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[14] [15] It is said to be an improper fraction, or sometimes top-heavy fraction, [16] if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, −3/4, and 4/9, whereas examples of improper fractions are 9/4, −4/3, and 3/3.
The calculation can be simplified by adding twelfths together and reducing the fraction beforehand: Rise of tide in three hours = ... 4 / 16: 0.25 12 / 16: 0.75 ...
Pairing up the terms of the series 1 / 2 − 1 / 4 + 1 / 8 − 1 / 16 + ⋯ results in another geometric series with the same sum, 1 / 4 + 1 / 16 + 1 / 64 + 1 / 256 + ⋯. This series is one of the first to be summed in the history of mathematics; it was used by Archimedes circa ...
Today, a more standard phrasing of Archimedes' proposition is that the partial sums of the series 1 + 1 / 4 + 1 / 16 + ⋯ are: + + + + = +. This form can be proved by multiplying both sides by 1 − 1 / 4 and observing that all but the first and the last of the terms on the left-hand side of the equation cancel in pairs.
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
Proofs of the mathematical result that the rational number 22 / 7 is greater than π (pi) date back to antiquity. One of these proofs, more recently developed but requiring only elementary techniques from calculus, has attracted attention in modern mathematics due to its mathematical elegance and its connections to the theory of Diophantine approximations.
Type 376 - 15/16", 1" and 1 1/16" bore diameters, replacing the 276 standard types; Type 389 - 1 1/8", 1 5/32" and 1 3/16" bore diameters, replacing the 289 standard types; The 375 was also available with a cast in air filter in 25/32" and 13/16" bore diameters. These were designated type 363. [17]
On February 25, 1913, Secretary of State Philander Knox proclaimed that the amendment had been ratified by three-fourths of the states and so had become part of the Constitution. [35] The Revenue Act of 1913 , which greatly lowered tariffs and implemented a federal income tax, was enacted shortly after the Sixteenth Amendment was ratified.