Integers Class 7 Maths Best Notes – Chapter 1

Q: 2. What is the difference between whole numbers and integers?

Whole numbers : Include 0 and positive numbers (0, 1, 2, 3, …). Integers : Include 0 and positive numbers with negative numbers (-3, -2, -1, 0, 1, 2, 3, …).

Q: 3. What is the smallest and largest integer?

There is no smallest or largest integer because integers extend infinitely in both directions on a number line.

Q: 4. Is zero a positive or negative integer?

Zero is neither positive nor negative . It is a neutral integer.

Q: 5. Can the sum of two negative integers be positive?

No, the sum of two negative integers is always negative . Example: (-5) + (-3) = -8

Q: 7. What is the result of multiplying two negative integers?

The product of two negative integers is always positive. Example: (-4) × (-3) = 12

Q: 8. Why is (-5) × (-3) = 15?

Multiplication follows the rule of signs : Negative × Negative = Positive Negative × Positive = Negative So, (-5) × (-3) = +15 .

Q: 9. How do we subtract integers?

Subtracting an integer means adding its opposite (change the sign and add). Example: (-7) – (+3) = -7 + (-3) = -10

Q: 10. Is division of integers commutative?

No, division is not commutative because a ÷ b ≠ b ÷ a . Example: (-10) ÷ 5 = -2, but 5 ÷ (-10) ≠ -2

Key Learning Outcomes

By the end of this lesson, readers will:

✔️ Understand the concept of positive and negative numbers.
✔️ Learn how to represent integers on a number line.
✔️ Perform addition, subtraction, multiplication, and division of integers.
✔️ Understand important properties of integers, such as commutative, associative, and distributive properties.
✔️ Solve real-life problems using integers, such as temperature, elevation, and financial transactions.

Click On The Name To Go To A Specific Topic:

Introduction to Integers

Integers are a set of numbers that include positive numbers, negative numbers, and zero but not fractions or decimals.

Positive Integers: Numbers greater than zero (1, 2, 3, 4,…).
Negative Integers: Numbers less than zero (-1, -2, -3, -4,…).
Zero (0): It is neither positive nor negative but plays a significant role in integer operations.

Representation of Integers on Number Line

Integers can be represented on a horizontal number line, where:

Positive numbers are on the right of zero.
Negative numbers are on the left of zero.
The greater the number, the more right it is on the number line.

Diagram: Integer Number Line

← -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 →

Operations on Integers

1. Addition of Integers

Rules for addition:

Same Sign: Add the absolute values and keep the common sign.
- Example: (-3) + (-5) = -8
Different Sign: Subtract the absolute values and take the sign of the greater number.

Example: (-7) + (+4) = -3

2. Subtraction of Integers

To subtract an integer, add the additive inverse (change the sign of the second number and then add).

Example: (-5) – (-3) = -5 + 3 = -2

3. Multiplication of Integers

Rules:

Positive × Positive = Positive → (+3) × (+4) = +12
Negative × Negative = Positive → (-3) × (-4) = +12
Positive × Negative = Negative → (+3) × (-4) = -12

integers-class-7-maths-best-notes(www.jngacademy.com)

4. Division of Integers

Rules:

Positive ÷ Positive = Positive → (+12) ÷ (+3) = +4
Negative ÷ Negative = Positive → (-12) ÷ (-3) = +4
Positive ÷ Negative = Negative → (+12) ÷ (-3) = -4

Properties of Integers

1. Closure Property

Addition & Multiplication: Always closed (result is always an integer).
Subtraction & Division: May not be closed.

2. Commutative Property

Addition & Multiplication: a + b = b + a, a × b = b × a.
Subtraction & Division: Not commutative.

3. Associative Property

Addition & Multiplication: (a + b) + c = a + (b + c).
Subtraction & Division: Not associative.

4. Distributive Property

a × (b + c) = (a × b) + (a × c).

Important Summary Points

✔️ Integers include positive numbers, negative numbers, and zero.
✔️ Addition and multiplication of integers follow commutative and associative properties.
✔️ The product of two negative numbers is always positive.
✔️ Subtraction means adding the opposite.
✔️ Integers are useful in daily life for temperature, banking, and elevation measurements.

Interesting Facts About Integers

💡 The word “integer” comes from Latin, meaning “whole.”
💡 In real life, integers are used in thermometers, elevators, bank transactions, and GPS coordinates.
💡 The smallest integer is negative infinity, and the largest integer is positive infinity.

Activity (Exercise):

Fill in the Blanks

(-9) + (____) = 0.
The product of two negative integers is always ____.
(-7) – (-3) = ____.

Short Answer Questions

Represent -3 to 5 on a number line.
Find the value of: (-12) + 7 – (-5) × 2.
Solve: (-25) ÷ (+5) + (-3) × (-4).

Word Problems

A submarine is 40 meters below sea level. It moves 15 meters deeper, then rises 20 meters. What is its new depth?
A company gained ₹5000 in January but lost ₹7000 in February. What is the overall profit or loss?

Quiz:

Coming Soon…

Supplementary Materials:

Provide downloadable materials for learners to review:

– PDF Guide: “Coming Soon”
– Cheat Sheet: “Coming Soon”
– Video Source: “JNG ACADEMY“
– Articles: “Blog Page“

FAQs:

1. What are integers?

Integers are a set of whole numbers that include positive numbers, negative numbers, and zero. They do not include fractions or decimals.

2. What is the difference between whole numbers and integers?

Whole numbers: Include 0 and positive numbers (0, 1, 2, 3, …).
Integers: Include 0 and positive numbers with negative numbers (-3, -2, -1, 0, 1, 2, 3, …).

3. What is the smallest and largest integer?

There is no smallest or largest integer because integers extend infinitely in both directions on a number line.

4. Is zero a positive or negative integer?

Zero is neither positive nor negative. It is a neutral integer.

5. Can the sum of two negative integers be positive?

No, the sum of two negative integers is always negative.
Example: (-5) + (-3) = -8

6. What happens when a positive and a negative integer are added?

Subtract the smaller absolute value from the larger one.
The answer takes the sign of the larger number.
Example: (+8) + (-5) = 3

7. What is the result of multiplying two negative integers?

The product of two negative integers is always positive.
Example: (-4) × (-3) = 12

8. Why is (-5) × (-3) = 15?

Multiplication follows the
rule of signs:
Negative × Negative = Positive
Negative × Positive = Negative
So, (-5) × (-3) = +15.

9. How do we subtract integers?

Subtracting an integer means adding its opposite (change the sign and add).
Example: (-7) – (+3) = -7 + (-3) = -10

10. Is division of integers commutative?

No, division is not commutative because a ÷ b ≠ b ÷ a.
Example: (-10) ÷ 5 = -2, but 5 ÷ (-10) ≠ -2

11. How are integers used in real life?

Integers are used in:
✔️ Banking (credits and debits)
✔️ Temperature measurement (positive for above freezing, negative for below)
✔️ Elevation levels (above and below sea level)

12. What is the absolute value of an integer?

The absolute value of an integer is its distance from zero, regardless of its sign.
Example: |−7| = 7

13. Can integers be decimal numbers?

No, integers cannot have decimal or fractional parts.

14. What is the additive inverse of an integer?

The additive inverse of an integer “x” is “-x” because its sum is zero.
Example: The additive inverse of 5 is -5 because 5 + (-5) = 0.

15. What is the distributive property of integers?

The distributive property states:
a × (b + c) = (a × b) + (a × c)
Example: 2 × (3 + 4) = (2 × 3) + (2 × 4) = 6 + 8 = 14