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.

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

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

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

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

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

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

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.