CBSE Notes Class 7 Maths Chapter 1 Integers

Chapter 1 of CBSE Class 7 Maths, Integers, introduces students to the extended number system, including positive and negative numbers along with zero. This chapter builds on the concept of whole numbers and expands students' understanding by incorporating negative values to represent real-world scenarios such as temperatures below zero or financial losses.  

Students will learn the properties of integers, including operations like addition, subtraction, multiplication, and division. The chapter also emphasizes important concepts like the commutative, associative, and distributive properties, which simplify calculations involving integers. Through this chapter, learners develop the ability to handle both positive and negative values effectively, which is essential for higher-level mathematics.

1.0Natural Numbers

Counting numbers 1, 2, 3, 4, 5, and so on are called natural numbers. The collection of natural numbers is expressed as N = {1, 2, 3, 4, 5,....} 

2.0Whole Numbers 

When 0 is included with the natural numbers, the resulting set is called whole numbers. It is represented as W = {0, 1, 2, 3, 4, 5,....}

3.0Integers 

Integers include all counting numbers, their negatives, and zero. The set of integers is represented as Z or I = {....., –4, –3, –2, –1, 0, 1, 2, 3, 4,......} 

1. Positive Integers : The set of all positive integers is denoted as I+ = {1, 2, 3, 4,....}. Positive integers are essentially the same as natural numbers. 
2. Negative Integers : The set I–= {....., –3, –2, –1} is the set of all negative integers. 0 is neither positive nor negative.

4.0Representation of Integers on Number Line 

The number line representing the integers is given below

1. Addition of Integers on the Number Line :

Rule 1: When we add two positive integers, we add their values, and the result will take the positive sign (Common sign of both the integers)

Rule 2: When we add a positive and a negative integer, we find the difference of their numerical values, regardless of their signs and give the sign of the integer which is greater.

2. Subtraction of integers on the number Line

Subtract the numbers 5 – (+6) by using the number line. 

Explanation 

Subtraction is just an addition. When we change the operation from subtraction to addition, we need to use the opposite sign for the number that follows. This allows us to rewrite the expression as: 5 – (+6) ⇒ 5 + (–6) 

We start at the first number, 5 and move 6 units towards the left side.

5.0Properties of Addition of Integers 

• Closure property of addition: The sum of any two integers will always be an integer. 
• Commutative law of addition: For any two integers a and b the equation a + b = b + a holds true.
• Associative law of addition: For any three integers a, b, c the equation (a + b) + c = a + (b + c) is valid.
• Existence of additive identity: For any integer a, we have: a + 0 = 0 + a = a. 0 is called the additive identity for integers.
• Existence of additive inverse: For any integer a, we have: a + (–a) = (–a) + a = 0. The opposite of an integer a is (–a). The sum of an integer and its opposite is 0. Additive inverse of a is (–a).

6.0Properties of Subtraction of Integers 

• Closure property for subtraction: If a and b are any 2 integers, then (a – b) is always an integer.
• Subtraction of integers is not commutative

7.0Properties of Multiplication of Integers 

• Closure property for multiplication: The product of any two integers is always an integer.
• Commutative law for multiplication: For any 2 integers a and b, (a × b) = (b × a)
• Associative law for multiplication: For any 3 integers a, b, c, (a × b) × c = a × (b × c)
• Distributive law for multiplication over addition: For any integers a, b, c, a × (b + c) = (a × b) + (a × c)
• Distributive of multiplication over Subtraction: If a, b and c are 3 integers, then a × (b – c) = (a × b) – (a × c)
• Existence of multiplicative identity: For every integer a, we have: (a × 1) = (1× a) = a 

The number 1 is referred to as the multiplicative identity for integers.

• Existence of multiplicative inverse: Multiplicative inverse of a non-zero integer 'a' is the number $\frac{1}{a}$, as a $\times (\frac{1}{a})=(\frac{1}{a})\times a=1$
• Property of zero 

For every integer a, we have: (a × 0) = (0 × a) = 0

8.0Properties and Operation of Division 

Rule 1: For dividing one integer by the other, the two having unlike signs, we divide their values regardless of their signs and give a minus sign to the quotient. 

Rule 2: For dividing one integer by the other having like signs, we divide their values regardless of their signs and give a plus sign to the quotient. 

9.0Properties of Division of Integers

• For integers a and b, then (a ÷ b) is not always an integer.
• If a is a non-zero integer, then a ÷ a = 1
• If a is an integer, then (a ÷ 1) = a
• If a is a non-zero integer, then (0 ÷ a) = 0 but (a ÷ 0) is undefined.
• If a, b, c are integers, then (a ÷ b) ÷ c ≠ a ÷ (b ÷ c), unless c = 1 Thus, division of integers is not associative.

10.0Mind Map for Integers

11.0Sample Question for Integers

1. What is the smallest and largest integer?  

Ans: Integers extend infinitely in both the positive and negative directions. Hence, there is no smallest or largest integer. We can write:

$-\infty <\mathbb{I}<+\infty$