Properties of operations of rational numbers, For any rational numbers a, b, c.-

(i) Rational numbers are closed under addition, multiplication and subtraction,i.e.,(a + b),(a x b) and (a - b) are also rational numbers.

(ii) Rational numbers follow the commutative law of addition and multiplication, i.e., a + b = b + a and a × b = b × a.

(iii) Rational numbers follow the associative law of addition and multiplication, i.e., (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).

(iv) Additive identity : 0 is the additive identity for rational numbers as a + 0 = 0 + a = 0.

(v) Multiplicative identity : 1 is the multiplicative identity for rational numbers as a × 1 = 1 × a = a.

(vi) Additive inverse : For every rational number 'a', there is a rational number '–a' such that a + (–a) = 0.

(vii) Multiplicative inverse : For every rational number 'a' except 0, there is a rational number \(\frac{1}{a}\) such that a x \(\frac{1}{a}\)=1.

(viii) Distributive property : Multiplication distributes over addition in rational numbers,i.e., a (b + c) = a × b + a × c.

(ix) Between any two different rational numbers, there are infinitely many rational numbers. Rational numbers between any two given rational numbers a and b are q₁ = \(\frac{1}{2}(a+b)\), q₂= \(\frac{1}{2}(q_1+b)\), q₃= \(\frac{1}{2}(q_2+b)\) and so on.