# NCERT Class 7 Maths Chapter 1 ‘Integers’: Notes and Solutions (Free PDF)

From our study of Class 6 Maths, we have understood that integers are a bigger set of numbers that contain whole numbers along with negative numbers. In Class 7 Maths Chapter 1, we will study a bit more on the topic of integers. We will study more about the integers, their properties, and their operations. Let us continue to read the NCERT Class 7 Maths Chapter 1 Integers Notes and Solutions (PDF) below.

## NCERT Class 7 Maths Chapter 1 Notes – PDF Available

Check the topic-wise notes for NCERT Maths Class 7 – Chapter 1 below. You can also download the PDF of the notes and take a printout to study later when you need quick revision before going to the exam hall.

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## Integers Class 7 Notes

Let us now look at Class 7 Maths Chapter 1 notes.

### Topic 1: Properties of Addition and Subtraction of Integers

1. Closure under Addition: For any two integers a and b, (a + b) is an integer. Since addition of integers gives integers, we say integers are closed under addition.
2. Closure under Subtraction: If a and b are two integers then, (a – b) is also an integer. Thus, we say that the integers are closed under subtraction.
3. Commutative Property: For any two integers a and b, we can say that (a + b) = (b + a). Thus, we say that addition is commutative for integers.
4. Associative Property: For any integers a, b, and c, we can say [a + (b + c)] = [(a + b) + c]. Thus, we say that addition is associative for integers.
5. Additive Identity: For any integer a, (a + 0) = a = (0 + a). Thus, we see that when we add zero to any integer, we get the same integer. Zero is an additive identity for integers.

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### Topic 2: Multiplication of Integers

We know that the result of the multiplication of two positive integers is a positive integer. Now let us see what happens when 1 or both the integers under the operation of multiplication are negative integers.

1. Multiplication of a Positive and a Negative Integer: While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer.

a × (– b) = (– a) × b = – (a × b)

1. Multiplication of two Negative Integers: We multiply the two negative integers as whole numbers and put the positive sign before the product. Thus, we say that the product of two negative integers is a positive integer.

(– a) × (– b) = (a × b)

### Topic 3: Product of more than Two Negative Integers

Let us see the following:

(–1) × (–1) = +1

(–1) × (–1) × (–1) = –1

(–1) × (–1) × (–1) × (–1) = +1

(–1) × (–1) × (–1) × (–1) × (–1) = –1

From the above products we observe that –

1. the product of two negative integers is a positive integer;
2. the product of three negative integers is a negative integer.
3. product of four negative integers is a positive integer.

This means that if the integer (–1) is multiplied an even number of times, the product is +1 and if the integer (–1) is multiplied an odd number of times, the product is –1.

Hence, it can be concluded that if the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd, then the product is a negative integer.

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### Topic 4: Properties of Multiplication of Integers

1. Closure under Multiplication: The product of two integers is again an integer. So, we can say that integers are closed under multiplication.

(a × b) is an integer, for all integers a and b

1. Commutativity of Multiplication: For any two integers a and b, (a × b) = (b × a). So, we say that multiplication is commutative for integers.
2. Multiplication by Zero: For any integer a, (a × 0) = (0 × a) = 0. Thus, the product of any positive or negative integer and zero is zero.
3. Multiplicative Identity: For any integer a, (a × 1) = a = (1 × a). Thus, we see that when we multiply one with any integer, we get the same integer. Thus, we say that  1 is the multiplicative identity for integers.

(a × 1) = (1 × a) = a

1. Associativity for Multiplication: for any three integers a, b and c, [(a × b) × c] = [a × (b × c)]. Thus, like whole numbers, the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers.
2. Distributive Property: For any three integers a, b and c, [a × (b + c)] = (a × b + a × c). Thus, we say that the multiplication of integers is distributive over addition.

### Topic 5: Division of Integers

When we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (–) before the quotient. We, thus, get a negative integer.

In general, for any two positive integers a and b,

a ÷ (– b) = (– a) ÷ b, where b ≠ 0

Similarly, when we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+). That is, we get a positive integer.

In general, for any two positive integers a and b,

(– a) ÷ (– b) = (a ÷ b), where b ≠ 0

### Topic 6: Properties of Division of Integers

1. Commutativity of Division: For any two integers a and b, (a ÷ b) ≠ (b ÷ a). So, we say that division is not commutative for integers.

For example, (-8) ÷ (-4) = 2, which is an integer; but (-4) ÷ (-8) = not an integer

1. For any integer a, (a ÷ 1) = a. Thus we say that a negative integer divided by 1 gives the same negative integer. So, any integer divided by 1 gives the same integer.
2. Associativity for Division: for any three integers a, b, and c, [(a ÷ b) ÷ c] ≠ [a ÷ (b ÷ c)]. Thus, the division of three integers depends upon the grouping of integers and thus division is not associative for integers.

## NCERT Solutions of Class 7 Maths Chapter 1:

Below we have provided solutions for NCERT Class 7 Maths Chapter 1, Integers. Go through for answers to some important questions.

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### Exercise 1.1 Solutions

Q 1. The following number line shows the temperature in degrees Celsius (°C) at different places on a particular day.

1. Observe this number line and write the temperature of the places marked on it.
2. What is the temperature difference between the hottest and the coldest places among the above?
3. What is the temperature difference between Lahulspiti and Srinagar?
4. Can we say the temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar?

Solutions. The answers are given below.

1. The temperature in Lahulspiti is -8 °C.

The temperature in Srinagar is -2 °C.

The temperature in Shimla is 5 °C.

The temperature in Ooty is 14 °C.

The temperature in Bengaluru is 22 °C.

1. The temperature of the hottest place, Bengaluru is 22 °C and of the coldest place, Lahulspiti is -8 °C.

Their difference ⇒ 22 °C – (-8 °C) = 30 °C

1. The difference between the temperatures of Lahulspiti and Srinagar is:

-8 °C – (-2 °C) = -8 °C + 2 °C = -6 °C

1. The temperature in Srinagar is -2 °C and in Shimla is 5 °C. Taken together, their temperature is ⇒ -2 °C + 5 °C = 3 °C.

So, it’s true that the temperature of Srinagar and Shimla taken together is less than the temperature at Shimla.

Now, as the temperature in Srinagar is -2 °C. So, the temperature of Srinagar and Shimla taken together is not less than the temperature at Srinagar.

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Q 2. Use the sign of >, < or = in the box to make the statements true.

(a) (– 8) + (– 4) ⛾ (–8) – (– 4)

(b) (– 3) + 7 – (19) ⛾ 15 – 8 + (– 9)

(c) 23 – 41 + 11 ⛾ 23 – 41 – 11

(d) 39 + (– 24) – (15) ⛾ 36 + (– 52) – (– 36)

(e) – 231 + 79 + 51 ⛾ –399 + 159 + 81

Solutions. The answers are given below.

1. (– 8) + (– 4) = -12 and (–8) – (– 4) = -4. We know that -12 < -4.

So, (– 8) + (– 4) [<] (–8) – (– 4).

1. (– 3) + 7 – (19) = -15 and 15 – 8 + (– 9) = -2. We know that -15 < -2

So, (– 3) + 7 – (19) [<] 15 – 8 + (– 9).

1. 23 – 41 + 11 = -7 and 23 – 41 – 11 = -29. We know that -7 > -29

So, 23 – 41 + 11 [>] 23 – 41 – 11

1. 39 + (– 24) – (15) = 0 and 36 + (– 52) – (– 36) = 20. We know that 0 < 20

So, 39 + (– 24) – (15) [<] 36 + (– 52) – (– 36)

1. – 231 + 79 + 51 = -101 and –399 + 159 + 81 = -159. We know that -101 > -159

So, – 231 + 79 + 51 [>] –399 + 159 + 81

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### Exercise 1.2 Solutions

Q 1. Write down a pair of integers whose:

1. sum is –7
2. difference is –10
3. sum is 0

Solutions. The answers are given below.

1. The pair of integers whose sum is -7 are -3 and -4; -3 + (-4) = -7.
2. The pair of integers whose difference is -10 are -20 and -10; -20 -(-10) = -10
3. The pair of integers whose sum is zero are -1 and 1; -1 + 1 = 0

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Q 2. Fill in the blanks to make the following statements true:

1. (–5) + (– 8) = (– 8) + (…………)
2. –53 + ………… = –53
3. 17 + ………… = 0
4. [13 + (– 12)] + (…………) = 13 + [(–12) + (–7)]

Solutions. The answers are given below.

1. Let us assume the missing integer to be “𝒙”.

(–5) + (– 8) = (– 8) + (𝒙)

∴ (–5) + (– 8) = -13

So, (– 8) + (𝒙) = -13 ⇒ 𝒙 = -5

The missing integer is -5.

1. In this case, let us assume the missing integer to be “𝒙”.

This gives us: -53 + 𝒙 = -53 ⇒ 𝒙 = -53 + 53 = 0

∴ 𝒙 = 0, the missing integer is 0.

1. Let us assume the missing integer to be 𝒙.

This gives us: 17 + 𝒙 = 0 ⇒ 𝒙 = -17.

Hence, the missing integer is -17.

1. Let us assume the missing integer to be 𝒙.

This gives us: [13 +(-12)] + 𝒙 = 13 + [(-12) + (-7)]

13 + [(-12) + (-7)] = -6

∴ [13 +(-12)] + 𝒙 = -6 ⇒ 𝒙 = -6 -13 + 12 = -7

Hence, the missing integer is -7.

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### Exercise 1.3 Solutions

Q 1. Give answers to the following questions.

1. For any integer a, what is [(–1) × a] equal to?
2. Determine the integer whose product with (–1) is
• –22
• 37
• 0

Solutions. The answers to each question are given below.

1. The product of [(–1) × a] results in -a.
2. When any integer is multiplied by -1, it will acquire a negative sign without any change in its magnitude.
• So, the product of -22 and -1:  -22 × -1 = 22 (two negative integers in multiplication result in a positive integer.)
• The product: 37 × -1 = -37
• The product of any number with 0 results in 0.

∴ 0 × -1 = 0

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Q 2. A certain freezing process requires that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Ans. The given temperature of the room is 40 °C.

Decrease in temperature per hour = 5 °C

Decrease in temperature in 10 hours = 5 × 10 °C = 50 °C

∴ the final temperature of the room after 10 hours = 40 °C – 50 °C = -10 °C

Q 3. A cement company earns a profit of ₹ 8 per bag of white cement sold and a loss of ₹ 5 per bag of grey cement sold.

1. The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
2. What is the number of white cement bags it must sell to have neither profit nor loss if the number of grey bags sold is 6,400 bags?

Ans. The answers are given below.

1. Profit on selling 1 bag of white cement = ₹ 8

Profit on selling 3000 bags of white cement = ₹ 8 × 3000 = ₹ 24,000

Now, loss on selling 1 bag of grey cement = -₹ 5

Loss on selling 5000 bags of grey cement = -₹ 5 × 5000 = -₹ 25,000

Since the total loss is greater than the total profit for the month, the company is in loss.

The value of loss: ₹ 24,000 + (-₹ 25,000) = – ₹ 1000

1. Loss on selling 6400 bags of grey cement = – ₹ 5 × 6400 = -₹ 32,000

To compensate for the loss of -₹ 32,000, the company should make a profit of ₹ 32,000.

Let us assume the number of bags of white cement to be sold is 𝒙.

So, total profit: ₹ 8 × 𝒙 = ₹ 32,000 ⇒ 𝒙 = 32000/8 = 4000 bags of white cement

∴ the company must sell 4,000 bags of white cement to have neither profit nor loss.

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## FAQs

Q.1. What is the definition of integers in Maths?

Ans: The integers are the collection of all positive numbers, negative numbers, and zero. Integers do not include fractions or decimal numbers.

Q.2. What is the additive identity of integers?

Ans: Zero is the additive identity for integers.

Q.3. Is division commutative for integers?

Ans: For any two integers a and b, (a ÷ b) ≠ (b ÷ a). So, we say that division is not commutative for integers.

For example, (-8) ÷ (-4) = 2, which is an integer; but (-4) ÷ (-8) = not an integer.

This was all about NCERT Class 7 Maths Chapter 1, Integers in which we studied the different properties of integers. Download the NCERT Class 7 Maths Chapter 1 Notes and Solutions PDF to ace your exam preparations. Follow the CBSE Class 7 Maths Solutions and Notes for more such chapter notes and important questions and answers for preparation for CBSE Class 7 Maths.