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.

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

Formal definition

The real numbers ω₁, ω₂, ... , ω_n are said to be rationally dependent if there exist integers k₁, k₂, ... , 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: ω₁, ω₂, ... , ω_n are rationally independent if the only n-tuple of integers k₁, k₂, ... , 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.

See also

• Linear flow on the torus

Bibliography

• Anatole Katok and Boris Hasselblatt (1996). Introduction to the modern theory of dynamical systems. Cambridge. ISBN 0-521-57557-5.