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Mathematics Grade 9

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Chapter 1: Number Systems

Grade 9 Mathematics — Chapter 1

Number Systems

From counting numbers to the vast real number line — explore rational numbers, irrational numbers, decimal expansions, surds, and laws of exponents with full proofs, diagrams, and practice.

✓ 10+ Worked Examples

✓ Practice Sets A – D

✓ Exam Quick-Check

Contents

1. Introduction

2. Number Families

3. Rational Numbers

4. Irrational Numbers

5. Real Numbers

6. Decimal Expansions

7. Operations on Surds

8. Rationalising Denominators

9. Laws of Exponents

10. Worked Examples (10+)

11. Practice Sets A–D

12. Chapter Summary & Exam Tips

Introduction to the Number Line

Imagine standing at zero on an infinite number line and walking in the positive direction. As far as you can see, numbers stretch on forever. Now picture picking up numbers as you walk — first the counting numbers, then zero, then negatives, then fractions — until your bag holds an entire family of numbers. This chapter tells the story of how that family grew from simple counting numbers all the way to the full real number system.

Understanding number systems is fundamental to all higher mathematics. Every equation you will ever solve, every graph you will ever draw, and every formula you will ever use lives inside the real number system explored in this chapter.

Diagram: The Real Number Line

←

|
−3

|
−2

|
−1

|
0

|
1

|
2

|
3

|
√2

→

Negative Integers

Zero

Positive Integers

Irrationals (e.g. √2)

The Number Families

Numbers are organised into families (sets), each one containing the previous. Think of them as nested bags, each bag fitting inside a larger one.

Diagram: Nested Number Sets (N ⊂ W ⊂ Z ⊂ Q ⊂ R)

𝐑 — REAL NUMBERS

Contains ALL rational AND irrational numbers: −√2, π, √5, 0.1011011101…, −3.7, 4/5 …

Irrationals

√2, √3, π, 0.10110111…

𝑄 — RATIONAL NUMBERS (p/q, q≠0)

3/4, −5/7, 0.333…, 2.556 …

ℤ — INTEGERS

…−3, −2, −1, 0, 1, 2, 3…

𝐖 — Whole Numbers: 0, 1, 2, 3…

ℕ — Natural Numbers: 1, 2, 3, 4…

Rational Numbers

Definition

A number r is called a rational number if it can be written in the form p/q, where p and q are integers and q ≠ 0.

The word rational comes from ratio, and the symbol Q comes from the Latin word quotient. Key facts about rational numbers:

Every natural number is rational: for example, 5 = 5/1.

Every integer is rational: for example, −7 = −7/1.

Zero is rational: 0 = 0/1.

Equivalent forms: 1/2 = 2/4 = 10/20 = 25/50. On the number line we always use the form where p and q share no common factor other than 1 (co-prime form).

Infinitely many rationals between any two rationals: Between any two rational numbers r and s, the number (r + s)/2 is always a rational number lying strictly between them. Repeating this process shows there are infinitely many such numbers.

Irrational Numbers

Definition

A number s is called an irrational number if it cannot be written in the form p/q, where p and q are integers and q ≠ 0.

The Pythagoreans of ancient Greece (around 400 BC) were the first to prove that √2 is irrational. Theodorus of Cyrene later showed √3, √5, √6, √7, √10, √11, √12, √13, √14, √15, and √17 are also irrational. The irrationality of π was proved only in the late 1700s by Lambert and Legendre.

Irrational Number	Decimal Expansion (partial)	Why Irrational?
√2	1.41421356237…	Non-terminating, non-recurring
√3	1.73205080757…	Non-terminating, non-recurring
π	3.14159265358979…	Non-terminating, non-recurring
0.10110111011110…	Constructed pattern	Pattern never repeats in a cycle

Important Remark

The symbol √ always means the positive square root. So √4 = 2, not −2, even though both 2 and −2 are square roots of 4. Also note: 22/7 is only an approximation of π; they are not equal (π ≠ 22/7).

Diagram: Locating √2 on the Number Line (Pythagoras Method)

Step 1: Draw unit square OABC

√2

Each side = 1 unit

Diagonal OB = √(1² + 1²) = √2

By Pythagoras theorem

Step 2: Transfer to number line

Place vertex O at zero. Use compass with centre O and radius OB = √2. Draw an arc that cuts the number line at point P. Point P represents √2 ≈ 1.414.

←

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0

|
√2

|
1.5

|
2

|
3

→

Diagram: Locating √3 on the Number Line (Extending the Method)

Starting from the √2 construction, erect a perpendicular BD of length 1 unit at point B (where OB = √2). By the Pythagorean theorem:

OD = √((√2)² + 1²) = √(2 + 1) = √3

Use compass with centre O and radius OD to mark Q on the number line. Q represents √3 ≈ 1.732. In general, √n can be located after √(n−1) has been located.

√2	1
O———A—P—Q
OQ = √3 ≈ 1.732

Real Numbers

Definition

The collection of all rational and irrational numbers together forms the set of Real Numbers, denoted by ℝ. Every real number corresponds to exactly one point on the number line, and every point on the number line represents exactly one real number.

This remarkable one-to-one correspondence between points and real numbers was formally established in the 1870s by German mathematicians Georg Cantor (1845–1918) and Richard Dedekind (1831–1916). Because of this, we call the number line the real number line.

A real number is either rational or irrational — there is no third type. Every real number can be represented as a decimal expansion (which is our topic in the next section).

Decimal Expansions of Real Numbers

The decimal expansion of a number reveals whether it is rational or irrational. There are exactly three types of decimal expansion:

Type 1

Terminating

Remainder becomes 0. Decimal ends after finite digits.

7/8 = 0.875

1/2 = 0.5

→ RATIONAL

Type 2

Non-terminating Recurring

Remainder never reaches 0. Digits repeat in a cycle.

10/3 = 3.333… = 3.3̅

1/7 = 0.142857̅

→ RATIONAL

Type 3

Non-terminating Non-recurring

Decimal goes on forever with NO repeating block.

√2 = 1.41421356…

π = 3.14159265…

→ IRRATIONAL

THEOREM: Characterisation of Rational and Irrational Numbers

The decimal expansion of a rational number is either terminating or non-terminating recurring.

Conversely, any number with a terminating or non-terminating recurring decimal expansion is a rational number.

The decimal expansion of an irrational number is non-terminating non-recurring, and any such number is irrational.

Diagram: Long Division Showing Remainder Patterns

7 ÷ 8 = 0.875 (terminates)

70 ÷ 8 = 8, rem 6

60 ÷ 8 = 7, rem 4

40 ÷ 8 = 5, rem 0 ← stops

Remainders: 6, 4, 0 ✓

10 ÷ 3 = 3.333̅ (recurring)

10 ÷ 3 = 3, rem 1

10 ÷ 3 = 3, rem 1 (repeats!)

Remainders: 1,1,1,… (cycle of 1)

1 ÷ 7 = 0.142857̅ (recurring)

Remainders: 3,2,6,4,5,1

Then: 3,2,6,4,5,1… repeats

Block of 6 digits repeats

Max repeating digits < divisor (7)

Operations on Real Numbers (Surds)

Irrational numbers satisfy commutative, associative, and distributive laws. However, operations between two irrationals do not always produce an irrational result.

Key Facts

(i) rational + irrational = irrational e.g. 2 + √3 is irrational

(ii) nonzero rational × irrational = irrational e.g. 2√3 is irrational

(iii) irrational ± irrational = may be rational OR irrational e.g. √6 + (−√6) = 0 (rational)

Fundamental Surd Identities (a, b > 0)

(i) √(ab) = √a · √b

(ii) √(a/b) = √a / √b

(iii) (√a + √b)(√a − √b) = a − b

(iv) (a + √b)(a − √b) = a² − b

(v) (√a + √b)(√c + √d) = √(ac) + √(ad) + √(bc) + √(bd)

(vi) (√a + √b)² = a + 2√(ab) + b

Rationalising the Denominator

When a fraction has an irrational denominator, we convert it to an equivalent fraction with a rational denominator. This process is called rationalising the denominator. We use the conjugate and surd identities.

Case 1: Single Surd Denominator

To rationalise 1/√a, multiply top and bottom by √a:

1/√2 × √2/√2 = √2/2

Case 2: Binomial Surd Denominator

To rationalise 1/(a + √b), multiply by conjugate (a − √b)/(a − √b):

1/(2+√3) × (2−√3)/(2−√3) = (2−√3)/(4−3) = 2−√3

Rationalising with Opposite Sign Conjugate

For 5/(√3 − √5), multiply by (√3 + √5)/(√3 + √5):

= 5(√3 + √5)/(3 − 5) = 5(√3 + √5)/(−2) = (−5/2)(√3 + √5)

Laws of Exponents for Real Numbers

We define the nth root of a positive real number: if a > 0 and n is a positive integer, then √ⁿa = b means bⁿ = a and b > 0. In exponential notation, √ⁿa = a^1/n.

For rational exponents m/n (where m and n are coprime integers, n > 0): a^m/n = (√ⁿa)^m = √ⁿ(a^m)

Extended Laws of Exponents (a > 0; p, q rational)

Law 1 a^p · a^q = a^p+q

Law 2 (a^p)^q = a^pq

Law 3 a^p / a^q = a^p−q

Law 4 a^p · b^p = (ab)^p

Special cases: a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^1/2 = √a, a^1/3 = √³a

✎

Worked Examples

Example 1True or False: Classification of Number Types

Question: State whether true or false: (i) Every whole number is a natural number. (ii) Every integer is a rational number. (iii) Every rational number is an integer.

(i) FALSE. Zero (0) is a whole number but is NOT a natural number. Natural numbers start at 1.

(ii) TRUE. Any integer m can be written as m/1 (where denominator = 1 ≠ 0), so it satisfies the definition of a rational number.

(iii) FALSE. Consider 3/5. It is rational but not an integer (it lies between 0 and 1).

Example 2Find Five Rational Numbers Between 1 and 2

Method 1 (Midpoint method): The midpoint of 1 and 2 is (1+2)/2 = 3/2. Then midpoints of [1, 3/2] and [3/2, 2] give more. Five rational numbers: 5/4, 3/2, 11/8, 13/8, 7/4.

Method 2 (Scale the denominator): Write 1 = 6/6 and 2 = 12/6. The five fractions 7/6, 8/6, 9/6, 10/6, 11/6 all lie between them. Simplified: 7/6, 4/3, 3/2, 5/3, 11/6.

Example 3Locate √2 on the Number Line

Draw a unit square with vertex at O (zero on the number line). By Pythagoras: diagonal OB = √(1²+1²) = √2. Using a compass centred at O with radius OB, draw an arc to the number line. The arc meets the line at point P, which represents √2 ≈ 1.414.

Example 4Locate √3 on the Number Line

From the √2 construction, erect BD (length 1 unit) perpendicular to OB at B. Then OD = √((√2)²+1²) = √3. Using compass, mark Q on the number line where OQ = √3 ≈ 1.732. This process extends to locate √n for any positive integer n.

Example 5Decimal Expansions of 10/3, 7/8, and 1/7

10/3 = 3.333… Remainders: 1, 1, 1… (divisor 3). The digit 3 repeats. Written as 3.3̅. Non-terminating recurring.

7/8 = 0.875. Remainders: 6, 4, 0. Decimal terminates. Terminating.

1/7 = 0.142857142857… Remainders: 3,2,6,4,5,1,3,2,6,4,5,1… (cycle of 6, < divisor 7). Written as 0.̅1̅4̅2̅8̅5̅7̅. Non-terminating recurring.

Example 6Express 3.142678 as p/q

A terminating decimal is easy to convert. Count the decimal places (6 here) and use 10⁶ as denominator:

3.142678 = 3142678 / 1000000

Example 7Express 0.333… as p/q

Let x = 0.3333… Multiply both sides by 10:

10x = 3.333… = 3 + x ⇒ 9x = 3 ⇒ x = 1/3

Example 8Express 1.272727… as p/q

Let x = 1.272727… Two digits repeat, so multiply by 100:

100x = 127.2727… = 126 + 1.2727… = 126 + x

99x = 126 ⇒ x = 126/99 = 14/11

Example 9Express 0.2353535… as p/q

Let x = 0.2353535… Note: 2 does not repeat, but 35 repeats. Multiply by 100:

100x = 23.53535… = 23.3 + 0.23535… = 23.3 + x

99x = 23.3 = 233/10 ⇒ x = 233/990

Example 10Find an Irrational Number Between 1/7 and 2/7

1/7 = 0.142857… and 2/7 = 0.285714… We need a non-terminating, non-recurring decimal between these values.

One valid answer: 0.150150015000150000… (the pattern 150, 1500, 15000 … adds one more zero each time — never a finite repeating block).

Example 11Simplify Surd Expressions

(i) (5+√7)(2+√5) = 10 + 5√5 + 2√7 + √35

(ii) (5+√5)(5−√5) = 25 − 5 = 20

(iii) (√3+√7)² = 3 + 2√21 + 7 = 10 + 2√21

(iv) (√11−√7)(√11+√7) = 11 − 7 = 4

Example 12Add Surd Expressions

Add (2√2 + 5√3) and (√2 − 3√3):

= (2√2 + √2) + (5√3 − 3√3) = (2+1)√2 + (5−3)√3 = 3√2 + 2√3

Group like surds, then add their coefficients — just like collecting like terms in algebra.

Example 13Rationalise 1/(2+√3)

Multiply numerator and denominator by conjugate (2−√3):

1/(2+√3) = (2−√3)/[(2+√3)(2−√3)] = (2−√3)/(4−3) = 2−√3

Example 14Rationalise 1/(7+3√2)

Multiply by conjugate (7−3√2):

= (7−3√2) / (49 − 18) = (7−3√2)/31

Example 15Simplify using Laws of Exponents

(i) 2^2/3 · 2^1/3 = 2^{(2/3 + 1/3)} = 2¹ = 2

(ii) (3^1/5)⁴ = 3^4/5

(iii) 7^1/5 / 7^1/3 = 7^{(1/5 − 1/3)} = 7^(3−5)/15 = 7^−2/15

(iv) 13^1/5 · 17^1/5 = (13×17)^1/5 = 221^1/5

Practice Set A — Rational Numbers & Number Families

1. Is zero a rational number? Write it in the form p/q where p, q are integers and q ≠ 0.

2. Find six rational numbers between 3 and 4.

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

4. State true or false with reasons: (i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is a whole number.

5. Which of the following are rational? 5/0, −7/1, 0/5, 11/4.

6. Between which two consecutive integers does √50 lie?

▶ Show Answers

1. Yes. 0 = 0/1 (p=0, q=1). Also 0 = 0/2, 0/7, etc.

2. Write 3 = 21/7, 4 = 28/7. Six rationals: 22/7, 23/7, 24/7, 25/7, 26/7, 27/7.

3. Write 3/5 = 18/30, 4/5 = 24/30. Five rationals: 19/30, 20/30, 21/30, 22/30, 23/30 (simplified: 19/30, 2/3, 7/10, 11/15, 23/30).

4. (i) TRUE — Natural numbers (1,2,3,…) are all whole numbers. (ii) FALSE — −3 is an integer but not a whole number. (iii) FALSE — 3/5 is rational but not whole.

5. 5/0 is NOT defined (q=0). −7/1, 0/5, 11/4 are all rational.

6. 7² = 49 and 8² = 64. So √50 lies between 7 and 8.

Practice Set B — Irrational Numbers & Decimal Expansions

1. State true or false: (i) Every irrational number is a real number. (ii) Every point on the number line is of the form √m where m is natural. (iii) Every real number is irrational.

2. Write the decimal form of 36/100, 1/11, 3/13, 2/11, 329/400.

3. Express as p/q: (i) 0.6̅ (ii) 0.4̅7̅ (iii) 0.̅0̅0̅1̅

4. Express 0.99999… as p/q. Are you surprised?

5. Write three numbers with non-terminating non-recurring decimal expansions.

6. Classify: (i) √23 (ii) √225 (iii) 0.3796 (iv) 7.478478… (v) 1.101001000100001…

▶ Show Answers

1. (i) TRUE (ii) FALSE — points like 1.5 exist (iii) FALSE — rational numbers like 1/2 are real.

2. 36/100=0.36 (terminating); 1/11=0.09̅ (recurring); 3/13=0.̅2̅3̅0̅7̅6̅9̅ (recurring); 2/11=0.̅1̅8̅ (recurring); 329/400=0.8225 (terminating).

3. (i) Let x=0.666… 10x=6.666…=6+x → 9x=6 → x=2/3. (ii) Let x=0.4747… 100x=47.47…=47+x → 99x=47 → x=47/99. (iii) Let x=0.001001… 1000x=1.001…=1+x → 999x=1 → x=1/999.

4. Let x=0.9999… 10x=9.999…=9+x → 9x=9 → x=1. Yes, 0.999…=1! This surprises many people but is mathematically exact.

5. Examples: 0.101001000100001…; 0.212212221…; 0.030030003…

6. (i) IRRATIONAL (23 is not a perfect square) (ii) RATIONAL (√225=15) (iii) RATIONAL (terminates) (iv) RATIONAL (recurring block 478) (v) IRRATIONAL (non-terminating, non-recurring)

Practice Set C — Operations on Surds & Rationalisation

1. Classify as rational or irrational: (i) 2−√5 (ii) (3+√23)−√23 (iii) 2√7/(7√7) (iv) 1/√2 (v) 2π

2. Simplify: (i) (3+√3)(2+√2) (ii) (3+√3)(3−√3) (iii) (√5+√2)² (iv) (√5−√2)(√5+√2)

3. Rationalise the denominators: (i) 1/√7 (ii) 1/(√7−√6) (iii) 1/(√5+√2) (iv) 1/(√7−2)

4. Check if 7√5, 7/√5, √2+21, π−2 are irrational.

5. Simplify: (i) 6√5 × 2√5 (ii) 8√15 ÷ 2√3

▶ Show Answers

1. (i) IRRATIONAL (ii) RATIONAL (=3) (iii) RATIONAL (=2/7) (iv) IRRATIONAL (v) IRRATIONAL

2. (i) 6+3√2+2√3+√6 (ii) 9−3=6 (iii) 5+2√10+2=7+2√10 (iv) 5−2=3

3. (i) √7/7 (ii) (√7+√6)/1=√7+√6 (iii) (√5−√2)/3 (iv) (√7+2)/3

4. All four are irrational (each has a non-terminating non-recurring decimal expansion).

5. (i) 6×2×5=60 (ii) (8/2)×(√15/√3)=4√5

Practice Set D — Laws of Exponents

1. Find: (i) 64^1/2 (ii) 32^1/5 (iii) 125^1/3

2. Find: (i) 9^3/2 (ii) 32^2/5 (iii) 16^3/4 (iv) 125^−1/3

3. Simplify: (i) 2^2/3·2^1/5 (ii) (1/3³)⁷ (iii) 11^1/2/11^1/4 (iv) 7^1/2·8^1/2

4. Evaluate: (i) (8/27)^2/3 (ii) (0.001)^1/3 (iii) (32)^0.2

5. If 2^x = 3^y = 6^z, show that 1/x + 1/y − 1/z = 0. [Challenge]

▶ Show Answers

1. (i) 8 (ii) 2 (iii) 5

2. (i) 9^3/2=(9^1/2)³=3³=27 (ii) 32^2/5=(2⁵)^2/5=2²=4 (iii) 16^3/4=(2⁴)^3/4=2³=8 (iv) 125^−1/3=1/5

3. (i) 2^2/3+1/5=2^13/15 (ii) 3⁻²¹=1/3²¹ (iii) 11^1/4 (iv) (56)^1/2=√56=2√14

4. (i) (8/27)^2/3=(2/3)²=4/9 (ii) (0.001)^1/3=(1/1000)^1/3=1/10 (iii) 32^1/5=(2⁵)^1/5=2

5. Let 2^x=3^y=6^z=k. Then 2=k^1/x, 3=k^1/y, 6=k^1/z. Since 6=2×3: k^1/z=k^1/x×k^1/y=k^1/x+1/y. So 1/z=1/x+1/y → 1/x+1/y−1/z=0. □

Chapter Summary

Number Sets

N ⊂ W ⊂ Z ⊂ Q ⊂ R. Natural numbers are the counting numbers. Adding zero gives whole numbers. Adding negatives gives integers. Adding fractions gives rationals. Adding irrationals gives reals.

Rational Numbers

Any number expressible as p/q (p,q integers, q≠0). Decimal expansion is terminating or non-terminating recurring. Infinitely many rationals exist between any two rationals.

Irrational Numbers

Cannot be written as p/q. Decimal expansion is non-terminating non-recurring. Examples: √2, √3, π. They fill the “gaps” between rationals on the number line.

Surd Identities

√(ab)=√a·√b; (√a+√b)(√a−√b)=a−b; (a+√b)(a−√b)=a²−b. Use conjugate to rationalise denominators.

Laws of Exponents

a^p·a^q=a^p+q; (a^p)^q=a^pq; a^p/a^q=a^p−q; a^pb^p=(ab)^p. Extended to rational exponents: a^m/n=(√ⁿa)^m.

Operations

rational+irrational=irrational. nonzero rational × irrational=irrational. irrational ± irrational can be either. Use decimal checking to verify.

⚡ 8-Point Exam Quick-Check

1Zero IS rational (0=0/1) but is NOT a natural number.

222/7 is only an approximation of π. Do NOT write π = 22/7.

3√4 = 2 (positive root only). Never write √4 = ±2.

4Recurring block length in p/q is always less than q (the divisor).

5To express recurring decimal x as p/q: multiply by 10ⁿ where n = number of recurring digits, then subtract x from both sides.

6Rationalise by multiplying numerator and denominator by the conjugate. The denominator becomes a rational using a²−b² identity.

7a^m/n = (√ⁿa)^m. Always simplify the root first, then raise to power — avoids large numbers.

8√(a+b) ≠ √a + √b. This is one of the most common errors. E.g. √(9+16) = √25 = 5, NOT 3+4 = 7.

This comprehensive Grade 9 Maths Number Systems guide covers all topics from the NCERT Chapter 1 curriculum including natural numbers, whole numbers, integers, rational numbers, irrational numbers, real numbers, the real number line, decimal expansions (terminating, non-terminating recurring, and non-terminating non-recurring), locating irrational numbers on the number line using the Pythagorean theorem, operations on real numbers and surds, rationalising the denominator, and laws of exponents for rational exponents. Students preparing for CBSE Class 9 Mathematics exams, board examinations, or competitive entrance tests will find full worked examples, practice exercises with answers, and exam technique tips in this guide. Topics such as converting recurring decimals to p/q form, simplifying surd expressions using conjugate identities, and applying extended laws of exponents to real number bases are explained step-by-step at a Grade 10 accessible level.