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An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered). [1]
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45°. [1]
In mathematics, the plastic ratio is a geometrical proportion close to 53/40.Its true value is the real solution of the equation x 3 = x + 1.. The adjective plastic does not refer to the artificial material, but to the formative and sculptural qualities of this ratio, as in plastic arts.
The gamma function is an important special function in mathematics.Its particular values can be expressed in closed form for integer and half-integer arguments, but no simple expressions are known for the values at rational points in general.
The harmonic seventh interval, also known as the septimal minor seventh, [2] [3] or subminor seventh, [4] [5] [6] is one with an exact 7:4 ratio [7] (about 969 cents). [8] This is somewhat narrower than and is, "particularly sweet", [9] "sweeter in quality" than an "ordinary" [10] just minor seventh, which has an intonation ratio of 9:5 [11] (about 1018 cents).
In the last two of the five versions of "Promenade" from Pictures at an Exhibition by Modest Mussorgsky, 7 4 is mixed irregularly with other metres: (4th Promenade) 5 4, 6 4, and 7 4, with a single 3 4 bar at the end; (5th Promenade) four pairs of regularly alternating 5 4 and 6 4, then an irregular mixture of 5 4, 6 4, and 7 4 to the end. [17]