Chapter 3 — Number Systems | Class 9 CBSE Maths 2026-27

Section 3.1

The Number System — Big Picture

All the numbers we use are part of the real number system ℝ. Understanding how these fit together is the foundation of this chapter.

Hierarchy of the real number system — every number you know fits somewhere here

Natural ℕ

1,2,3…

Counting numbers. Start from 1. Used for counting objects.

Whole 𝕎

0,1,2,3…

Natural numbers + zero. Whole numbers include 0.

Integer ℤ

…−2,−1,0,1,2…

Whole numbers + negative numbers. No fractions.

Rational ℚ

p/q

Can be written as fraction p/q where p,q ∈ ℤ and q ≠ 0.

Irrational

√2, π

Cannot be written as p/q. Non-terminating, non-repeating decimals.

Real ℝ

ℚ ∪ Irrat.

All rational AND irrational numbers together form ℝ.

Section 3.2

Rational Numbers

A number is rational if it can be expressed as a fraction p/q where p and q are integers and q ≠ 0.

📐 Definition — Rational Number

A number r is rational if r = p/q where p, q ∈ ℤ and q ≠ 0

Every integer n is rational: n = n/1 · Every terminating decimal is rational · Every repeating decimal is rational

✅ Examples of Rational Numbers

3 = 3/1 | −5 = −5/1 | 0 = 0/1 | ½ = 1/2
0.75 = 3/4 | 0.3̄ = 1/3 | 1.6̄ = 5/3 | −2/7

📌 Between any two rationals, there is always another rational

Given rationals a and b (a < b), the number (a+b)/2 is also rational and lies strictly between a and b. This can be repeated infinitely — there are infinitely many rationals between any two distinct rationals. This is called the density property.

The real number line — green dots are rational, coloured dots are irrational. Irrationals fill the "gaps".

Section 3.3

Irrational Numbers

Irrational numbers cannot be expressed as p/q. Their decimal expansions are non-terminating and non-repeating — they go on forever with no pattern.

📌 Definition — Irrational Number

A real number is irrational if it CANNOT be written in the form p/q where p, q are integers and q ≠ 0.

Equivalently: its decimal expansion is non-terminating AND non-repeating.

Examples: √2 = 1.41421356…, √3 = 1.73205080…, π = 3.14159265…, e = 2.71828182…

✅ Rational — Recognise

0.5 = ½ (terminates)
0.333… = 1/3 (repeats)
0.142857142857… = 1/7 (repeats)
√4 = 2 (whole number)
√9 = 3 (whole number)
√(25/16) = 5/4 (fraction)

❌ Irrational — Recognise

√2 = 1.41421356… (no repeat)
√3 = 1.73205080… (no repeat)
√5 = 2.23606797… (no repeat)
π = 3.14159265… (no repeat)
e = 2.71828182… (no repeat)
∛2 = 1.25992105… (no repeat)

⚠️

Common Mistake: √4 = 2 is rational (NOT irrational) because 4 is a perfect square. Only √n where n is NOT a perfect square gives an irrational number.

Section 3.4

Decimal Representation

Every real number has a decimal expansion. The type of decimal tells you whether the number is rational or irrational.

Decimal Type	Example	As Fraction	Rational?
Terminating (ends)	0.75	3/4	✓ Rational
Non-term. repeating (recurring)	0.333…= 0.3̄	1/3	✓ Rational
Non-term. repeating (long block)	0.142857̄	1/7	✓ Rational
Non-term. NON-repeating	1.41421356…	Cannot	✗ Irrational

Example 1

Convert 0.6̄ (= 0.6666…) to a fraction. [2 Marks]

1

Let x = 0.6666…

2

Multiply both sides by 10:
10x = 6.6666…

3

Subtract original from this:
10x − x = 6.6666… − 0.6666…
9x = 6

4

Solve: x = 6/9 = 2/3

0.6̄ = 2/3

Example 2

Convert 0.2̄3̄ (= 0.232323…) to p/q form. [2 Marks]

1

Let x = 0.232323… (repeating block has 2 digits)

2

Multiply by 100 (2 digits in block):
100x = 23.2323…

3

Subtract:
100x − x = 23.2323… − 0.2323…
99x = 23

4

Solve: x = 23/99

0.2̄3̄ = 23/99

💡

Rule of thumb: If the repeating block has n digits, multiply by 10ⁿ then subtract. 1 digit → ×10, 2 digits → ×100, 3 digits → ×1000.

Section 3.5 — Most Important Proof in this Chapter

Proof: √2 is Irrational

This is the most frequently tested proof in CBSE Class 9. It uses a powerful technique called proof by contradiction. Learn every step.

🔑 Technique: Proof by Contradiction

To prove something is true, assume the opposite is true, then show that assumption leads to a logical impossibility (contradiction). Since the opposite cannot be true, the original statement must be true.

Key Proof

Prove that √2 is irrational. [5 Marks — Full Proof]

1

Assume the opposite (contradiction hypothesis):
Assume √2 is rational. Then √2 = p/q where p, q are integers, q ≠ 0, and p and q have no common factor (HCF = 1) — i.e., the fraction is in lowest terms.

2

Square both sides:
√2 = p/q → 2 = p²/q²
∴ p² = 2q²
💡 This means p² is even (divisible by 2).

3

Deduce p is even:
If p² is even, then p must be even (because if p were odd, p² would also be odd).
So let p = 2m for some integer m.

4

Substitute p = 2m back:
(2m)² = 2q²
4m² = 2q²
∴ q² = 2m²
💡 This means q² is even, so q is also even.

5

Contradiction found!
We have shown both p and q are even. But we assumed p/q is in lowest terms (HCF = 1). If both are even, their HCF ≥ 2 — CONTRADICTION

6

Conclusion:
Our assumption that √2 is rational must be false. Therefore, √2 is irrational. □

√2 is irrational — proved by contradiction. If √2 = p/q (lowest terms), both p and q turn out to be even — contradiction. ∴ √2 ∉ ℚ

Also Proved

Proof that √3 is irrational — same technique [3 Marks]

1

Assume √3 = p/q (lowest terms, HCF(p,q) = 1)

2

Square: 3 = p²/q² → p² = 3q² → p² divisible by 3 → p divisible by 3

3

Let p = 3m: 9m² = 3q² → q² = 3m² → q divisible by 3

4

Both p and q divisible by 3 → HCF ≥ 3 → contradicts HCF = 1 → √3 is irrational □

√3 is irrational — same contradiction argument with factor 3 instead of 2.

⭐

CBSE Exam Strategy: This proof is worth 3–5 marks. Write every step clearly. The most commonly lost marks are: (1) forgetting to state "HCF(p,q) = 1", and (2) not explaining why "p² even → p even". Both must be stated explicitly.

Section 3.6

Density of Rational Numbers

Between any two distinct rational numbers, there is always another rational number. In fact, there are infinitely many.

Example 3

Find 3 rational numbers between 1/3 and 1/2. [2 Marks]

1

Method: Use equivalent fractions with a common denominator.
1/3 = 2/6 and 1/2 = 3/6 — only one fraction between them (5/12 etc). Let's use larger denominator.

2

Convert to 10ths:
1/3 = 10/30 and 1/2 = 15/30

3

Fractions between 10/30 and 15/30:
11/30, 12/30 = 2/5, 13/30

3 rationals between 1/3 and 1/2: 11/30, 2/5, 13/30

The midpoint of two rationals is always rational — and this can be repeated infinitely

📌 General Method — n rationals between a and b

To find n rational numbers between a and b:
Write a = a×(n+1)/(n+1) and b = b×(n+1)/(n+1), then list the n fractions between them.
Alternatively: d = (b−a)/(n+1), then the n numbers are: a+d, a+2d, a+3d, …, a+nd

Section 3.7

The Square Root Spiral

A beautiful geometric construction that produces √2, √3, √4, √5, … using right triangles. Each new hypotenuse is the next square root.

Square Root Spiral — each hypotenuse gives √2, √3, √4, √5, … Building on a right triangle with 1-unit height each time

📌 How the spiral works

Step 1: Draw a right angle with both legs = 1. The hypotenuse = √(1²+1²) = √2
Step 2: Use √2 as the base. Add a perpendicular of height 1. New hypotenuse = √(√2)²+1² = √3
Step 3: Use √3 as base, height 1. Hypotenuse = √4 = 2
General: If previous hypotenuse = √n, next = √(n+1)
This gives us every irrational √2, √3, √5, √6, √7, √8, √10, … geometrically.

NCERT Style · CBSE Pattern

Worked Examples

These cover all examination question types — matching the 2–5 mark CBSE format.

Example 4

Classify each: (a) 22/7 (b) 0 (c) −√5 (d) √16 (e) 3.14159… [2 Marks]

a

22/7 = p/q form, p=22, q=7 → RationalNote: 22/7 ≠ π; it's only an approximation

b

0 = 0/1 → Rational (also Whole & Integer)

c

−√5: √5 = 2.2360679… (non-terminating non-repeating) → −√5 is also irrational → Irrational

d

√16 = 4 = 4/1 → Rational (also Natural, Whole, Integer)

e

3.14159… (π): non-terminating non-repeating → Irrational

Example 5

Show that 3−√2 is irrational given that √2 is irrational. [3 Marks]

1

Assume 3−√2 is rational.
Then 3−√2 = p/q for some integers p, q with q ≠ 0.

2

Rearrange:
√2 = 3 − p/q = (3q−p)/q

3

Since p, q are integers, (3q−p)/q is rational. So √2 would be rational. CONTRADICTION — we know √2 is irrational.

4

Conclusion: Our assumption is false. ∴ 3−√2 is irrational. □

3 − √2 is irrational (proved by contradiction using the known irrationality of √2)

Example 6

Convert 1.3̄1̄ (= 1.313131…) to p/q form. [2 Marks]

1

Let x = 1.313131… (2-digit repeat)

2

Multiply by 100: 100x = 131.313131…

3

Subtract: 100x − x = 131.313… − 1.313…
99x = 130

4

x = 130/99

1.3̄1̄ = 130/99

NCERT Exercise

Exercise 3.1 — Fully Solved

All standard exercise questions with complete working in CBSE-expected format.

Q1 · 1 Mark

Is zero (0) a rational number? Justify.

1

A rational number is of the form p/q, p,q ∈ ℤ, q ≠ 0.

2

0 = 0/1 = 0/2 = 0/5 — it can be written in p/q form with many choices of q.

Yes, 0 is rational: 0 = 0/1.

Q2 · 2 Marks

Find 5 rational numbers between 3/5 and 4/5.

1

Convert to 60ths: 3/5 = 36/60, 4/5 = 48/60

2

Choose 5 fractions between 36/60 and 48/60: 37/60, 38/60, 40/60=2/3, 42/60=7/10, 45/60=3/4

5 rationals: 37/60, 19/30, 2/3, 7/10, 3/4

Q3 · 3 Marks

Prove that 5+√3 is irrational.

1

Assume 5+√3 is rational. Then 5+√3 = p/q (integers, q≠0, HCF=1).

2

Rearrange: √3 = p/q − 5 = (p−5q)/q

3

RHS is rational (integers), so √3 would be rational. But √3 is irrational → Contradiction

4

∴ 5+√3 is irrational. □

Q4 · 3 Marks

Express 0.47̄ (= 0.4777…) as a fraction.

1

Let x = 0.4777… The repeating block is just "7" (1 digit), but there's a non-repeating part too.

2

Multiply by 10: 10x = 4.777…
Multiply by 100: 100x = 47.777…

3

Subtract: 100x − 10x = 47.777… − 4.777…
90x = 43 → x = 43/90

0.47̄ = 43/90

Smart Study

10 Study Tips for Number Systems

These tips target the most common CBSE exam mistakes in this chapter.

1

Every integer is rational

3 = 3/1, −5 = −5/1. Integers are a subset of rationals. When asked "is 7 rational?" — YES.

2

√(perfect square) is rational

√4=2, √9=3, √16=4, √25=5 are rational. Only √n where n is NOT a perfect square is irrational.

3

22/7 ≠ π

22/7 is rational (a fraction). π is irrational. They are only approximately equal. Never say "π = 22/7".

4

Proof: state HCF=1 first

In the √2 proof, always write "let √2 = p/q where HCF(p,q)=1". Forgetting this loses 1 mark.

5

Always say WHY p is even

"p² is even → p is even" needs justification: "if p were odd, p² = odd×odd = odd — contradiction." Write this out.

6

Conversion rule: n repeating digits → ×10ⁿ

1 digit repeating → ×10 and ×1 subtract. 2 digits → ×100 and ×1. 3 digits → ×1000 and ×1.

7

Sum/product of irrationals can be rational

√2 + (−√2) = 0 (rational). √2 × √2 = 2 (rational). So the sum of two irrationals isn't always irrational.

8

Finding n rationals: multiply denominator by (n+1)

Want 4 rationals between 1/2 and 3/4? Multiply: 5/10 and 15/20 → use 20ths: 11/20, 12/20=3/5, 13/20, 14/20=7/10.

9

Spiral: each hyp = √(next number)

In the square root spiral: Hyp 1 = √2, Hyp 2 = √3, Hyp 3 = √4 = 2, Hyp n = √(n+1).

10

Irrational proof template: 4 steps

(1) Assume rational = p/q. (2) Rearrange to isolate the known irrational. (3) Show it would then be rational. (4) State contradiction → proved irrational.

Quick Reference

Chapter 3 — Formula & Fact Sheet

Concept	Rule / Fact	Example
Rational number	p/q, q≠0, p,q∈ℤ	3/4, −2/5, 0.75
Irrational number	Not p/q; non-term, non-repeat decimal	√2, π, e, √5
Terminating decimal	Ends after finite digits → rational	0.75 = 3/4
Recurring decimal → p/q	n repeating digits → multiply by 10ⁿ	0.3̄ = 1/3
Density of rationals	Between any two rationals → another rational	(a+b)/2 always works
√2 irrational	Proved by contradiction: HCF(p,q)=1 violated	√2 = 1.41421356…
a+b irrational	If a rational, b irrational → a+b irrational	3+√2 is irrational
Square root spiral	Hyp n = √(n+1) using 1-unit perpendicular	Hyp 1=√2, Hyp 2=√3

CBSE Pattern Practice

50 Practice Questions

MCQ · 1 Mark · 2 Marks · 3 Marks · 5 Marks · Case-Based — all with full step-by-step solutions.