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.

**Download NCERT Class 7 Maths Chapter 1 Integers Notes and Solutions PDF**

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Table of Contents

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

**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.**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.**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.**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.**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.

**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)

**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 –

- the product of two negative integers is a positive integer;
- the product of three negative integers is a negative integer.
- 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.

### Topic 4: Properties of Multiplication of Integers

**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

**Commutativity of Multiplication:**For any two integers a and b, (a × b) = (b × a). So, we say that multiplication is commutative for integers.**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.**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

**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.**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

**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

- 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.
**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.

**Download NCERT Class 7 Maths Chapter 1 Integers Notes and Solutions PDF**

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Chapter 1 | Chapter 2 | Chapter 3 | Chapter 4 | Chapter 5 |

Chapter 6 | Chapter 7 | Chapter 8 | Chapter 9 | Chapter 10 |

Chapter 11 | Chapter 12 | Chapter 13 |

## NCERT Solutions of Class 7 Maths Chapter 1:

## Integers Solutions- Free PDF Download

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.**

**Observe this number line and write the temperature of the places marked on it.****What is the temperature difference between the hottest and the coldest places among the above?****What is the temperature difference between Lahulspiti and Srinagar?****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.

- 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.

- 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

- The difference between the temperatures of Lahulspiti and Srinagar is:

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

- 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.

**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.

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

So, (– 8) + (– 4) **[<]** (–8) – (– 4).

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

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

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

So, 23 – 41 + 11 **[>]** 23 – 41 – 11

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

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

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

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

### Exercise 1.2 Solutions

**Q 1. Write down a pair of integers whose:**

**sum is –7****difference is –10****sum is 0**

**Solutions. **The answers are given below.

- The pair of integers whose sum is -7 are -3 and -4; -3 + (-4) = -7.
- The pair of integers whose difference is -10 are -20 and -10; -20 -(-10) = -10
- 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:**

**(–5) + (– 8) = (– 8) + (…………)****–53 + ………… = –53****17 + ………… = 0****[13 + (– 12)] + (…………) = 13 + [(–12) + (–7)]**

**Solutions.** The answers are given below.

- 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.

- 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.

- Let us assume the missing integer to be 𝒙.

This gives us: 17 + 𝒙 = 0 ⇒ 𝒙 = -17.

Hence, the missing integer is -17.

- 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.

### Exercise 1.3 Solutions

**Q 1. Give answers to the following questions.**

**For any integer a, what is [(–1) × a] equal to?****Determine the integer whose product with (–1) is**

**–22****37****0**

**Solutions. **The answers to each question are given below.

- The product of [(–1) × a] results in -a.
- 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.**

**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?****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.

- 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

- 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.