How to prove that real numbers are dependent over Q? I have the following question.
Are the following numbers:
$$
Y = 1-\sqrt{2}\\
X= 2+7\sqrt{2}
$$
Dependent over $R$
Are they dependent over $Q$?
I don§t understand what does it mean "numbers are dependent" how can numbers be dependent?
 A: I note that the linear-algebra tag was on the original Question.  We can consider the nonzero real numbers $X,Y$ as elements of a vector space over $\mathbb{R}$ or over $\mathbb{C}$.
In either case $X$ and $Y$ are linearly dependent since $X,Y \in \mathbb{R} \subset \mathbb{C}$ and $XY - YX = 0$ is a linear dependence relation.
In the edited version of the Question we are asked if $X,Y$ are linearly dependent over $\mathbb{Q}$.  They are not, rather they are linear independent.  For if $rX + sY = 0$ for rational numbers $r,s$ not both zero, then $rX = -sY$ and (since now both sides are nonzero):
$$-s/r = X/Y = (2+7\sqrt{2})/(1-\sqrt{2}) = -(16+9\sqrt{2})$$
by rationalizing the denominator (multiply numerator and denominator by $1+\sqrt{2}$).  This is clearly impossible since $\sqrt{2}$ is irrational.
A: To show they are dependent over $\mathbb Q$, you have to show that
$$r=\frac YX=\frac {1-\sqrt{2}}{2+7\sqrt{2}}$$
is in $\mathbb Q$, so you assume $r=\frac pq$ for coprime integers $p$ and $q$, and solve for $p$ and $q$. Probably you end up with something like $2p^2=q^2$, but that is not possible, because a square has each prime factor an even number of times in its factorization.
