Properties of Rational Numbers
Closure, commutativity and associativity of rational numbers.
Properties of Rational Numbers — The Complete Picture
The six core properties of rational numbers — closure, commutativity, associativity, the role of identity elements, the existence of inverses, and distributivity — govern every arithmetic operation you perform. In Class 8 Mathematics, mastering these properties means you can justify every algebraic step you take.
Property-by-Property Breakdown
- Closure: Q is closed under +, −, × but NOT ÷ by zero — division by zero is undefined, not irrational.
- Commutativity: a+b = b+a and a×b = b×a hold for all rationals; a−b ≠ b−a and a÷b ≠ b÷a in general.
- Associativity: (a+b)+c = a+(b+c) and (a×b)×c = a×(b×c) hold; this is why bracket placement doesn't matter for +/×.
- Additive identity is 0 (a+0=a); multiplicative identity is 1 (a×1=a) — every rational has a unique additive and multiplicative inverse.
- Distributive law a×(b+c) = a×b + a×c links multiplication and addition — used constantly in algebraic expansion.
Board Exam Strategy
Property questions appear in every CBSE, Telangana and Andhra Pradesh board exam as 1-mark identification, 2-mark verification, and 4-mark proof tasks. Practise writing out both sides of each property equation separately and showing they are equal. One minute of practice per property per day will make you unbeatable on this topic.
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