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 .
The real numbers ÃÂ<sub>1</sub>, ÃÂ<sub>2</sub>, ... , ÃÂ<sub>n</sub> are said to be rationally dependent if there exist integers k<sub>1</sub>, k<sub>2</sub>, ... , k<sub>n</sub>, 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: ÃÂ<sub>1</sub>, ÃÂ<sub>2</sub>, ... , ÃÂ<sub>n</sub> are rationally independent if the only n-tuple of integers k<sub>1</sub>, k<sub>2</sub>, ... , k<sub>n</sub> such that
is the trivial solution in which every k<sub>i</sub> 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.