Birinci soru için ipucu:
$x,y\in\mathbb{Q}\Rightarrow (\exists (a,b),(c,d)\in\mathbb{Z}\times\mathbb{Z}\setminus \{0\})\left(x=\frac{a}{b}\right)\left(y=\frac{c}{d}\right)$
$$\frac{a}{b}+\frac{c}{d}:=\frac{a\cdot d+b\cdot c}{b\cdot d}\ldots (1)$$
$$a,d\in\mathbb{Z}\Rightarrow a\cdot d\in\mathbb{Z}\ldots (2)$$
$$b,c\in\mathbb{Z}\Rightarrow b\cdot c\in\mathbb{Z}\ldots (3)$$
$$b,d\in\mathbb{Z}\setminus \{0\}\Rightarrow b\cdot d\in\mathbb{Z}\setminus \{0\}\ldots (4)$$
$$a\cdot d\in\mathbb{Z}, b\cdot c\in\mathbb{Z}\Rightarrow a\cdot d+b\cdot c\in\mathbb{Z}\ldots (5)$$
$(1)$ nolu ifade tanım diğerleri teorem.