Rational and Irrational Numbers

Introduction

Numbers form the foundation of mathematics. In this chapter, we explore two fundamental categories: rational numbers and irrational numbers. Understanding their properties and differences is essential for ICSE Class 9.

Rational Numbers

A rational number is any number that can be expressed in the form p/q, where p and q are integers and q ≠ 0.

Examples: 3/4, -5/7, 2 (which is 2/1), 0.25 (which is 1/4), 0.333... (which is 1/3)

Key Properties:

Rational numbers can be positive, negative, or zero
Every integer is a rational number
Every fraction is a rational number
Terminating decimals are rational numbers
Non-terminating repeating decimals are rational numbers

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

Irrational Numbers

An irrational number is a number that cannot be expressed in the form p/q, where p and q are integers with q ≠ 0.

Examples: √2, √3, √5, π (pi), e (Euler's number)

Key Properties:

Irrational numbers have non-terminating, non-repeating decimal expansions
The set of irrational numbers is not closed under any operation
Sum of a rational and an irrational number is always irrational
Product of a non-zero rational and an irrational number is always irrational

<ICSEExample title="Classify as Rational or Irrational"> Classify each number: (i) √9 (ii) √7 (iii) 0.142857142857... (iv) 2.10110111011110... <Solution> (i) √9 = 3, which is rational (can be written as 3/1) (ii) √7 = 2.64575131... non-terminating non-repeating, hence irrational (iii) 0.142857142857... is a repeating decimal (block 142857), hence rational = 1/7 (iv) 2.10110111011110... shows no repeating pattern, hence irrational </Solution> </ICSEExample>

Representation on the Number Line

Every real number corresponds to a unique point on the number line.

Method: To represent an irrational number like √2 on the number line:

Mark point O at 0 and point A at 1 on the number line
Construct a perpendicular AB of length 1 unit at A
Join OB. By Pythagoras theorem, OB = √(1²+1²) = √2
With O as centre and OB as radius, draw an arc intersecting the number line at P
Point P represents √2

Surds

A surd is an irrational number containing a root symbol (√) that cannot be simplified to a rational number.

Examples: √2, √3, ³√5, √7 Not a surd: √4 = 2 (rational), √9 = 3 (rational)

Types of Surds

Type	Description	Example
Pure surd	Entirely under the root	√7
Mixed surd	Has a rational coefficient	3√5
Quadratic surd	Square root	√11
Cubic surd	Cube root	³√10

Simplification of Surds

Rule: √ab = √a × √b and √(a/b) = √a / √b

Example: Simplify √72 = √(36 × 2) = √36 × √2 = 6√2

Rationalisation of Denominators

Rationalisation is the process of eliminating a surd from the denominator of a fraction.

Method: Multiply both numerator and denominator by the conjugate of the denominator.

The conjugate of (a + √b) is (a — √b).

Proofs That √2, √3, √5 Are Irrational

Proof That √2 Is Irrational

Step 1: Assume √2 is rational. Then √2 = p/q, where p and q are integers with no common factors (coprime) and q ≠ 0.

Step 2: Squaring both sides: 2 = p²/q²

Step 3: Therefore, p² = 2q²

Step 4: This means p² is divisible by 2, so p is divisible by 2. Let p = 2k.

Step 5: Substituting: (2k)² = 2q², so 4k² = 2q², or q² = 2k²

Step 6: This means q² is divisible by 2, so q is also divisible by 2.

Step 7: But if both p and q are divisible by 2, they have a common factor of 2, contradicting our assumption that p and q are coprime.

Conclusion: Our assumption is false. Therefore, √2 is irrational.

The same method can be used to prove √3 and √5 are irrational by replacing 2 with 3 or 5 in the proof.

Common Mistakes With Fixes

Mistake	Correction
Confusing surds with rational numbers	A surd always gives an irrational value (e.g., √4 = 2 is not a surd)
Forgetting to rationalise	Always remove surds from denominator
Simplifying √(a+b) incorrectly	√(a+b) ≠ √a × √b (only products work under a single root)
Adding surds directly	Only like surds can be added (e.g., 2√3 + 5√3 = 7√3)

ICSE Exam Focus

Topic	Marks (approx.)	Frequency
Rational vs Irrational classification	2-3 marks	Very common
Simplification of surds	3-4 marks	Common
Rationalisation	3-4 marks	Very common
Proof that √2 is irrational	2-3 marks	Frequently asked

Self-Test

Q1: Classify as rational or irrational: (i) √25 (ii) 2π (iii) 3.1416 (iv) 0.121221222...

Q2: Simplify: √50 + √18 — √8

Q3: Rationalise: 1/(√7 — √6)

Q4: Prove that √5 is irrational.

Q5: If x = 3 + 2√2, find the value of x + 1/x.

Q6: Express 0.666... in the form p/q.

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