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.
:$\begin \mbox\qquad\\ \underbrace\\ \mbox\\ \end$
Because if we let $x=3, y=\sqrt$, then $1+\sqrt=\fracx+\fracy$.

Formal definition

The real numbers ω₁, ω₂, ... , 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: ω₁, ω₂, ... , 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.

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

* {{cite book , author=Anatole Katok and Boris Hasselblatt , title= Introduction to the modern theory of dynamical systems , publisher= Cambridge , year= 1996 , isbn=0-521-57557-5

Dynamical systems