Class 7 Integers

In class 6, we learn about integers and various operations on them. Here we will learn various properties satisfied by various operations on integers.

Till now we have covered Natural numbers, Whole numbers, and integers.

Natural Numbers

Counting numbers are known as natural numbers.
1, 2, 3, 4, 5, 6, ... are all natural numbers.

Whole Numbers

All natural numbers together with zero are known as whole numbers.
0, 1, 2, 3, 4, 5, 6, ... are whole numbers.

Integers

All natural numbers, zero and negative counting numbers are known as integers.

..., -5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... are all integers.
Here 1, 2, 3, 4, 5, ... are all Positive Integers.
−1, −2, −3, −4, −5 are all Negative Integers.
Zero is an integer which is neither positive nor negative.

Properties of Addition of Integers

Here we are going to discuss following properties of integer addition.

Closure property of addition
Commutative law of addition
Associative law of addition
Additive identity
Additive inverse

Closure property of addition

The sum of two integers is always an integer. Let's see some examples.

Example 1. 3 + 2 = 5, here number 5 is a positive integer.

Example 2. 5 + (−9) = −4, is a negative integer

Example 3. −4 + (−5) = −9, is a negative integer

Example 4. 12 + (−7) = 5, is a positive integer

Example 5. −5 + 5 = 0, is an integer

Commutative law of addition

If 'a' and 'b' are any two integers, then a + b = b + a. Let's see some examples.

Example 1. −3 + 7 = 4 and 7 + (−3) = 4
Hence, −3 + 7 = 7 + (−3).

Example 2. (−5) + (−7) = −12 and (−7) + (−5) = −12
Hence, (−5) + (−7) = (−7) + (−5).

Associative law of addition

If a, b, c are any three integers, then (a + b) + c = a + (b + c). Let's see some examples.

Example 1. Consider three integers 4, −5, −7.
{4 + (−5)} + (−7) = (4 − 5) −7 = −1 −7 = −8
And, 4 + {(−5) + (−7)} = 4 + (−12) = −8

Additive Identity

For any integer 'p', we have: p + 0 = 0 + p = p.
0 is called the additive identity for integers. Let's see some examples.

Example 1. 8 + 0 = 0 + 8 = 8.

Example 2. (−5) + 0 = 0 + (−5) = −5

Additive inverse

For any integer 'p', we have: p + (−p) = (−p) + p = 0.
The opposite of an integer 'p' is (−p).
The sum of an integer and it's opposite is 0.
Additive inverse of p is (−p).
Similarly, additive inverse of (−p) is p. Let's see some examples.

Example 1. 4 + (−4) = (−4) + 4 = 0 So, the additive inverse of 4 is (−4). Additive inverse of (−4) is 4.

Properties of Subtraction of Integers

Here we are going to discuss following properties of integer subtraction.

Closure properties of subtraction
Subtraction of integers is not commutative
Subtraction of integers is not associative

Closure Properties of Subtraction

If 'a' and 'b' are any two integers, then (a − b) is always an integer. Let's see some examples.

Example 1. 5 − 4 = 5 + (−4) = 1, which is an integer.

Example 2. − 3 − 6 = (−3) + (−6) = −9, which is an integer.

Example 3. − 5 − (−7) = −5 + 7 = 2, which is an integer.

Example 4. 4 − (−8) = 4 + 8 = 12, which is an integer.

Subtraction of Integers is not Commutative

If 'a' and 'b' are any two integers, then a − b ≠ b − a. Let's see some examples.

Examples 1. Consider two integers 4 and 9.
4 − 9 = 4 + (−9) = −5
9 − 4 = 9 + (−4) = 5
Hence, 4 − 9 ≠ 9 − 4

Example 2. Consider two integers −3 and 6.
(−3) − 6 = (−3) + (−6) = −9
6 − (−3) = 6 + 3 = 9
Hence, (−3) − 6 ≠ 6 − (−3)

Subtraction of Integers is not Associative

If a, b, c are any three integers, then (a − b) − c ≠ a − (b − c). Let's see some examples.

Example 1. Consider three integers as 5, 7, and −2.
(5 − 7) − (−2) = −2 + 2 = 0
And 5 − {7 − (−2)} = 5 − (7 + 2) = 5 − 9 = −4
Hence, (5 − 7) − (−2) ≠ 5 − {7 − (−2)}

Example 2. Consider three integers as −10, 8, and −5.
{(−10) − 8} − (−5) = (−18) + 5 = −13
And (−10) − {8 − (−5)} = (−10) − (8 + 5) = (−10) − 13 = −23
Hence, {(−10) − 8} − (−5) ≠ (−10) − {8 − (−5)}

Multiplication of Integers

We have four different ways to multiply integers, they are mentioned below.

Multiplication of two positive integers
Multiplication of positive and negative integers
Multiplication of two negative integers
Multiplication by zero

Multiplication of Two Positive Integers

To multiply two positive integers, multiply them as natural numbers and the product is a positive integer. Let's see some examples.

Example 1. Multiply 5 and 6.
5 x 6 = 30, which is a positive integer.

Example 2. Multiply 6 and 10.
6 x 10 = 60, which is a positive integer.

Multiplication of Positive and Negative Integers

To multiply a positive integer and negative integer, we multiply them as natural numbers and put the minus sign before the result. So, we get a negative integer. Let's see some examples.

Example 1. Multiply 5 and (-8).
5 x (−8) = −40, which is a negative integer.

Example 2. Multiply (−8) and 10.
(−8) x 10 = −80, which is a negative integer.

Multiplication of Two Negative Integers

To multiply two negative integers, we multiply them as natural numbers and put the positive sign before the result. Let's see some examples.

Example 1. Multiply (−4) and (−5). (−4) x (−5) = 20

Example 2. Multiply (−5) and (−8). (−5) x (−8) = 40

Multiplication by Zero

If any integer multiplied by zero, then the result will be zero.

Example 1. Multiply 5 by 0.
5 x 0 = 0

Example 2. Multiply (−9) by 0.
(−9) x 0 = 0

Properties of Multiplication of Integers

There are seven types properties of multiplication, they are mentioned below.

Closure property of multiplication
Commutative law of multiplication
Associative law of multiplication
Distributive law of multiplication over addition
Multiplicative identity
Multiplicative inverse
Property of Zero

Closure Property of Multiplication

The product of two integers is always an integer.

Example 1. Multiply 4 and 5.
4 x 5 = 20, which is an integer.

Example 2. Multiply (−6) and 9
(−6) x 9 = −54, which is an integer.

Example 3. Multiply 3 and (−8)
3 x (−8) = −24, which is an integer.

Example 4. Multiply (−5) and (−6)
(−5) x (−6) = 30, which is an integer.

Commutative Law of Multiplication

For any two integers 'a' and 'b', (a x b) = (b x a)

Example 1. Consider 2 and 5 as two integers.
2 x 5 = 10 and 5 x 2 = 10
Hence, 2 x 5 = 5 x 2.

Example 2. Consider (−4) and 8 as two integers.
(−4) x 8 = −32 and 8 x (−4) = −32
Hence, (−4) x 8 = 8 x (−4)

Example 3. Consider (−5) and (−7) as two integers.
(−5) x (−7) = 35 and (−7) x (−5) = 35 Hence, (−5) x (−7) = (−7) x (−5)

Associative Law of Multiplication

For any three integers 'a', 'b', 'c', (a x b) x c = a x (b x c)

Example 1. Consider three integers 4, (−5), and 6.
{4 x (−5)} x 6 = (−20) x 6 = −120
4 x {(−5) x 6} = 4 x (−30) = −120
Hence, {4 x (−5)} x 6 = 4 x {(−5) x 6}

Example 2. Consider three integers (−3), (−4) and (−5).
{(−3) x (−4)} x (−5) = 12 x (−5) = −60
(−3) x {(−4) x (−5)} = (−3) x 20 = 60
Hence, {(−3) x (−4)} x (−5) = (−3) x {(−4) x (−5)}

Distributive Law of Multiplication Over Addition

For any three integers a, b, c, a x (b + c) = (a x b) + (a x c).

Example 1. Consider three integers 2, (−3) and (−5).
2 x {(−3) + (−5)} = 2 x (−8) = −16
{2 x (−3)} + {2 x (−5)} = (−6) + (−10) = −16
Hence, 2 x {(−3) + (−5)} = {2 x (−3)} + {2 x (−5)}

Multiplicative Identity

For every integer 'a', we have (a x 1) = (1 x a) = a
1 is called multiplicative identity for integers.

Multiplicative Inverse

Multiplicative inverse of a nonzero integer 'a' is the number ¹⁄_a.
a x ¹⁄_a = ¹⁄_a x a = 1

Example 1. Multiplicative inverse of 5 is ¹⁄₅.

Example 2. Multiplicative inverse of −8 is -¹⁄₈.

Property of Zero

If any integer multiplied by zero then the result will be zero.
(a x 0) = (0 x a) = 0

Division of Integers Having Like Signs

For dividing one integer by the other having like signs, we divide their values and give a plus sign to the quotient. Let's see some examples.

Example 1. Divide 95 by 5.
95 ÷ 5 = ⁹⁵⁄₅ = 19

Example 2. Divide (−64) by (−8).
(−64) ÷ (−8) = ^-64⁄_-8 = 8

Division of Integers Having Unlike Signs

For dividing two integers having unlike signs, we divide their values and give a minus sign to the quotient. Let's see some examples.

Example 1. Divide (−150) by 15.
(−150) ÷ 15 = ^-150⁄₁₅ = −10

Example 2. Divide 75 by (−15).
75 ÷ (−15) = ⁷⁵⁄_-15 = −5

Properties of Division of Integers

1. If 'a' and 'b' are integers then a ÷ b is not always an integer. Let's consider two integers as 15 and 6, but (15 ÷ 6) is not an integer.
2. If 'a' is an integer, then (a ÷ 1) is equal to 'a'. For example, 5 ÷ 1 = 5.
3. If 'a' is an integer and a ≠ 0, then a ÷ a = 1. For example, 11 ÷ 11 = 1.
4. If 'a' is an integer and a ≠ 0, then (0 ÷ a) = 0 but (a ÷ 0) has no value. For example, 0 ÷ 5 = 0, but 5 ÷ 0 has no value.
5. If 'a', 'b', 'c' are three integers, then (a ÷ b) ÷ c ≠ a ÷ (b ÷ c), unless c =1. For example, a = 15, b= 5 and c = −3.
(15 ÷ 5) ÷ (−3) = 3 ÷ (−3) = −1
15 ÷ {5 ÷ (−3)} = 15 ÷ ⁵⁄_-3 = 15 x ^-3⁄₅ = −9
Hence, (15 ÷ 5) ÷ (−3) ≠ 15 ÷ {5 ÷ (−3)}

Class-7 Integers Test

Integers Test - 1

Integers Test - 2

Class-7 Integers Worksheet

Integers Worksheet - 1

Integers Worksheet - 2

Integers Worksheet - 3

Integers Worksheet - 4

Answer Sheet

Integers-AnswerDownload the pdf

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