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In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x, denoted ⌈x⌉ or ceil(x). [1]
For example, 10 is a multiple of 5 because 5 × 2 = 10, so 10 is divisible by 5 and 2. Because 10 is the smallest positive integer that is divisible by both 5 and 2, it is the least common multiple of 5 and 2. By the same principle, 10 is the least common multiple of −5 and −2 as well.
Least-width integer types that are guaranteed to be the smallest type available in the implementation, that has at least specified number n of bits. Guaranteed to be specified for at least N=8,16,32,64. Fastest integer types that are guaranteed to be the fastest integer type available in the implementation, that has at least specified number n ...
Rational numbers (): Numbers that can be expressed as a ratio of an integer to a non-zero integer. [3] All integers are rational, but there are rational numbers that are not integers, such as −2/9. Real numbers (): Numbers that correspond to points along a line. They can be positive, negative, or zero.
The nth Ramanujan prime is the least integer R n for which () (/), for all x ≥ R n. [2] In other words: Ramanujan primes are the least integers R n for which there are at least n primes between x and x/2 for all x ≥ R n. The first five Ramanujan primes are thus 2, 11, 17, 29, and 41.
Then, by the well-ordering principle, there is a least element ; cannot be prime since a prime number itself is considered a length-one product of primes. By the definition of non-prime numbers, n {\displaystyle n} has factors a , b {\displaystyle a,b} , where a , b {\displaystyle a,b} are integers greater than one and less than n ...
Suppose, to the contrary, there is an integer that has two distinct prime factorizations. Let n be the least such integer and write n = p 1 p 2... p j = q 1 q 2... q k, where each p i and q i is prime. We see that p 1 divides q 1 q 2... q k, so p 1 divides some q i by Euclid's lemma. Without loss of generality, say p 1 divides q 1.
One integer selected randomly from the first B integers U+003F ? QUESTION MARK: Ceiling ⌈B: Least integer greater than or equal to B: U+2308 ⌈ LEFT CEILING: Floor ⌊B: Greatest integer less than or equal to B: U+230A ⌊ LEFT FLOOR: Shape, Rho ⍴B: Number of components in each dimension of B: U+2374 ⍴ APL FUNCTIONAL SYMBOL RHO: Not ...