Rational dependence
# 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.

    ${\begin{matrix}{\mbox{independent}}\qquad \\\underbrace {\overbrace {3,\quad {\sqrt {8}}\quad } ,1+{\sqrt {2}}} \\{\mbox{dependent}}\\\end{matrix}}$

Because if we let $x=3,y={\sqrt {8}}$, then $1+{\sqrt {2}}={\frac {1}{3}}x+{\frac {1}{2}}y$.

## Formal definition

The real numbers $ω_1,$ $ω_2,$ ... , $ω_n$ are said to be rationally dependent if there exist integers $k_1,$ $k_2,$ ... , $k_n,$ not all of which are zero, such that

    $k_{1}\omega _{1}+k_{2}\omega _{2}+\cdots +k_{n}\omega _{n}=0.$

If such integers do not exist, then the vectors are said to be rationally independent. This condition can be reformulated as follows: $ω_1,$ $ω_2,$ ... , $ω_n$ are rationally independent if the only n-tuple of integers $k_1,$ $k_2,$ ... , $k_n$ such that

    $k_{1}\omega _{1}+k_{2}\omega _{2}+\cdots +k_{n}\omega _{n}=0$

is the trivial solution in which every $k_i$ 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.

