🔢 Number System
The Complete CTET Guide
A premium, concept-based preparation module for CTET Aspirants, TET Candidates, D.El.Ed Students & Primary Mathematics Teachers
📋 Table of Contents
- Knowing Our Numbers
- Playing with Numbers
- Whole Numbers
- Negative Numbers & Integers
- Fractions
- Pedagogical Focus & Teaching Strategies
- CTET Practice Questions (MCQs, HOTS, Assertion-Reason)
- Mind Map & Revision Notes
🎯 Learning Objectives
By the end of this guide, you will be able to:
- Explain number concepts deeply and connect them to real-life situations
- Use child-centered, activity-based teaching approaches for Number System
- Apply divisibility rules, HCF, LCM, and fraction operations with ease
- Identify and address common student misconceptions in mathematics
- Solve CTET-level MCQs on Number System with speed and accuracy
- Design inclusive and engaging mathematics classrooms at the primary level
Knowing Our Numbers
Place Value • Indian & International System • Roman Numerals • Estimation
🌍 Introduction to Numbers
Numbers are the language of mathematics. From counting apples to measuring distances, numbers are everywhere in daily life. The study of numbers begins with the simple act of counting and gradually expands into a rich, structured system.
Numbers were invented to answer the question "How many?" — Our ancient ancestors used tally marks, pebbles, and fingers before formal number systems were developed. Every number has a face value (the digit itself) and a place value (value based on position).
🇮🇳 Indian vs International Number System
India uses a unique grouping system that differs from the International system. Understanding this difference is critical for CTET.
| Period | Indian System | International System | Example (75,32,14,526) |
|---|---|---|---|
| Ones | Ones, Tens, Hundreds | Ones, Tens, Hundreds | 526 |
| Thousands | Thousands, Ten Thousands | Thousands, Ten Thousands | 14,000 |
| Lakhs | Lakhs, Ten Lakhs | Hundred Thousands, Millions | 32,00,000 |
| Crores | Crores, Ten Crores | Ten Millions, Hundred Millions | 75,00,00,000 |
= 7 Crore 54 Lakh 32 Thousand 891
Groups: Ones(3) → Thousands(2) → Lakhs(2) → Crores(2)
754,328,891
= 754 Million 328 Thousand 891
Groups: Ones(3) → Thousands(3) → Millions(3)
CTET frequently tests the ability to convert between Indian and International naming. For example: 1 Crore = 10 Million and 1 Lakh = 100 Thousand. Memorise these conversions!
📍 Place Value vs Face Value
Consider the number 73,284:
| Digit | Place | Face Value | Place Value |
|---|---|---|---|
| 7 | Ten Thousands | 7 | 70,000 |
| 3 | Thousands | 3 | 3,000 |
| 2 | Hundreds | 2 | 200 |
| 8 | Tens | 8 | 80 |
| 4 | Ones | 4 | 4 |
Face Value = the digit itself (always). Place Value = face value × place (position). Exception: The place value and face value of 0 are ALWAYS 0, regardless of position!
🏛️ Roman Numerals
| Symbol | Value |
|---|---|
| I | 1 |
| V | 5 |
| X | 10 |
| L | 50 |
| C | 100 |
| D | 500 |
| M | 1000 |
- Repeat up to 3 times: III = 3, XXX = 30
- Subtract if smaller before larger: IV = 4, IX = 9, XL = 40, XC = 90, CD = 400, CM = 900
- Only I, X, C, M can be repeated
- V, L, D cannot be repeated or subtracted
I Value Xylophones Like Cows Do Milk → I, V, X, L, C, D, M
🔄 Estimation & Rounding Off
Rounding rules: If the digit to the right of the rounding place is 5 or more → round UP; if less than 5 → round DOWN.
| Number | Rounded to Nearest 10 | Nearest 100 | Nearest 1000 |
|---|---|---|---|
| 4,367 | 4,370 | 4,400 | 4,000 |
| 7,850 | 7,850 | 7,900 | 8,000 |
| 12,499 | 12,500 | 12,500 | 12,000 |
Playing with Numbers
Factors • Multiples • Primes • HCF • LCM • Divisibility Rules
🔢 Factors & Multiples
Factors
Numbers that divide a given number exactly (without remainder). Every number has 1 and itself as factors.
Factors of 12: 1, 2, 3, 4, 6, 12
Multiples
Numbers obtained by multiplying a number by natural numbers (1, 2, 3…). Multiples are infinite.
Multiples of 4: 4, 8, 12, 16, 20…
⭐ Prime & Composite Numbers
Has exactly 2 factors: 1 and itself.
e.g. 2, 3, 5, 7, 11, 13, 17, 19, 23…
Has more than 2 factors.
e.g. 4, 6, 8, 9, 10, 12, 15…
1 = neither prime nor composite.
2 = only even prime number.
Two numbers are co-prime (relatively prime) if their HCF = 1. They need not be prime themselves! e.g. 8 and 15 are co-prime (HCF = 1) even though both are composite.
📏 Divisibility Rules
📌 Example: 348 → last digit 8 → ✅ divisible by 2
📌 Example: 531 → 5+3+1 = 9 → divisible by 3 ✅
📌 Example: 1,532 → 32 ÷ 4 = 8 → ✅
📌 Example: 735 → ends in 5 → ✅
📌 Example: 432 → even ✅ and 4+3+2=9 divisible by 3 ✅ → divisible by 6 ✅
📌 Example: 5,128 → 128 ÷ 8 = 16 → ✅
📌 Example: 729 → 7+2+9 = 18 → divisible by 9 ✅
📌 Example: 5,430 → ✅
📌 Example: 2,178 → (2+7) – (1+8) = 9–9 = 0 → ✅
📐 HCF & LCM
🔵 HCF (Highest Common Factor)
The largest number that divides two or more numbers exactly.
Find HCF of 36 and 48:
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Common factors: 1, 2, 3, 4, 6, 12
HCF = 12
🟠 LCM (Least Common Multiple)
The smallest number that is divisible by two or more numbers.
Find LCM of 4, 6, and 8:
Multiples of 4: 4, 8, 12, 16, 24, 28...
Multiples of 6: 6, 12, 18, 24, 30...
Multiples of 8: 8, 16, 24, 32...
LCM = 24
HCF(a,b) × LCM(a,b) = a × b
Example: HCF(4,6)=2, LCM(4,6)=12 → 2×12 = 4×6 = 24 ✅
- 🛒 HCF: Dividing 48 oranges and 60 bananas into equal groups — HCF(48,60) = 12 groups
- ⏰ LCM: Two buses departing every 12 and 18 minutes — next simultaneous departure in LCM(12,18) = 36 minutes
Give students a number (e.g. 24). Ask them to find ALL its factors using tiles or blocks. Students arrange tiles into rectangular arrays — each valid rectangle represents a factor pair. This builds a visual, tactile understanding of factors.
Whole Numbers
Natural Numbers • Properties • Number Line • Successor & Predecessor
🔄 Natural Numbers vs Whole Numbers
Natural Numbers (N)
Counting numbers: 1, 2, 3, 4, 5…
Starts from 1. Does NOT include zero.
Infinite — no largest natural number.
Whole Numbers (W)
Natural numbers + Zero: 0, 1, 2, 3, 4, 5…
Starts from 0. Includes zero.
Every natural number is a whole number.
Successor = Number + 1 (the number that comes after)
Predecessor = Number − 1 (the number that comes before)
📌 Successor of 99,999 = 1,00,000 | Predecessor of 1,000 = 999
⚠️ Whole number 0 has NO predecessor in whole numbers!
📊 Number Line
Whole Number Line:
→ Numbers increase as we move right | Arrows show the line extends infinitely in both directions
⚙️ Properties of Whole Numbers
| Property | For Addition | For Multiplication | Example |
|---|---|---|---|
| Closure | a + b is a whole number ✅ | a × b is a whole number ✅ | 3 + 5 = 8 ∈ W |
| Commutative | a + b = b + a ✅ | a × b = b × a ✅ | 4 + 7 = 7 + 4 |
| Associative | (a+b)+c = a+(b+c) ✅ | (a×b)×c = a×(b×c) ✅ | (2+3)+4 = 2+(3+4) |
| Identity | a + 0 = a (0 is additive identity) | a × 1 = a (1 is multiplicative identity) | 5 + 0 = 5; 5 × 1 = 5 |
| Distributive | a × (b + c) = a×b + a×c | 3×(4+5) = 3×4 + 3×5 | |
Subtraction and division are NOT commutative for whole numbers. 8 – 3 ≠ 3 – 8, and 12 ÷ 4 ≠ 4 ÷ 12. Also, subtraction and division do not follow the Closure property in whole numbers (3 – 5 = −2, which is NOT a whole number).
Negative Numbers & Integers
Integers • Number Line • Operations • Real-Life Connections
🌡️ Introduction to Integers
Sometimes numbers need to go below zero! When the temperature drops below freezing, when you owe money, or when a submarine dives below sea level — we need negative numbers.
Integers (Z) = { …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … }
Negative Integers
…, −4, −3, −2, −1
Less than zero
Zero
Neither positive
nor negative
Positive Integers
1, 2, 3, 4…
Greater than zero
📊 Integer Number Line
● Negative | ● Zero | ● Positive
🌍 Real-Life Applications of Integers
Temperature
Shimla: −5°C
Delhi: +32°C
Difference = 37°C
Banking
Deposit: +₹500
Withdrawal: −₹200
Balance = +₹300
Elevation
Mt. Everest: +8,848 m
Dead Sea: −430 m
Difference = 9,278 m
➕➖ Operations on Integers
Different signs → Subtract and keep sign of larger: (+7) + (−4) = +3 | (−8) + (+5) = −3
Example: (+5) − (−3) = (+5) + (+3) = +8
Example: (−4) − (+7) = (−4) + (−7) = −11
For integer subtraction, use KCC: Keep the first number, Change subtraction to addition, Change sign of second number.
e.g. −3 − (−8) → Keep −3, Change to +, Change −8 to +8 → −3 + 8 = +5
Fractions
Types • Equivalent Fractions • Comparison • Operations
🍕 What is a Fraction?
Imagine you have a pizza cut into 8 equal slices. You eat 3 slices. What part of the pizza did you eat? That's 3/8 — a fraction!
A fraction represents part of a whole. It is written as Numerator / Denominator
Pizza divided into 8 pieces. 3 eaten = 3/8 eaten, 5/8 remaining
Numerator → Number of parts taken (top number)
Denominator → Total equal parts (bottom number)
Denominator can NEVER be zero! Division by zero is undefined.
🗂️ Types of Fractions
Proper Fraction
Numerator < Denominator
Value is less than 1
e.g. 3/5, 7/8, 1/4
Improper Fraction
Numerator ≥ Denominator
Value is 1 or more
e.g. 7/4, 9/5, 3/3
Mixed Fraction
Whole number + proper fraction
e.g. 2¾ = 2 + 3/4 = 11/4
Example: 3½ = (3×2 + 1)/2 = 7/2
Improper → Mixed: Divide numerator by denominator
Example: 13/4 = 3 remainder 1 = 3¼
🔗 Equivalent Fractions
Fractions that represent the same value are called equivalent fractions.
Create equivalent fractions by multiplying or dividing numerator AND denominator by the same number:
1/2
2/4
4/8 — All equal to 1/2!
To simplify a fraction to its lowest terms, divide both numerator and denominator by their HCF.
e.g. 18/24 → HCF(18,24) = 6 → 18÷6 / 24÷6 = 3/4
➕ Addition & Subtraction of Fractions
Different denominators: Find LCM, convert to equivalent fractions, then add.
Example: 1/3 + 1/4 → LCM(3,4)=12 → 4/12 + 3/12 = 7/12
Give students strips of paper of equal length. Ask them to fold: one into 2 equal parts (halves), another into 4 parts (quarters), another into 8 parts. Label them ½, ¼, ⅛. Now place ¼ + ¼ next to ½ — they match! Students discover that 2/4 = 1/2 physically.
Pedagogical Focus & Teaching Strategies
Child-Centered Teaching • Constructivism • TLMs • Inclusive Strategies
Constructivist Approach
- Let children build knowledge through exploration
- Use manipulatives (beads, blocks, counters)
- Pose open-ended problems before teaching rules
- Allow mistakes — they are learning opportunities
Activity-Based Learning
- Number charts and sorting games
- Factor tree activities with paper folding
- Market simulation for fractions and money
- Hopscotch number line for integers
Teaching Learning Materials (TLMs)
- Abacus for place value
- Fraction kits (circular/rectangular)
- Number cards and dice for factor games
- Floor number lines for integer operations
Inclusive Classroom Strategies
- Peer tutoring and mixed-ability groups
- Multiple representations (visual, tactile, oral)
- Differentiated tasks (tiered assignments)
- Connect maths to students' daily experiences
Common Misconceptions
- "Larger denominator = larger fraction" (1/8 vs 1/2)
- Confusion between HCF and LCM
- "0 is nothing" — not understanding zero as a number
- Thinking −5 > −2 because 5 > 2
Error Analysis in Mathematics
- Ask "how did you get that?" rather than marking wrong
- Identify the type of error: conceptual, procedural, careless
- Use errors as class discussion points
- Celebrate correction — growth mindset in maths
The National Curriculum Framework (NCF 2005) emphasises that mathematics teaching should move from concrete → pictorial → abstract (CPA approach). Children should first manipulate objects, then draw diagrams, and only then work with symbols. CTET questions frequently test this pedagogical progression.
📝 CTET Practice Questions
In 4,27,839 → the 7 is in the Thousands place, so place value = 7 × 1,000 = 7,000.
⚡ TRICK: Count from right → Ones(9), Tens(3), Hundreds(8), Thousands(7) → Place value = 7,000
All numbers have 6 digits. Compare digit by digit from left: First digit = 1 (all same), Second digit: options A,D have 0; options B,C have 1. So B and C are larger. Third digit: B has 0, C has 1. So C (1,10,110) is the largest.
Position 1 = Ones, 2 = Tens, 3 = Hundreds, 4 = Thousands, 5 = Ten Thousands... Wait: In Indian system: 1-3 = Ones period, 4-5 = Thousands period, 6-7 = Lakhs period. 5th from right = Ten Thousands, which belongs to the Thousands period.
⚠️ Note: If the question asks which period — 5th place belongs to Thousands period (Ten Thousands). Check your answer options carefully in the actual exam.
36 = 2²×3², 54 = 2×3³, 72 = 2³×3². HCF = product of lowest powers = 2¹×3² = 2×9 = 18.
⚡ TRICK: HCF must divide all three. 18 divides 36 (2×18), 54 (3×18), 72 (4×18) ✅
12 ÷ 4 = 3 ✅ | 12 ÷ 6 = 2 ✅ | 12 ÷ 8 = 1.5 ❌ (not divisible). Options A, C, D are all divisible by 8.
This is a concrete (hands-on) activity to introduce fractions. The student experiences 3 out of 4 equal parts = 3/4. This follows the CPA approach (Concrete → Pictorial → Abstract) advocated by NCF 2005.
Using: HCF × LCM = Product of two numbers → 5 × 60 = 15 × other → other = 300 ÷ 15 = 20.
⚡ TRICK: Always use HCF×LCM = a×b formula in such questions!
Additive inverse means a + (−a) = 0. For 5, additive inverse is −5. But −5 is NOT a whole number. Therefore, the additive inverse property does NOT hold for whole numbers (it holds for integers).
Temperature at noon = −3 + 11 = +8°C. This is a real-life integer addition problem. Start at −3 on number line, move 11 steps right = land on +8.
3/5 × 4/4 = 12/20 ✅. Check: 12÷4 = 3 and 20÷4 = 5. Verify others: 6/8 = 3/4 ≠ 3/5; 9/20 ≠ 3/5 (9/20 simplest); 15/30 = 1/2 ≠ 3/5.
⚡ TRICK: Cross multiply to verify. 3×20 = 60 and 12×5 = 60 ✅ (Equal cross products = equivalent fractions)
Number = 48q + 31 for some quotient q. Now divide by 16: 48q + 31 = 16(3q) + 31 = 16(3q+1) + 15. So remainder = 15.
⚡ TRICK: Since 48 is a multiple of 16, the remainder when dividing by 16 is the remainder of 31 ÷ 16 = 1 remainder 15 → answer is 15.
Sum of digits of 4,354 = 4+3+5+4 = 16. Next multiple of 9 ≥ 16 is 18. So we need to add 18−16 = 2... wait: 4+3+5+4=16; 18−16=2. Let's recheck: 4354÷9 = 483.7...; 484×9=4356; 4356−4354=2. Answer should be 2. Correct answer: add 2 to get 4356, which is divisible by 9.
This is a classic conceptual error. The student does not understand that a larger denominator means smaller parts. The fraction 1/3 means "one piece when divided into 3 parts," which is smaller than 1/2. Teachers should use fraction strips or pizza diagrams to address this.
Primes 1–50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 = 15 prime numbers.
⚡ MEMORY: Use Sieve of Eratosthenes. Cross out multiples of 2, 3, 5, 7 — remaining numbers are prime.
Let other integer = x. Then 8 + x = −5 → x = −5 − 8 = −13.
Reason (R): A prime number has exactly 2 factors, and 1 has only 1 factor (itself).
The assertion that 1 is neither prime nor composite is correct. The reason correctly explains why: a prime must have exactly 2 factors (1 and itself), but 1 has only one factor. Therefore R correctly explains A.
Reason (R): Natural numbers start from 1, while whole numbers start from 0.
Assertion A is false: 0 is a whole number but NOT a natural number. The correct statement is: "Every natural number is a whole number." Reason R is true: Natural numbers start from 1; whole numbers start from 0.
Smallest 5-digit number = 10,000. Predecessor = 10,000 − 1 = 9,999 (a 4-digit number).
Column A
i. XLII
ii. XCIX
iii. CDLV
iv. MCMXC
Column B
P. 1990
Q. 42
R. 455
S. 99
XLII = 40+2 = 42 (Q) | XCIX = 90+9 = 99 (S) | CDLV = 400+50+5 = 455 (R) | MCMXC = 1000+900+90 = 1990 (P)
Kinaesthetic learning engages the body in the learning process. Jumping on a floor number line makes integer operations physically felt rather than just mentally calculated. This is aligned with the constructivist and activity-based approach advocated in NCF 2005 and CTET pedagogy.
HCF(7,13) = 1 (both are primes and different), so 7/13 is already in simplest form. Others: 14/21 = 2/3 (÷7); 18/24 = 3/4 (÷6). To check simplest form, verify HCF(numerator, denominator) = 1.
Sum of known digits: 6+4+8+3+1+5 = 27. For divisibility by 9, total sum must be divisible by 9. 27 + x must be divisible by 9. 27 is already divisible by 9, so x = 0 OR 9. But checking options... 27+0=27 ✅. Hmm, let's re-examine: 27+6=33 (not divisible by 9); 27+0=27 ✅. Correct answer = 0 (A). Check the actual digits carefully in exam.
An abacus or Dienes Blocks (Base-10 blocks) are concrete manipulatives that allow young learners to physically see and handle ones, tens, and hundreds. This is the most effective TLM for place value as it follows the CPA (Concrete-Pictorial-Abstract) framework. Worksheets and blackboards are pictorial/abstract, less effective for beginners.
Smallest 2-digit prime = 11. Largest 2-digit prime = 97. Difference = 97 − 11 = 86.
⚡ TRICK: Is 99 prime? No (9×11). Is 97 prime? Yes (not divisible by 2,3,5,7 — 97÷7≈13.8). So 97 is the largest 2-digit prime.
Factors of 84 (possible piece lengths > 5 cm): 6, 7, 12, 14, 21, 28, 42, 84. Corresponding number of pieces: 14, 12, 7, 6, 4, 3, 2, 1. To maximise pieces, use smallest valid length = 6 cm → 84÷6 = 14 pieces. Answer: 14 pieces of 6 cm each.
🧠 MIND MAP — Number System
Place Value • Indian/International • Roman
HCF • LCM • Primes • Divisibility
Closure • Commutative • Associative
Negative • Temperature • Banking
Proper • Improper • Mixed • Equivalent
CPA • TLMs • Constructivism • NCF 2005
⚡ Quick Revision Notes
✅ 1 Crore = 10 Million
✅ 1 is neither prime nor composite
✅ 2 is the only even prime
✅ HCF × LCM = Product of two numbers
✅ Denominator cannot be zero
✅ Whole numbers start from 0; Natural from 1
✅ Place value of 0 is always 0
✅ Subtraction & Division are NOT commutative
✅ Additive identity = 0; Multiplicative identity = 1
✅ Integers include all whole numbers + negatives
✅ Simplify fractions by dividing by HCF
✅ Primes 1–50: there are 15 prime numbers
🌟 You've Got This, Future Teacher!
Mathematics is not about memorising formulas — it's about understanding the why behind every number. When you understand deeply, you teach confidently. When you teach confidently, your students love mathematics.
Remember: Every great teacher was once a determined learner. Your passion for mathematics will light the spark in thousands of young minds. Keep going!
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