The number system is the foundation of mathematics. In Class 9 we expand our understanding from integers and fractions all the way to irrational and real numbers.
Natural numbers (N): 1, 2, 3, ... Counting numbers.
Whole numbers (W): 0, 1, 2, 3, ...
Integers (Z): ..., -2, -1, 0, 1, 2, ...
Rational numbers (Q): numbers of the form p/q where p and q are integers and q is not 0. Their decimal expansions are either terminating (e.g. 3/4 = 0.75) or non-terminating repeating (e.g. 1/3 = 0.333...).
Irrational numbers: numbers that cannot be written as p/q. Their decimal expansions are non-terminating and non-repeating. Examples: root(2), root(3), pi.
Real numbers (R): the union of rational and irrational numbers. Every point on the number line represents a real number.
Key fact: Between any two rational numbers there exists another rational number (dense property).
Is 0.3535... (0.35 repeating) rational or irrational?
Let x = 0.353535...
Then 100x = 35.3535...
Subtracting: 99x = 35, so x = 35/99. Since it can be written as p/q, it is rational.
Locate root(2) on the number line.
Draw a right-angled triangle with both legs = 1 unit. The hypotenuse = root(12 + 12) = root(2). Using a compass, transfer this length to the number line from 0.
Simplify root(45) - 3 · root(20) + 4 · root(5).
root(45) = 3 · root(5), root(20) = 2 · root(5).
So: 3 · root(5) - 3 · (2 · root(5)) + 4 · root(5) = 3 · root(5) - 6 · root(5) + 4 · root(5) = root(5).
Rationalise 1 / (root(3) - root(2)).
Multiply numerator and denominator by (root(3) + root(2)):
= (root(3) + root(2)) / (3 - 2) = root(3) + root(2).
Express 0.47474... as p/q.
Let x = 0.474747...
100x = 47.4747...
99x = 47, so x = 47/99.
Find two irrational numbers between 1 and 2.
root(2) ≈ 1.414... and root(3) ≈ 1.732... Both lie between 1 and 2 and are irrational.
Evaluate (root(5) + root(3))2.
= (root(5))2 + 2 · root(5) · root(3) + (root(3))2 = 5 + 2 · root(15) + 3 = 8 + 2 · root(15).
- Key Formulas:
- (a + b)(a - b) = a2 - b2 (used for rationalisation)
- Laws of exponents: am · an = am+n; (am)n = amn; a1/n = n-th root of a.
Common mistakes
Students often assume that the sum of two irrational numbers is always irrational. In fact (root(2)) + (-root(2)) = 0, which is rational. Also, do not confuse terminating decimals with non-terminating repeating decimals — both are rational.
Summary
Real numbers include all rationals and irrationals. Rational numbers have terminating or repeating decimal forms. Irrational numbers cannot be expressed as p/q. The real number line is complete — every point corresponds to exactly one real number.