Class 9 • Mathematics • Ganita Manjari
🌍 The World of Numbers
Chapter 3 — From ancient tally bones to irrational mysteries: the complete story of how numbers evolved
Natural Numbers
Zero & Integers
Rational Numbers
Irrational Numbers
Real Numbers
Decimal Expansions
Proof by Contradiction
Zero & Integers
Rational Numbers
Irrational Numbers
Real Numbers
Decimal Expansions
Proof by Contradiction
📋 Contents
- 3.1 The Dawn of Mathematics — Natural Numbers
- 3.2 The Revolution of Śhūnya — Zero
- 3.3 Integers — Expanding the Horizon
- 3.4 Rational Numbers — Fractions & Number Line
- 3.5 Irrational Numbers & Proof by Contradiction
- 3.6 Real Numbers — Decimals & Cyclic Patterns
- The Complete Number System Hierarchy
- Chapter Summary
- Exam Practice Questions
3.1 The Dawn of Mathematics — Natural Numbers
Mathematics did not begin in a classroom — it began in the dirt, on tree bark, and on bones. The very first mathematical concept humanity needed was simple: counting.
One-to-One Correspondence: The Birth of Counting
Imagine a herder thousands of years ago on the banks of the Saraswati river. Each morning he placed one pebble in a clay pot for every cow that left, and removed one pebble for every cow that returned. If the pot was empty — the herd was safe! If pebbles remained — cows were missing. This matching of one object to another is called one-to-one correspondence — the birth of Natural Numbers.
Imagine a herder thousands of years ago on the banks of the Saraswati river. Each morning he placed one pebble in a clay pot for every cow that left, and removed one pebble for every cow that returned. If the pot was empty — the herd was safe! If pebbles remained — cows were missing. This matching of one object to another is called one-to-one correspondence — the birth of Natural Numbers.
Natural Numbers: ℕ = {1, 2, 3, 4, 5, …}
🦴 History Written in Bone
🪨 Lebombo Bone (~35,000 years old)
Found in South Africa/Swaziland. Contains 29 carved notches — believed to be a lunar phase counter or calendar. Earliest physical evidence of humans recording numbers.
Found in South Africa/Swaziland. Contains 29 carved notches — believed to be a lunar phase counter or calendar. Earliest physical evidence of humans recording numbers.
🧮 Ishango Bone (~20,000 BCE)
Found near the Nile in Congo. One column groups notches into 11, 13, 17, 19 — the prime numbers between 10 and 20! Another column demonstrates doubling. Abstract concept of numbers is 20,000+ years old!
Found near the Nile in Congo. One column groups notches into 11, 13, 17, 19 — the prime numbers between 10 and 20! Another column demonstrates doubling. Abstract concept of numbers is 20,000+ years old!
The Ishango Bone — Prime numbers 11, 13, 17, 19 carved ~20,000 years ago!
🏛️ The Indian Context
In the Indus Valley cities of Lothal and Harappa, standardised weights were essential for trade. Indian philosophers gave names to all powers of 10 up to 10¹² (parārdha) in the Vedas, and up to 10⁵³ (tallakṣhaṇa) by Buddha’s time. This obsession with large numbers paved the way for the most important invention in mathematics: zero.
Properties of Natural Numbers
Natural numbers are closed under addition and multiplication (result is always a natural number). They are NOT closed under subtraction (e.g., 3 − 5 = −2, which is not a natural number).
Natural numbers are closed under addition and multiplication (result is always a natural number). They are NOT closed under subtraction (e.g., 3 − 5 = −2, which is not a natural number).
3.2 The Revolution of Śhūnya — When Nothing Became Something
For thousands of years, the number line started at 1. If you gave away all your apples, you had no number to represent your state — only a void. The Babylonians and Mayans used placeholder symbols for empty columns, but they never treated “nothing” as an actual number.
From Philosophy to Mathematics: Śhūnyatā
In Indian philosophy (Upanishads, Buddhist literature, Patanjali’s Yoga Sutras), Śhūnya (meaning zero/empty) described the meditative state of emptying the mind of all vṛttis (fluctuations). Indian thinkers deeply revered this concept of “nothingness,” which gave them the conceptual foundation to eventually bring zero into mathematics.
In Indian philosophy (Upanishads, Buddhist literature, Patanjali’s Yoga Sutras), Śhūnya (meaning zero/empty) described the meditative state of emptying the mind of all vṛttis (fluctuations). Indian thinkers deeply revered this concept of “nothingness,” which gave them the conceptual foundation to eventually bring zero into mathematics.
📜 Bakhśhālī Manuscript & Brahmagupta
📄 Bakhśhālī Manuscript (early centuries CE)
Shows the physical transition from a blank space to a symbol — a bold dot (bindu) used to represent zero. The earliest known use of zero as a written symbol.
Shows the physical transition from a blank space to a symbol — a bold dot (bindu) used to represent zero. The earliest known use of zero as a written symbol.
🧮 Brahmagupta (628 CE)
In his Brāhmasphuṭasiddhānta, he formally defined zero as: a − a = 0. He was the first to give rules for arithmetic with zero.
In his Brāhmasphuṭasiddhānta, he formally defined zero as: a − a = 0. He was the first to give rules for arithmetic with zero.
📐 Brahmagupta’s Rules for Zero
a + 0 = a | a − 0 = a | a × 0 = 0
Division by Zero is Undefined!
Brahmagupta tried but could not give a satisfying rule for a ÷ 0. Division by zero is undefined — it breaks mathematics! This is why in every definition involving fractions, we always state denominator ≠ 0.
Brahmagupta tried but could not give a satisfying rule for a ÷ 0. Division by zero is undefined — it breaks mathematics! This is why in every definition involving fractions, we always state denominator ≠ 0.
🌟
India’s Greatest Gift to the World
The decimal place-value system with zero, developed in India and later transmitted through Arabic mathematicians to Europe, is considered by historians as one of the most transformative inventions in human history. Without zero, modern computing, banking, and science would be impossible!
The decimal place-value system with zero, developed in India and later transmitted through Arabic mathematicians to Europe, is considered by historians as one of the most transformative inventions in human history. Without zero, modern computing, banking, and science would be impossible!
3.3 Integers — Expanding the Horizon
Brahmagupta asked: if 5 − 5 = 0, what is 3 − 5? To answer this, he looked at the real world of commerce: debts and fortunes.
💰 Fortunes (Dhana) = Positive Numbers
Wealth, assets, money you have. Moving right on the number line.
Wealth, assets, money you have. Moving right on the number line.
💸 Debts (Ṛiṇa) = Negative Numbers
Loans, losses, money you owe. Moving left on the number line.
Loans, losses, money you owe. Moving left on the number line.
Integers: ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}
ℤ comes from German word Zahlen meaning “numbers”
📐 Brahmagupta’s Laws for Integer Arithmetic
| Rule (Original Words) | Mathematical Form | Real-Life Meaning |
|---|---|---|
| Fortune + Fortune = Fortune | 5 + 4 = 9 | Save ₹5 + earn ₹4 = have ₹9 |
| Debt + Debt = Debt | (−5) + (−4) = −9 | Owe ₹5 + borrow ₹4 = owe ₹9 |
| Fortune − Zero = Fortune | 7 − 0 = 7 | Nothing changes |
| Debt × Fortune = Debt | (−3) × 4 = −12 | Take on 4 debts of ₹3 each |
| Debt × Debt = Fortune | (−3) × (−4) = +12 | Someone removes 4 of your ₹3-debts → you’re ₹12 richer! |
Why Negative × Negative = Positive?
Think of it this way: if a negative number means debt, then multiplying by a negative means removing that debt. If someone takes away (−) four of your debts worth ₹3 each (−3), you are ₹12 richer! So (−3) × (−4) = +12.
Think of it this way: if a negative number means debt, then multiplying by a negative means removing that debt. If someone takes away (−) four of your debts worth ₹3 each (−3), you are ₹12 richer! So (−3) × (−4) = +12.
3.4 Rational Numbers — Fractions & Number Line
As civilisation grew, counting wasn’t enough — we also needed to measure. How do you share a field equally among 3 children? How do you measure half a cup? This need gave rise to fractions and rational numbers.
Rational Number: p/q, where p, q are integers and q ≠ 0
ℚ stands for quotient
📌 Key Properties of Rational Numbers
- All natural numbers, whole numbers, and integers are rational numbers (e.g., 5 = 5/1).
- Rational numbers have infinitely many equivalent forms: −1/3 = −2/6 = −3/9 = … (equivalent fractions)
- Standard form requires p and q to be co-prime (no common factor other than 1) and q > 0.
- Rational numbers are closed under +, −, × and ÷ (except division by zero).
- Both addition and multiplication are commutative and follow the distributive law.
🧮 Laws of Rational Number Arithmetic
| Operation | Rule | Example |
|---|---|---|
| Equality | a/b = c/d if ad = bc | 2/3 = 4/6 since 2×6 = 3×4 = 12 ✓ |
| Addition | a/b + c/b = (a+c)/b | 2/5 + 3/5 = 5/5 = 1 |
| Subtraction | a/b − c/b = (a−c)/b | 7/8 − 3/8 = 4/8 = 1/2 |
| Multiplication | (a/b) × (c/d) = ac/bd | 2/3 × 3/4 = 6/12 = 1/2 |
| Division | (a/b) ÷ (c/d) = ad/bc | 2/3 ÷ 4/5 = 10/12 = 5/6 |
📍 Representing Rational Numbers on the Number Line
To place p/q on the number line, divide each unit interval into q equal parts, then move p parts from 0 (right if positive, left if negative).
Rational numbers fill the gaps between integers — and between each other!
💡 Absolute Value
|x| = Distance of x from 0 on the number line
|5/3| = 5/3 | |−5/3| = 5/3 | |0| = 0
Always non-negative: |x| ≥ 0 for any rational number x.
Distance between a and b on number line = |a − b|
|5/3| = 5/3 | |−5/3| = 5/3 | |0| = 0
Always non-negative: |x| ≥ 0 for any rational number x.
Distance between a and b on number line = |a − b|
🔮 The Density of Rational Numbers
There are Infinitely Many Rationals Between Any Two Rationals!
No matter how close two rational numbers are, you can always find another by taking their average:
Rational number between a and b = (a + b) / 2
Example: Between 1 and 3/2, average = (1 + 3/2)/2 = 5/4 ✓
No matter how close two rational numbers are, you can always find another by taking their average:
Rational number between a and b = (a + b) / 2
Example: Between 1 and 3/2, average = (1 + 3/2)/2 = 5/4 ✓
3.5 Irrational Numbers & Proof by Contradiction
For centuries, mathematicians believed every measurable length could be expressed as a fraction. Then came a shocking discovery: the diagonal of a unit square has a length that no fraction can ever represent.
Baudhāyana & The Crisis
When Indian mathematician Baudhāyana (~800 BCE) wrote his Śhulbasūtra (manual for constructing geometric fire altars), he encountered lengths that defied all fractions. A unit square has a diagonal of length √2 — and √2 cannot be written as p/q!
When Indian mathematician Baudhāyana (~800 BCE) wrote his Śhulbasūtra (manual for constructing geometric fire altars), he encountered lengths that defied all fractions. A unit square has a diagonal of length √2 — and √2 cannot be written as p/q!
Irrational Number: Cannot be expressed as p/q (where p, q are integers, q ≠ 0)
🔬 Proof that √2 is Irrational (Proof by Contradiction)
This proof, first given by Hippasus (~400 BCE), is one of the most elegant proofs in mathematics. We assume the opposite of what we want to prove, and show it leads to a contradiction.
1
Assume √2 is rational. So √2 = p/q in lowest terms (p, q are co-prime integers, q ≠ 0).
2
Square both sides: 2 = p²/q²
3
Multiply by q²: 2q² = p² → p² is even → p must be even. Let p = 2k.
4
Substitute p = 2k: 2q² = (2k)² = 4k² → q² = 2k² → q² is even → q must be even.
5
CONTRADICTION! 💥 Both p and q are even, so they share factor 2. But we said they were co-prime (no common factors)! Our assumption was WRONG.
✓
Conclusion: √2 cannot be expressed as p/q. Therefore, √2 is irrational.
Same method works for √3, √5, √7, √10…
Any √n where n is not a perfect square is irrational. Perfect squares (4, 9, 16, 25…) give rational square roots.
Any √n where n is not a perfect square is irrational. Perfect squares (4, 9, 16, 25…) give rational square roots.
📐 Constructing √2 on the Number Line
- On the number line, mark O (origin) and A at 1 unit. Draw a perpendicular at A.
- Mark point B on the perpendicular so that AB = 1 unit. Join O to B. By Pythagoras: OB = √2.
- With O as centre and OB as radius, draw an arc cutting the number line at P.
- P represents √2 on the number line!
Geometric construction to locate √2 on the number line
🔵 The Story of π and Mādhava’s Infinite Series
Another famous irrational number is π (pi) — the ratio of a circle’s circumference to its diameter. Āryabhaṭa (499 CE) gave the approximation 3927/1250 = 3.1416, but noted it was only an asanna (approximation).
Since π is irrational (proven by Lambert in 1761), no single fraction can ever express it exactly. Mādhava of Sangamagrama (14th century), founder of the Kerala School, discovered the first exact infinite series for π:
π = 4 × (1 − 1/3 + 1/5 − 1/7 + 1/9 − …)
🎯
Why is Mādhava’s Series Revolutionary?
This was the first known infinite series in history — predating European discoveries of similar series by 200+ years. It shows that irrational numbers require infinitely many terms to express exactly, not just a single fraction!
This was the first known infinite series in history — predating European discoveries of similar series by 200+ years. It shows that irrational numbers require infinitely many terms to express exactly, not just a single fraction!
3.6 Real Numbers — Decimals & Cyclic Patterns
Real Numbers (ℝ) = Rational Numbers + Irrational Numbers = the complete, unbroken number line. Every physical measurement in the universe has a home on this line.
Real Numbers ℝ = Rational (ℚ) ∪ Irrational (𝕀)
🔍 How to Tell Rational vs Irrational: Decimal Expansion
✅ Rational → Terminating OR Repeating decimal
3/8 = 0.375 (terminates)
5/11 = 0.454545… = 0.4̄5̄ (repeats)
1/7 = 0.142857142857… (repeats)
3/8 = 0.375 (terminates)
5/11 = 0.454545… = 0.4̄5̄ (repeats)
1/7 = 0.142857142857… (repeats)
❌ Irrational → Non-terminating, Non-repeating
√2 = 1.41421356237…
π = 3.14159265358…
No pattern, no loop, never ends!
√2 = 1.41421356237…
π = 3.14159265358…
No pattern, no loop, never ends!
Predicting Terminating Decimals (without long division!)
For p/q in lowest terms, the decimal terminates if and only if the prime factors of q are only 2, only 5, or both 2 and 5.
Example: 3/20 = 3/(2²×5) → terminates → 0.15 ✓
Example: 4/15 = 4/(3×5) → has factor 3 → non-terminating!
For p/q in lowest terms, the decimal terminates if and only if the prime factors of q are only 2, only 5, or both 2 and 5.
Example: 3/20 = 3/(2²×5) → terminates → 0.15 ✓
Example: 4/15 = 4/(3×5) → has factor 3 → non-terminating!
🔄 Converting Repeating Decimals to p/q Form
Three Cases to Know
Case 1: Pure Repeating (repeating starts right after decimal)
Case 2: General Repeating (some non-repeating digits first)
Case 1: Pure Repeating (repeating starts right after decimal)
Case 2: General Repeating (some non-repeating digits first)
📝 Case 1 — Pure Repeating: Convert 0.6̄ to p/q
Step 1: Let x = 0.666…
Step 2: 1 digit repeats → multiply by 10: 10x = 6.666…
Step 3: Subtract: 10x − x = 6.666… − 0.666…
9x = 6
Step 4: Solve: x = 6/9 = 2/3
Step 2: 1 digit repeats → multiply by 10: 10x = 6.666…
Step 3: Subtract: 10x − x = 6.666… − 0.666…
9x = 6
Step 4: Solve: x = 6/9 = 2/3
📝 Case 2 — Two-digit repeat: Convert 0.4̄5̄ to p/q
Let x = 0.454545…
2 digits repeat → multiply by 100: 100x = 45.4545…
Subtract: 100x − x = 45.4545… − 0.4545…
99x = 45
Solve: x = 45/99 = 5/11
2 digits repeat → multiply by 100: 100x = 45.4545…
Subtract: 100x − x = 45.4545… − 0.4545…
99x = 45
Solve: x = 45/99 = 5/11
📝 Case 3 — General Repeating: Convert 0.16̄ to p/q
Let x = 0.1666… (1 non-repeating, 1 repeating digit)
Step 1: Multiply by 10 (for 1 non-repeating): 10x = 1.666…
Step 2: Multiply by 10 again (for 1 repeating): 100x = 16.666…
Step 3: Subtract first from second:
100x − 10x = 16.666… − 1.666…
90x = 15
Solve: x = 15/90 = 1/6
Step 1: Multiply by 10 (for 1 non-repeating): 10x = 1.666…
Step 2: Multiply by 10 again (for 1 repeating): 100x = 16.666…
Step 3: Subtract first from second:
100x − 10x = 16.666… − 1.666…
90x = 15
Solve: x = 15/90 = 1/6
General Rule for Conversion
If there are m non-repeating and n repeating digits after the decimal:
Multiply by 10m and 10m+n, then subtract. The denominator will be (10n−1) × 10m.
If there are m non-repeating and n repeating digits after the decimal:
Multiply by 10m and 10m+n, then subtract. The denominator will be (10n−1) × 10m.
🔮 The Magic of Cyclic Numbers
The decimal of 1/7 = 0.142857 — and the block 142857 is a cyclic number:
142857 × 1 = 142857
142857 × 2 = 285714 ← same digits, shifted!
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
142857 × 7 = 999999 ← beautiful!
142857 × 2 = 285714 ← same digits, shifted!
142857 × 3 = 428571
142857 × 4 = 571428
142857 × 5 = 714285
142857 × 6 = 857142
142857 × 7 = 999999 ← beautiful!
🌀
Mind-Blowing Fact!
0.9̄ (which means 0.9999…) is exactly equal to 1! Proof: Let x = 0.999… → 10x = 9.999… → 9x = 9 → x = 1. So 0.999… = 1. Many students find this hard to believe, but it’s mathematically true!
0.9̄ (which means 0.9999…) is exactly equal to 1! Proof: Let x = 0.999… → 10x = 9.999… → 9x = 9 → x = 1. So 0.999… = 1. Many students find this hard to believe, but it’s mathematically true!
The Complete Number System Hierarchy
Our number system grew in layers over thousands of years. Each new type of numbers includes all the previous ones (except Irrational, which is separate):
ℝ Real Numbers (Rational + Irrational) — Everything on the number line
𝕀 Irrational: √2, √3, π, e … (non-terminating, non-repeating decimals)
ℚ Rational: p/q (terminating or repeating decimals)
ℤ Integers: …, −2, −1, 0, 1, 2, …
Whole Numbers: 0, 1, 2, 3, …
ℕ Natural Numbers: 1, 2, 3, …
| Number System | Symbol | What it includes | Examples |
|---|---|---|---|
| Natural Numbers | ℕ | Counting numbers | 1, 2, 3, 4, 100 |
| Whole Numbers | W | Natural + Zero | 0, 1, 2, 3 |
| Integers | ℤ | Whole + Negatives | −3, −1, 0, 5 |
| Rational Numbers | ℚ | Integers + Fractions (p/q) | 3/4, −2/5, 0.333…, 1.5 |
| Irrational Numbers | 𝕀 | Non-rational reals | √2, √3, π, e, √10 |
| Real Numbers | ℝ | ℚ ∪ 𝕀 (everything) | All of the above |
Beyond Real Numbers: Imaginary Numbers
What is √(−1)? No real number squared gives a negative result. Mathematicians invented Imaginary Numbers (denoted i, where i² = −1) to solve this. They are essential for electronics, quantum mechanics, and the technology in your mobile phone!
What is √(−1)? No real number squared gives a negative result. Mathematicians invented Imaginary Numbers (denoted i, where i² = −1) to solve this. They are essential for electronics, quantum mechanics, and the technology in your mobile phone!
Chapter Summary
🦴 Natural Numbers (ℕ)
{1, 2, 3, …}. Born from one-to-one correspondence. Evidenced in Ishango Bone (20,000 BCE). Closed under + and ×, not −.
{1, 2, 3, …}. Born from one-to-one correspondence. Evidenced in Ishango Bone (20,000 BCE). Closed under + and ×, not −.
0️⃣ Zero (Śhūnya)
Formalised by Brahmagupta (628 CE) inspired by Indian philosophy. Rules: a+0=a, a−0=a, a×0=0. Division by 0 is undefined.
Formalised by Brahmagupta (628 CE) inspired by Indian philosophy. Rules: a+0=a, a−0=a, a×0=0. Division by 0 is undefined.
↔️ Integers (ℤ)
{…, −2, −1, 0, 1, 2, …}. Brahmagupta’s Dhana (fortunes) and Ṛiṇa (debts). Key rule: (−)×(−) = (+).
{…, −2, −1, 0, 1, 2, …}. Brahmagupta’s Dhana (fortunes) and Ṛiṇa (debts). Key rule: (−)×(−) = (+).
½ Rational Numbers (ℚ)
p/q form, q≠0. Dense (infinitely many between any two). Decimal = terminating or repeating. Include all integers.
p/q form, q≠0. Dense (infinitely many between any two). Decimal = terminating or repeating. Include all integers.
√ Irrational Numbers (𝕀)
Cannot be p/q. Decimal = non-terminating, non-repeating. √2 proved irrational by Hippasus (~400 BCE) using Proof by Contradiction.
Cannot be p/q. Decimal = non-terminating, non-repeating. √2 proved irrational by Hippasus (~400 BCE) using Proof by Contradiction.
ℝ Real Numbers
ℝ = ℚ ∪ 𝕀. Every point on the number line. Terminating/repeating decimal → rational. Non-repeating → irrational.
ℝ = ℚ ∪ 𝕀. Every point on the number line. Terminating/repeating decimal → rational. Non-repeating → irrational.
Quick Formula Sheet
• Rational number between a and b = (a + b) / 2
• Decimal terminates iff denominator has only 2s and 5s as prime factors
• √n is irrational if n is not a perfect square
• |a − b| = distance between a and b on number line
• Converting repeating decimal: use algebra (multiply by power of 10, subtract)
• Rational number between a and b = (a + b) / 2
• Decimal terminates iff denominator has only 2s and 5s as prime factors
• √n is irrational if n is not a perfect square
• |a − b| = distance between a and b on number line
• Converting repeating decimal: use algebra (multiply by power of 10, subtract)
Exam Practice Questions
📝 Short Answer (1–2 Marks)
Q1. Classify: (i) √81 (ii) √12 (iii) 0.33333… (iv) 1.01001000100001…
Ans: (i) √81 = 9 → Rational (ii) √12 = 2√3 → Irrational (iii) 0.3̄ = 1/3 → Rational (iv) Pattern increases by one extra 0 each time — never a repeating block → Irrational
Q2. Without long division, determine if 7/20 terminates or repeats.
Ans: 20 = 2² × 5. Prime factors are only 2 and 5. So 7/20 is a terminating decimal → 7/20 = 0.35 ✓
Q3. Calculate: (i) (−12) × 5 (ii) (−8) × (−7) (iii) 0 − (−14) (iv) (−20) ÷ 4
Ans: (i) −60 (ii) +56 (iii) 0 + 14 = 14 (iv) −5
Q4. Find 3 rational numbers between −1/2 and 1/4.
Ans: Convert to same denominator: −2/4 and 1/4. Three rationals between them: −3/8, 0, 1/8 (or use average method repeatedly).
🧮 Application Questions (3–4 Marks)
Q5. Convert 0.4̄5̄3̄7̄ (digits 45 non-repeating, 37 repeating) to p/q form.
Ans: Let x = 2.4537̄. Multiply by 100: 100x = 245.37̄. Multiply by 10000: 10000x = 24537.37̄. Subtract: 9900x = 24292. So x = 24292/9900 = 6073/2475.
Q6. Prove that √5 is irrational using proof by contradiction.
Ans: Assume √5 = p/q (co-prime). Then 5 = p²/q², so p² = 5q² → p² divisible by 5 → p divisible by 5 → p = 5k. Then 5q² = 25k² → q² = 5k² → q divisible by 5. Both p and q divisible by 5 — contradiction! Therefore √5 is irrational.
Q7. The temperature in Ladakh is 4°C at noon. By midnight it drops 15°C. What is the midnight temperature?
Ans: Midnight temp = 4 + (−15) = 4 − 15 = −11°C
Q8. Show that 0.9̄ = 1 using algebra.
Ans: Let x = 0.9999… Multiply by 10: 10x = 9.9999… Subtract: 10x − x = 9.999… − 0.999… → 9x = 9 → x = 1. Therefore 0.9̄ = 1. ✓
📊 Higher Order Questions (5 Marks)
Q9. Write all steps to construct √3 on the number line using ruler and compass.
Ans: Step 1: Construct √2 first (OA=1, AB=1 perp, OB=√2). Step 2: Now from P (where √2 is marked), draw a perpendicular of length 1. Call the endpoint C. Step 3: OC = √(√2² + 1²) = √(2+1) = √3. Step 4: With O as centre and OC as radius, draw arc to mark √3 on number line.
Common Exam Mistakes to Avoid
① √4 = 2 is rational, but √2 is irrational — always check if it’s a perfect square!
② 0.333… = 1/3 is RATIONAL (it repeats). Only non-repeating, non-terminating is irrational.
③ Denominator of 0 is ALWAYS undefined — never write a/0.
④ In proof by contradiction, you must clearly state what was assumed and where the contradiction arises.
⑤ When converting general repeating decimal, count non-repeating and repeating digits separately!
① √4 = 2 is rational, but √2 is irrational — always check if it’s a perfect square!
② 0.333… = 1/3 is RATIONAL (it repeats). Only non-repeating, non-terminating is irrational.
③ Denominator of 0 is ALWAYS undefined — never write a/0.
④ In proof by contradiction, you must clearly state what was assumed and where the contradiction arises.
⑤ When converting general repeating decimal, count non-repeating and repeating digits separately!
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