<Dd> ⋯ = ((− 2 m, − 2 n)) = ((− m, − n)) = ((m, n)) = ((2 m, 2 n)) = ⋯ . (\ displaystyle \ cdots = ((- 2m, - 2n)) = ((- m, - n)) = ((m, n)) = ((2m, 2n)) = \ cdots .) </Dd> <P> The canonical choice for ((m, n)) is chosen so that n is positive and gcd (m, n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime . For example, we would write ((1, 2)) instead of ((2, 4)) or ((− 12, − 24)), even though ((1, 2)) = ((2, 4)) = ((− 12, − 24)). </P> <P> We can also define a total order on Q. Let ∧ be the and - symbol and ∨ be the or - symbol . We say that ((m, n)) ≤ ((m, n)) if: </P> <Dl> <Dd> (n 1 n 2> 0 ∧ m 1 n 2 ≤ n 1 m 2) ∨ (n 1 n 2 <0 ∧ m 1 n 2 ≥ n 1 m 2). (\ displaystyle (n_ (1) n_ (2)> 0 \ \ land \ m_ (1) n_ (2) \ leq n_ (1) m_ (2)) \ \ lor \ (n_ (1) n_ (2) <0 \ \ land \ m_ (1) n_ (2) \ geq n_ (1) m_ (2)).) </Dd> </Dl>

Write and explain different properties of rational numbers