Musical instrument first device considered a musical instrument is disputed. The oldest object that some scholars refer to as a musical instrument, a simple flute, dates back as far as 67,000 years. Some consensus dates early flutes to about 37,000 years ago. However, most historians believe that determining a specific time of musical instrument invention is impossible due to the subjectivity of the definition and the relative instability of materials used to make them. Many early musical instruments were made from animal skins, bone, wood, and other non-durable materials. Musical instruments developed independently in many populated regions of the world. However, contact
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