Rational dependence explained

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} independent \\ \underbrace{ \overbrace{ 3 , \sqrt{ 8 } }, 1+ \sqrt{ 2 } }\\ dependent \\ \end{matrix} Because if we let x =3 , y = \sqrt{ 8 } , then 1+ \sqrt{ 2 } = (1)/(x + 3) (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} + … + 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} + … + 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

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