Integers

Integers Class 6 CBSE Maths Notes Examples, MCQs and Practice Questions

Welcome to our comprehensive guide on Integers for Class 6 CBSE Mathematics. This chapter introduces students to the world of negative numbers and expands their understanding beyond whole numbers. Whether you're preparing for your exams or looking for NCERT solutions, this guide covers everything you need to master integers.

1. What are Integers? – Definition and Meaning

1.1 Integer Meaning and Definition

Integers are a collection of all counting numbers (natural numbers), zero, and the negatives of counting numbers. In simple terms, integers include all positive numbers, negative numbers, and zero without any fractions or decimals.

Mathematical Definition

I = Z = {…, –3, –2, –1, 0, +1, +2, +3, …}

The set of integers is denoted by 'I' or 'Z' (from German 'Zahlen' meaning numbers)

1.2 Types of Integers

Integers can be classified into three main categories:

Type	Description	Examples
Positive Integers	Numbers greater than zero; located to the right of zero on the number line	+1, +2, +3, +4, +5, ...
Zero	Neither positive nor negative; the reference point on the number line	0
Negative Integers	Numbers less than zero; located to the left of zero on the number line	–1, –2, –3, –4, –5, ...

1.3 Integer Examples

Here are some examples to help you identify integers:

• Integers: –100, –25, –7, 0, 4, 15, 99, 1000
• NOT Integers: ½, 0.5, 3.14, –2.7, ¾ (these are fractions or decimals)

2. Difference Between Integers and Whole Numbers

Understanding the difference between integers and whole numbers is crucial for Class 6 students. This is one of the most commonly asked questions in CBSE exams.

Aspect	Whole Numbers	Integers
Definition	Natural numbers including zero	Whole numbers and their negatives
Set Notation	W = {0, 1, 2, 3, ...}	Z = {..., –2, –1, 0, 1, 2, ...}
Negative Numbers	NOT included	Included
Smallest Number	0 (zero)	No smallest integer
Relationship	Subset of integers	Contains whole numbers

Note: Every whole number is an integer, but not every integer is a whole number. For example, –5 is an integer but NOT a whole number.

3. Representing Integers on a Number Line

A number line is a visual representation that helps us understand the position and relationship between integers. It extends infinitely in both directions.

3.1 How to Draw a Number Line for Integers

1. Draw a horizontal line with arrows on both ends (indicating it extends infinitely)
2. Mark equal intervals (points) along the line
3. Place zero (0) at the center point
4. Write positive integers (+1, +2, +3, ...) to the RIGHT of zero
5. Write negative integers (–1, –2, –3, ...) to the LEFT of zero

Number Line Representation

←───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───→
 –6 –5 –4 –3 –2 –1 0 +1 +2 +3 +4 +5 +6

3.2 Key Points About Number Line

• Moving RIGHT: Numbers increase in value
• Moving LEFT: Numbers decrease in value
• Any number to the RIGHT is GREATER than the number to its left
• All negative integers are less than zero
• All positive integers are greater than zero

4. Operations on Integers – Rules and Methods

4.1 Rules for Adding Integers

Addition of integers follows specific rules based on the signs of the numbers involved:

Rule	Method	Example
Positive + Positive	Add and keep positive sign	(+5) + (+3) = +8
Negative + Negative	Add and keep negative sign	(–5) + (–1) = –6
Positive + Negative (or vice versa)	Subtract smaller from larger; take sign of larger	(–7) + (+8) = +1

Addition on Number Line

To add integers on a number line:

1. Start at zero
2. Move to the first number
3. For adding positive: Move RIGHT
4. For adding negative: Move LEFT

Example: 2 + 4 = ? → Start at 0, move 2 right to reach 2, then move 4 more right to reach 6. Answer: 6

4.2 Rules for Subtracting Integers

Subtraction of integers can be converted to addition using the additive inverse:

Rule: a – b = a + (–b)

To subtract an integer, add its additive inverse (change the sign and add)

Examples of Subtraction:

1. 7 – 2 = 7 + (–2) = 5
2. –10 – (–5) = –10 + 5 = –5
3. 5 – (–3) = 5 + 3 = 8

4.3 Rules for Multiplying Integers

Signs	Result
(+) × (+)	Positive (+)
(–) × (–)	Positive (+)
(+) × (–)	Negative (–)
(–) × (+)	Negative (–)

Memory Trick: Same signs = Positive result | Different signs = Negative result

4.4 Rules for Dividing Integers

Division of integers follows the same sign rules as multiplication:

1. (+) ÷ (+) = Positive: (+12) ÷ (+3) = +4
2. (–) ÷ (–) = Positive: (–12) ÷ (–3) = +4
3. (+) ÷ (–) = Negative: (+12) ÷ (–3) = –4
4. (–) ÷ (+) = Negative: (–12) ÷ (+3) = –4

5. Properties of Integers

Understanding the properties of integers helps solve problems more efficiently:

Property	For Addition	For Multiplication
Closure	a + b is always an integer	a × b is always an integer
Commutative	a + b = b + a	a × b = b × a
Associative	(a + b) + c = a + (b + c)	(a × b) × c = a × (b × c)
Identity	a + 0 = a (Zero is identity)	a × 1 = a (One is identity)
Distributive	a × (b + c) = (a × b) + (a × c)

5.1 Additive Inverse

The additive inverse of an integer is the number that when added to it gives zero.

• Additive inverse of +5 is –5 (because +5 + (–5) = 0)
• Additive inverse of –7 is +7 (because –7 + 7 = 0)
• Additive inverse of 0 is 0 (because 0 + 0 = 0)

6. Real-World Applications of Integers

Integers are used extensively in daily life. Here are some practical examples:

Application	Positive (+)	Negative (–)
Temperature	Above freezing (+25°C)	Below freezing (–10°C)
Banking	Deposit / Credit (+₹500)	Withdrawal / Debit (–₹200)
Altitude/Depth	Above sea level (+8848m)	Below sea level (–400m)
Profit/Loss	Profit (+₹1000)	Loss (–₹500)
Floor Levels	Floors above ground (+3)	Basement levels (–2)

7. Formulas and Quick Reference Table

Concept	Formula/Rule	Example
Additive Inverse	a + (–a) = 0	7 + (–7) = 0
Subtraction Rule	a – b = a + (–b)	5 – 8 = 5 + (–8) = –3
Same Signs Product	(+)(+) or (–)(–) = +	(–3)(–4) = +12
Different Signs Product	(+)(–) or (–)(+) = –	(–3)(+4) = –12
Zero Property	a × 0 = 0	(–99) × 0 = 0

8. Word Problems on Integers with Solutions

Problem 1: Temperature Change

Question: Today's temperature is 2°C, which is 5°C greater than yesterday. What was yesterday's temperature?

Solution: Yesterday's temperature = Today's temperature – 5°C = 2°C – 5°C = –3°C

Problem 2: Remainder Problem

Question: A whole number n when divided by 5 gives remainder 3. What will be the remainder when 2n is divided by 5?

Solution: If n = 5p + 3, then 2n = 10p + 6. When 10p + 6 is divided by 5, 10p gives remainder 0 and 6 gives remainder 1. So the answer is 1.

Problem 3: Sum of Integers

Question: Find the sum of (–10), 82, (–39), and 68.

Solution: Group same signs: (–10) + (–39) + 82 + 68 = –49 + 150 = 101

9. Important MCQs with Answers (CBSE 2025-26)

Q1. An integer which is neither positive nor negative is:

(a) 0

(b) 1

(c) –1

(d) None of these

Answer: (a) 0

Q2. The sum (–7) + (–9) + 14 + 6 is equal to:

(a) –4

(b) 24

(c) –6

(d) 4

Answer: (d) 4 [Solution: –7 – 9 + 20 = –16 + 20 = 4]

Q3. Which number will we reach if we move 4 numbers to the right of –2 on number line?

(a) 2

(b) 3

(c) 4

(d) 5

Answer: (a) 2 [Solution: –2 + 4 = 2]

Q4. –51 × 9 + 15 × 9 is equal to:

(a) 224

(b) –324

(c) 324

(d) –224

Answer: (b) –324 [Solution: 9 × (–51 + 15) = 9 × (–36) = –324]

Q5. 888 ÷ 3 ÷ 37 = ?

(a) 8

(b) 3

(c) 37

(d) 296

Answer: (a) 8 [Solution: 888 ÷ 3 = 296, 296 ÷ 37 = 8]

10. Common Mistakes and How to Avoid Them

1. Confusing subtraction with addition of negative → Fix: Remember: 5 – (–3) = 5 + 3 = 8, NOT 5 – 3
2. Wrong sign in multiplication → Fix: Use the rule: Same signs = +, Different signs = –
3. Moving wrong direction on number line →  Fix: Positive = RIGHT, Negative = LEFT (always!)
4. Thinking –5 > –3 → Fix: The number closer to zero is greater. –3 > –5
5. Forgetting that all negative integers < 0 → Fix: Zero is always greater than any negative integer

11. Tips to Score Full Marks in Integers Chapter

1. Master the Number Line: Practice drawing and using number lines for all operations
2. Memorize Sign Rules: Create flashcards for multiplication and division sign rules
3. Practice Word Problems: Convert real-world scenarios to integer operations
4. Use Additive Inverse: Convert all subtractions to additions for easier calculations
5. Check Your Signs: Always double-check the sign of your final answer
6. Group Similar Terms: Group positive and negative integers separately before calculating