Rational numbers :
Numbers that can be expressed in the form , where p and q are integers and q ≠ 0, are known as rational numbers. The collection of rational numbers is denoted by Q.
These rational numbers satisfies various laws or properties that are listed below:
Rational numbers are closed under addition, subtraction and multiplication. If a, b are any two rational numbers, then and the sum, difference and product of these rational numbers is also a rational number, then we say that rational numbers satisfy the closure law.
Rational numbers are commutative under addition and multiplication. If a, b are rational numbers, then:
Commutative law under addition: a + b = b + a
Commutative law under multiplication: a x b = b x a
Rational numbers are associative under addition and multiplication. If a, b, c are rational numbers, then:
Associative law under addition: a + (b + c) = (a + b) + c
Associative law under multiplication: a(bc) = (ab)c
0 is the additive identity for rational numbers.
1 is the multiplicative identity for rational numbers.
The additive inverse of a rational number is - , and the additive inverse of - is .
If x = 1, then is the reciprocal or multiplicative inverse of , and vice versa.
For all rational numbers, p, q and r, p(q + r ) = pq + pr and p(q - r ) = pq - pr , is known as the distributive property.