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The simplest fraction 3 / y with a three-term expansion is 3 / 7 . A fraction 4 / y requires four terms in its greedy expansion if and only if y ≡ 1 or 17 (mod 24), for then the numerator −y mod x of the remaining fraction is 3 and the denominator is 1 (mod 6). The simplest fraction 4 / y with a four-term ...
This order is usually determined by the order in which the elements are added to the structure, but the elements can be rearranged in some contexts, such as sorting a list. For a structure that isn't ordered, on the other hand, no assumptions can be made about the ordering of the elements (although a physical implementation of these data types ...
In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers.The tree is rooted at the number 1, and any rational number q expressed in simplest terms as the fraction a / b has as its two children the numbers 1 / 1+1/q = a / a + b and q + 1 = a + b / b .
The greatest common divisor (GCD) of integers a and b, at least one of which is nonzero, is the greatest positive integer d such that d is a divisor of both a and b; that is, there are integers e and f such that a = de and b = df, and d is the largest such integer.
The notion of irreducible fraction generalizes to the field of fractions of any unique factorization domain: any element of such a field can be written as a fraction in which denominator and numerator are coprime, by dividing both by their greatest common divisor. [7] This applies notably to rational expressions over a field. The irreducible ...
For example, this approach is used in some efficient FFT implementations, where the base cases are unrolled implementations of divide-and-conquer FFT algorithms for a set of fixed sizes. [12] Source-code generation methods may be used to produce the large number of separate base cases desirable to implement this strategy efficiently.
Python: The standard library includes a Fraction class in the module fractions. [6] Ruby: native support using special syntax. Smalltalk represents rational numbers using a Fraction class in the form p/q where p and q are arbitrary size integers. Applying the arithmetic operations *, +, -, /, to fractions returns a reduced fraction. With ...
Upward planarity testing [8] Hospitals-and-residents problem with couples; Knot genus [38] Latin square completion (the problem of determining if a partially filled square can be completed) Maximum 2-satisfiability [3]: LO5 Maximum volume submatrix – Problem of selecting the best conditioned subset of a larger matrix.