Class 7 Maths Integers Notes

Introduction:
The chapter “Integers” in Class 7 Maths focuses on understanding the set of integers, their properties, and how to perform operations like addition, subtraction, multiplication, and division with them. Integers are a fundamental concept in mathematics and are used to represent both positive and negative numbers. This chapter forms the foundation for more advanced topics in mathematics.

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Key Concepts Covered:

1. What are Integers?

• Integers are the set of whole numbers and their negatives, including zero. In other words, integers consist of:
  • Positive integers: Numbers greater than zero (1, 2, 3, 4, …).
  • Negative integers: Numbers less than zero (-1, -2, -3, -4, …).
  • Zero (0): Zero is considered neither positive nor negative but is included as part of the integers.
• Symbolically: The set of integers is represented as:
  $\mathbb{Z} = \{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots \}$Z={…,−3,−2,−1,0,1,2,3,…}

2. Representation of Integers on a Number Line:

• Integers can be represented on a number line, where positive integers are located to the right of zero, and negative integers are located to the left of zero.
• The number line is a useful tool for comparing integers, performing arithmetic operations, and visualizing the relationship between numbers.

Example:

• On the number line:
  • Zero (0) is at the center.
  • Positive integers (1, 2, 3, …) are to the right.
  • Negative integers (-1, -2, -3, …) are to the left.

3. Addition of Integers:

• Addition of two positive integers results in a positive integer.
• Addition of two negative integers results in a negative integer.
• Addition of a positive and a negative integer:
  • If the positive integer is greater than the negative integer, the result will be positive.
  • If the negative integer is greater, the result will be negative.
  • If both integers are equal, the result is zero.

Rules:

• + + → + (Positive + Positive = Positive)
• – + → – or + – → – (Negative + Positive = Negative or Positive + Negative = Negative)
• – – → + (Negative + Negative = Negative, and the result is more negative)

Example:
$3 + (-5) = -2 \quad \text{or} \quad (-3) + 5 = 2$3+(−5)=−2or(−3)+5=2

4. Subtraction of Integers:

• Subtracting an integer is the same as adding its opposite.
• To subtract an integer, you change the sign of the number to be subtracted and then add.

Example:$$7 – 3 = 7 + (-3) = 4 \quad \text{or} \quad (-5) – 6 = -5 + (-6) = -11$$7−3=7+(−3)=4or(−5)−6=−5+(−6)=−11

5. Multiplication of Integers:

• Multiplying two positive integers results in a positive integer.
• Multiplying two negative integers results in a positive integer.
• Multiplying a positive integer by a negative integer results in a negative integer.
• Multiplying zero by any integer results in zero.

Rules:

• + × + → + (Positive × Positive = Positive)
• – × – → + (Negative × Negative = Positive)
• + × – → – (Positive × Negative = Negative)
• 0 × any integer = 0 (Multiplying by Zero)

Example:$$4 × (-3) = -12 \quad \text{or} \quad (-4) × (-3) = 12$$4×(−3)=−12or(−4)×(−3)=12

6. Division of Integers:

• Dividing two positive integers results in a positive integer.
• Dividing two negative integers results in a positive integer.
• Dividing a positive integer by a negative integer results in a negative integer.
• Dividing by zero is undefined (division by zero is not possible).

Rules:

• + ÷ + → + (Positive ÷ Positive = Positive)
• – ÷ – → + (Negative ÷ Negative = Positive)
• + ÷ – → – (Positive ÷ Negative = Negative)
• – ÷ + → – (Negative ÷ Positive = Negative)

Example:$$6 ÷ (-2) = -3 \quad \text{or} \quad (-6) ÷ (-2) = 3$$6÷(−2)=−3or(−6)÷(−2)=3

7. Properties of Integers:

• Commutative Property: For addition and multiplication, the order of numbers does not affect the result.
  • Addition: $a + b = b + a$a+b=b+a
  • Multiplication: $a \times b = b \times a$a×b=b×a
• Associative Property: For addition and multiplication, the grouping of numbers does not affect the result.
  • Addition: $(a + b) + c = a + (b + c)$(a+b)+c=a+(b+c)
  • Multiplication: $(a \times b) \times c = a \times (b \times c)$(a×b)×c=a×(b×c)
• Distributive Property: Multiplication distributes over addition and subtraction.
  • $a \times (b + c) = a \times b + a \times c$a×(b+c)=a×b+a×c

8. Absolute Value of an Integer:

• The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always non-negative.
• Notation: $|a|$∣a∣ represents the absolute value of integer $a$a.

Examples:

• $|3| = 3$∣3∣=3
• $|-5| = 5$∣−5∣=5

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Important Questions with Answers:

1. What are integers?
   • Answer: Integers are the set of whole numbers and their negatives, including zero. They consist of positive integers, negative integers, and zero.
2. How do you add two negative integers?
   • Answer: When adding two negative integers, the result is negative. Add their absolute values and attach the negative sign.
3. What is the absolute value of an integer?
   • Answer: The absolute value of an integer is its distance from zero on the number line, regardless of direction, and is always non-negative.
4. How do you subtract integers?
   • Answer: Subtraction of integers is the same as adding the opposite of the integer to be subtracted.
5. What is the product of two negative integers?
   • Answer: The product of two negative integers is a positive integer.
6. What happens when you multiply a positive integer by a negative integer?
   • Answer: When you multiply a positive integer by a negative integer, the result is a negative integer.
7. What is the quotient of two integers with different signs?
   • Answer: The quotient of two integers with different signs is negative.