Ch 1 of Class 8 formalises the properties of rational numbers. Students revisit closure, commutativity, associativity, and distributivity for all four operations, and learn about additive and multiplicative inverses and the density property.
Rational numbers are closed under addition, subtraction, and multiplication (results are always rational). Addition and multiplication are commutative (a+b = b+a) and associative ((a+b)+c = a+(b+c)). Multiplication distributes over addition: a(b+c) = ab + ac.
Additive identity is 0 (a + 0 = a). Multiplicative identity is 1 (a × 1 = a). Additive inverse of a/b is −a/b (their sum = 0). Multiplicative inverse (reciprocal) of a/b is b/a (their product = 1).
Between any two rational numbers, there are infinitely many rational numbers. To find rationals between a and b, use (a+b)/2 or convert to equivalent fractions with a larger denominator.
Download: https://ncert.nic.in/textbook/pdf/hemh101.pdf | Complete book: https://ncert.nic.in/textbook/pdf/hemh1ps.zip
Division by zero is undefined — it doesn't give a rational number (or any number). So the set of rational numbers is not closed under division. If we exclude division by zero, the result of dividing two rationals is always rational.
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