- #1
flyingpig
- 2,579
- 1
Homework Statement
Q = rational numbers
My professor proved that it is closed under addition yesterday. I kinda understood a bit...
Proof
Let [tex]r_1, r_2, \in \mathbb{Q}[/tex]
[tex]r_1 + r_2 = \frac{m_1}{n_1} + \frac{m_2}{n_2} = \frac{m_1 n_2 + n_1 m_2}{n_1 n_2} \in \mathbb{Q}[/tex]
By letting [tex]m_1 n_2 + n_1 m_2 = m_3[/tex] and [tex]n_3 = n_1 n_2[/tex]
Since [tex]m_1, m_2, n_1, n_2[/tex] are integers, [tex]m_1 n_2 + n_1 m_2, n_1 n_2[/tex] are also integers.
So that
[tex]\frac{m_1 n_2 + n_1 m_2}{n_1 n_2} \in \mathbb{Q}[/tex]
How the heck does it work for when n_1 or n_2 = 0?