Number System

Overview

The Number System forms the bedrock of primary mathematics and carries significant weight in CG TET Paper I. This topic tests your understanding of how numbers are structured, named, and manipulated—skills essential for teaching Classes 1–5. Expect 3–5 direct questions on whole numbers, integers, place value, factors and multiples.

Mastery here is non-negotiable because nearly every other arithmetic topic (fractions, percentages, mensuration) depends on number sense. Questions range from basic place-value identification to finding LCM/HCF through factor trees, and occasionally include word problems involving divisibility rules. The pedagogy angle may also ask how to teach these concepts to young learners using concrete materials.

Your goal: internalize the definitions, memorize divisibility rules, and practice factor-multiple problems until they become automatic.

Key Concepts

• **Natural Numbers (N)**: Counting numbers starting from 1 → {1, 2, 3, 4, ...}. Zero is NOT included.

• **Whole Numbers (W)**: Natural numbers plus zero → {0, 1, 2, 3, ...}. Every natural number is a whole number, but 0 is only a whole number.

• **Integers (Z)**: Whole numbers extended to include negatives → {..., −3, −2, −1, 0, 1, 2, 3, ...}. The number line extends infinitely in both directions.

• **Place Value vs Face Value**: In 5,847, the digit 8 has face value 8 but place value 800 (8 × 100). Face value never changes; place value depends on position.

• **Factors**: Numbers that divide a given number exactly. Factors of 12 → {1, 2, 3, 4, 6, 12}. Always finite and include 1 and the number itself.

• **Multiples**: Numbers obtained by multiplying a given number by natural numbers. Multiples of 4 → {4, 8, 12, 16, ...}. Always infinite.

• **Prime Numbers**: Numbers with exactly two factors—1 and itself. Examples: 2, 3, 5, 7, 11. Note: 2 is the only even prime; 1 is NOT prime.

• **Composite Numbers**: Numbers with more than two factors. Examples: 4, 6, 8, 9. Note: 1 is neither prime nor composite.

Formulas / Key Facts

**Place Value System (Indian)** | Place | Value | |-------|-------| | Unit | 1 | | Tens | 10 | | Hundreds | 100 | | Thousands | 1,000 | | Ten Thousands | 10,000 | | Lakhs | 1,00,000 | | Ten Lakhs | 10,00,000 | | Crores | 1,00,00,000 |

**Divisibility Rules**

• By 2 → Last digit is 0, 2, 4, 6, or 8
• By 3 → Sum of digits divisible by 3
• By 4 → Last two digits form a number divisible by 4
• By 5 → Last digit is 0 or 5
• By 6 → Divisible by both 2 and 3
• By 8 → Last three digits divisible by 8
• By 9 → Sum of digits divisible by 9
• By 10 → Last digit is 0
• By 11 → Difference of sum of alternate digits is 0 or divisible by 11

**Properties of Integers**

• Addition of two negative integers → Negative integer
• Subtraction: a − (−b) = a + b
• Multiplication: Negative × Negative = Positive; Negative × Positive = Negative
• Division follows same sign rules as multiplication

**Key Number Facts**

• Smallest whole number: 0
• Smallest natural number: 1
• Smallest prime number: 2
• Prime numbers up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 (10 primes)
• Co-prime numbers: Two numbers whose HCF is 1 (e.g., 8 and 15)

Worked Examples

**Example 1: Place Value Problem** *In the number 7,09,436, find the difference between the place value and face value of 9.*

Solution:

• Face value of 9 = 9
• Place of 9 = Thousands place
• Place value of 9 = 9 × 1,000 = 9,000
• Difference = 9,000 − 9 = **8,991**

**Example 2: Finding All Factors** *Find all factors of 36.*

Solution: Start dividing from 1 and pair factors:

• 36 ÷ 1 = 36 → factors: 1, 36
• 36 ÷ 2 = 18 → factors: 2, 18
• 36 ÷ 3 = 12 → factors: 3, 12
• 36 ÷ 4 = 9 → factors: 4, 9
• 36 ÷ 6 = 6 → factors: 6, 6

Factors of 36 = {1, 2, 3, 4, 6, 9, 12, 18, 36} → **9 factors**

**Example 3: Integer Operation** *Evaluate: (−15) + 8 − (−12) + (−5)*

Solution:

• (−15) + 8 = −7
• −(−12) = +12, so −7 + 12 = 5
• 5 + (−5) = **0**

**Example 4: Divisibility Check** *Is 2,574 divisible by 6?*

Solution: Check for 2: Last digit is 4 (even) → Yes Check for 3: Sum of digits = 2 + 5 + 7 + 4 = 18, which is divisible by 3 → Yes Since divisible by both 2 and 3 → **Yes, 2,574 is divisible by 6**

Common Mistakes

• **Confusing factors and multiples** → Factors divide into the number (finite, always ≤ the number); multiples are products (infinite, always ≥ the number). Fix: Factors are "family members that fit inside"; multiples are "the number's extended family going outward."

• **Treating 1 as prime** → 1 has only one factor (itself), so it fails the "exactly two factors" rule. Fix: Prime means exactly TWO distinct factors.

• **Forgetting 0 in whole numbers** → Students often say "whole numbers start from 1." Fix: Natural starts from 1; Whole starts from 0.

• **Sign errors with integers** → Subtracting a negative number means adding. Students write (−5) − (−3) = −8 instead of −2. Fix: Draw a number line and physically trace the movement.

• **Place value in Indian system** → Mixing up lakhs (5 zeros) with millions (6 zeros). Fix: Indian system uses 2-digit grouping after thousands (10,00,000 = ten lakhs, not one million).

Quick Reference

• Natural: {1, 2, 3, ...} | Whole: {0, 1, 2, ...} | Integers: {..., −2, −1, 0, 1, 2, ...}

• Place value = Face value × Position value

• Divisibility by 6 = Check both 2 AND 3

• Prime numbers have exactly 2 factors; 1 is NOT prime; 2 is the only even prime

• Negative × Negative = Positive; Negative × Positive = Negative

• Sum of first n natural numbers = n(n + 1)/2