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- 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: axb = bxa
- 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
- 1 is the multiplicative identity for rational
- The additive inverse of a is
and the additive inverse of .
- If ,
the reciprocal or multiplicative inverse of ,
and vice versa.
- For all rational numbers, p, q and r, and
is known as the distributive property.