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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 ...
A 3-manifold is P²-irreducible if it is irreducible and contains no 2-sided (real projective plane). An irreducible fraction (or fraction in lowest terms) is a vulgar fraction in which the numerator and denominator are smaller than those in any other equivalent fraction.
The Ford circle associated with the fraction / is denoted by [/] or [,]. There is a Ford circle associated with every rational number . In addition, the line y = 1 {\displaystyle y=1} is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity , which is the case p = 1 , q = 0. {\displaystyle p=1,q=0.}
The gamma function evaluated at one half is the square root of pi. It has two different decimal representations in base ten , the familiar 0.5 {\displaystyle 0.5} and the recurring 0.4 9 ¯ {\displaystyle 0.4{\overline {9}}} [ dubious – discuss ] , with a similar pair of expansions in any even base ; while in odd bases, one half has no ...
In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions , such as 5 / 6 = 1 / 2 + 1 / 3 .
Another well-known example is the polynomial X 2 − X − 1, whose roots are the golden ratio φ = (1 + √5)/2 and its conjugate (1 − √5)/2 showing that it is reducible over the field Q[√5], although it is irreducible over the non-UFD Z[√5] which has Q[√5] as field of fractions.
Consider the polynomial Q(x) = 3x 4 + 15x 2 + 10.In order for Eisenstein's criterion to apply for a prime number p it must divide both non-leading coefficients 15 and 10, which means only p = 5 could work, and indeed it does since 5 does not divide the leading coefficient 3, and its square 25 does not divide the constant coefficient 10.
In particular, the unique factorization property of polynomials can be stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials. Vieta's formulas are simpler in the case of monic polynomials: The i th elementary symmetric function of the roots of a monic ...