Rational dependence

In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationally dependent. For instance we have the following example. In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients. A collection of numbers which is not rationally independent is called rationally dependent. For instance we have the following example. Because if we let x = 3 , y = 8 {displaystyle x=3,y={sqrt {8}}} , then 1 + 2 = 1 3 x + 1 2 y {displaystyle 1+{sqrt {2}}={frac {1}{3}}x+{frac {1}{2}}y} . The real numbers ω1, ω2, ... , ωn are said to be rationally dependent if there exist integers k1, k2, ... , kn, not all of which are zero, such that If such integers do not exist, then the vectors are said to be rationally independent. This condition can be reformulated as follows: ω1, ω2, ... , ωn are rationally independent if the only n-tuple of integers k1, k2, ... , kn such that is the trivial solution in which every ki is zero. The real numbers form a vector space over the rational numbers, and this is equivalent to the usual definition of linear independence in this vector space.