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.
independent
3
,
8
⏞
,
1
+
2
⏟
dependent
{\displaystyle {\begin{matrix}{\mbox{independent}}\qquad \\\underbrace {\overbrace {3,\quad {\sqrt {8}}\quad } ,1+{\sqrt {2}}} \\{\mbox{dependent}}\\\end{matrix}}}
Because if we let
x
=
3
,
y
=
8
{\displaystyle x=3,y={\sqrt {8}}}
, then
1
+
2
=
1
3
x
+
1
2
y
{\displaystyle 1+{\sqrt {2}}={\frac {1}{3}}x+{\frac {1}{2}}y}
.
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