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.

$\begin{array}{c}\text{independent}\\ \underbrace{\overbrace{3,\sqrt{8}},1+\sqrt{2}}\\ \text{dependent}\\ \end{array}$

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 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

• Baker's theorem
• Dehn invariant
• Gelfond–Schneider theorem
• Hamel basis
• Hodge conjecture
• Lindemann–Weierstrass theorem
• Linear flow on the torus
• Schanuel's conjecture

Bibliography

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