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, then.

Formal definition

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.