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.
- [math]\displaystyle{ \begin{matrix} \mbox{independent}\qquad\\ \underbrace{ \overbrace{ 3,\quad \sqrt{8}\quad }, 1+\sqrt{2} }\\ \mbox{dependent}\\ \end{matrix} }[/math]
Because if we let [math]\displaystyle{ x=3, y=\sqrt{8} }[/math], then [math]\displaystyle{ 1+\sqrt{2}=\frac{1}{3}x+\frac{1}{2}y }[/math].
Formal definition
The real numbers ω1, ω2, ... , ωn are said to be rationally dependent if there exist integers k1, k2, ... , kn, not all of which are zero, such that
- [math]\displaystyle{ k_1 \omega_1 + k_2 \omega_2 + \cdots + k_n \omega_n = 0. }[/math]
If such integers do not exist, then the vectors are said to be rationally independent. This condition can be reformulated as follows: ω1, ω2, ... , ωn are rationally independent if the only n-tuple of integers k1, k2, ... , kn such that
- [math]\displaystyle{ k_1 \omega_1 + k_2 \omega_2 + \cdots + k_n \omega_n = 0 }[/math]
is the trivial solution in which every ki 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.
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