Class 9 · Chapter 1

Number Systems

Rational and irrational numbers, decimal expansions, representation on the number line, and laws of exponents for real numbers.

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Detailed theory

A rational number is any number that can be written as

p/q

where

p, q

are integers and

q\ne 0

. Examples:

2 = 2/1, -3/4, 0.5 = 1/2

. Rational numbers are closed under addition, subtraction, multiplication, and division (except by zero).

Key concepts & formulas

Rational:

p/q

where

q \ne 0

; irrational: cannot be so expressed.

Rationals have terminating or non-terminating recurring decimals.

Irrationals have non-terminating, non-recurring decimals.

\sqrt{2}, \sqrt{3}, \sqrt{5}

are irrational.

For real

a > 0

a^m \cdot a^n = a^{m+n}

(a^m)^n = a^{mn}

Between any two rationals lie infinitely many rationals (and irrationals).