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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.
Formal definition
The real numbers ?1, ?2, ... , ?n are said to be rationally dependent if there exist integers k1, k2, ... , kn not all zero, such that
If such integers do not exist, then the vectors are said to be rationally independent. This condition can be reformulated as follows: ?1, ?2, ... , ?n are rationally independent if whenever k1, k2, ... , kn are integers such that
we have ki = 0 for i = 1, 2, ..., n, i.e. only the trivial solution exists on the integers.
See also
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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