to signal
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