Chapter 1 — Rational Numbers

Chapter 1 of the Class 8 Maths NCERT textbook, "Rational Numbers", teaches what rational numbers are (numbers written as p/q where p and q are integers and q ≠ 0), and explores their key properties including closure, commutativity, associativity, and distributivity across addition, subtraction, multiplication, and division operations.

• A richer number system — Rational numbers extend whole numbers and integers to include fractions, giving a system able to solve equations the earlier systems could not — every rational number can be written as p/q with a non-zero denominator.
• How operations behave — The chapter systematically tests closure, commutativity and associativity across the four operations, finding that addition and multiplication behave well while subtraction and division break these rules.
• Identities and density — Zero acts as the additive identity and one as the multiplicative identity, multiplication distributes over addition and subtraction, and between any two rationals lie infinitely many more — the numbers are densely packed.

1. 01A rational number is any number that can be written as p/q where p and q are integers and q ≠ 0. Examples: 2/3, −6/7, 9/−5, and even 0, −2, and 4 (written as 0/1, −2/1, 4/1).
2. 02Rational numbers are closed under addition, subtraction, and multiplication (the sum, difference, and product of any two rational numbers is always a rational number), but NOT closed under division (division by zero is undefined).
3. 03Addition and multiplication of rational numbers are both commutative (a + b = b + a and a × b = b × a), but subtraction and division are NOT commutative.
4. 04Addition and multiplication of rational numbers are both associative (a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c), but subtraction and division are NOT associative.
5. 05Zero is the additive identity (a + 0 = a for any rational number a) and 1 is the multiplicative identity (a × 1 = a for any rational number a).
6. 06Distributivity: Multiplication distributes over addition and subtraction for rational numbers: a(b + c) = ab + ac and a(b − c) = ab − ac.
7. 07Between any two rational numbers there are infinitely many other rational numbers; the mean of two rational numbers always lies between them.