Class 7 Mathematics

Number System – Term 1: Integers

Concepts • Rules • Worked Examples • Daily-life Applications

1. Understanding Numbers

Natural Numbers

Counting numbers are called natural numbers.

1, 2, 3, 4, 5, …

Whole Numbers

Whole numbers include zero and all natural numbers.

0, 1, 2, 3, 4, 5, …

Integers

Integers include negative whole numbers, zero and positive whole numbers. Fractions and decimals are not integers.

…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …

Examples: −8, −1, 0, 12 are integers. The numbers 2.5 and ¾ are not integers.

2. Positive Numbers, Negative Numbers and Zero

Positive integers are greater than zero, such as +2, +15 and +100.

Negative integers are less than zero, such as −3, −12 and −50.

Zero is neither positive nor negative.

Daily-life examples: A temperature of 6°C above zero is +6°C; 4°C below zero is −4°C. A deposit of ₹500 may be represented as +500 and a debt of ₹500 as −500.

3. Opposite Integers

Two integers that are at the same distance from zero but on opposite sides of the number line are called opposites.

Opposite of +7 = −7
Opposite of −9 = +9

Zero is its own opposite. The sum of an integer and its opposite is zero.

−12 + 12 = 0

4. Number Line and Integer Representation

A number line is a straight line on which numbers are placed in order. Zero is the reference point. Positive integers lie to the right of zero and negative integers lie to the left.

← −5 −4 −3 −2 −1 0 1 2 3 4 5 →

Moving right represents increasing value.
Moving left represents decreasing value.
Each equal step represents one unit.

Example: Starting at −2 and moving 5 steps right gives 3.

5. Comparing and Ordering Integers

On a number line, the integer farther to the right is greater.

−2 > −7 and −5 < 1

Remember: Among negative integers, the number closer to zero is greater. For example, −3 is greater than −8.

Ascending and Descending Order

Ascending: −6, −2, 0, 3, 8
Descending: 8, 3, 0, −2, −6

6. Absolute Value

The absolute value of an integer is its distance from zero on the number line. It is never negative.

|+6| = 6 |−6| = 6 |0| = 0

The distance between −4 and 0 is 4 units, so |−4| = 4.

7. Addition of Integers

Same signs

Add their absolute values and keep the common sign.

(+5) + (+3) = +8
(−6) + (−4) = −10

Different signs

Subtract the smaller absolute value from the larger absolute value. Keep the sign of the number with the larger absolute value.

(+9) + (−4) = +5
(−12) + (+7) = −5

Adding zero

a + 0 = 0 + a = a

−13 + 0 = −13

8. Subtraction of Integers

To subtract an integer, add its opposite.

a − b = a + (−b)

Example 1: 8 − 12 = 8 + (−12) = −4
Example 2: −5 − 7 = −5 + (−7) = −12
Example 3: −6 − (−9) = −6 + 9 = 3

Subtracting a negative number is equivalent to adding its positive opposite.

9. Multiplication of Integers

Multiply the absolute values first, then determine the sign.

Signs	Product sign	Example
Positive × Positive	Positive	(+4) × (+3) = +12
Negative × Negative	Positive	(−4) × (−3) = +12
Positive × Negative	Negative	(+4) × (−3) = −12
Negative × Positive	Negative	(−4) × (+3) = −12

Any integer × 0 = 0

10. Division of Integers

For non-zero divisor, divide the absolute values and apply the sign rule.

Signs	Quotient sign	Example
Same signs	Positive	(−24) ÷ (−6) = +4
Different signs	Negative	(+24) ÷ (−6) = −4

Division by zero is not defined. Zero divided by a non-zero integer is zero: 0 ÷ 5 = 0.

11. Properties of Integer Operations

Closure Property

Adding, subtracting or multiplying two integers always gives an integer.

−4 + 9 = 5; 3 − 8 = −5; (−2) × 6 = −12.

Integers are not closed under division, because 5 ÷ 2 = 2.5, which is not an integer.

Commutative Property

Changing the order does not change the result for addition and multiplication.

a + b = b + a
a × b = b × a

Subtraction and division are not commutative: 7 − 3 ≠ 3 − 7.

Associative Property

Changing the grouping does not change the result for addition and multiplication.

(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)

Subtraction and division are not associative.

Distributive Property

a × (b + c) = (a × b) + (a × c)

3 × (−2 + 5) = 3 × 3 = 9
(3 × −2) + (3 × 5) = −6 + 15 = 9

Identity Elements

Zero is the additive identity: a + 0 = a. One is the multiplicative identity: a × 1 = a.

Additive Inverse

The additive inverse of a is −a, because a + (−a) = 0.

The additive inverse of −15 is +15.

Zero Property

Multiplying any integer by zero gives zero. Zero has no multiplicative inverse.

12. Simplifying Expressions

Follow the order of operations: brackets first, then multiplication/division from left to right, then addition/subtraction from left to right.

Example: 6 + (−3) × 4
= 6 + (−12)
= −6

Example: [−8 + 3] − (−4)
= −5 + 4
= −1

13. Finding Missing Integers

Example 1: x + (−7) = 5
x = 5 + 7 = 12

Example 2: x − 4 = −9
x = −9 + 4 = −5

Example 3: (−3) × x = 18
x = 18 ÷ (−3) = −6

14. Word Problems Using Integers

Temperature

At dawn the temperature was −3°C. By noon it rose by 8°C.

Final temperature = −3 + 8 = 5°C.

Bank balance

A person has a debt of ₹350 and deposits ₹500.

Balance = −350 + 500 = ₹150 (credit).

Elevation

A diver is 12 m below sea level and rises 5 m.

New position = −12 + 5 = −7 m, still 7 m below sea level.

Gain and loss

A team loses 4 points and then gains 9 points.

Net change = −4 + 9 = +5 points.

15. Practice Questions

Write the opposite of −17.
Which is greater: −9 or −4?
Find: (−8) + 13.
Find: 7 − (−6).
Find: (−5) × (−7).
Find: 42 ÷ (−7).
Evaluate: 4 + (−3) × 5.
Find x: x + 9 = −2.
The temperature is −6°C and rises by 11°C. What is the new temperature?
State whether integers are closed under division. Give an example.

Show Answer Key

+17
−4
5
13
35
−6
−11
−11
5°C
No. For example, 5 ÷ 2 = 2.5, not an integer.

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7TH - MATHS - NUMBERS - BASIC CONCEPTS