Integers Class 7 Maths Chapter Notes

Key Points: Integers at a Glance

• Integers are a collection of positive numbers (1, 2, 3...), negative numbers (-1, -2, -3...), and zero (0).
• Zero is neither positive nor negative.
• On a number line, numbers to the right are greater than numbers to the left.
• Adding a positive integer means moving to the right on the number line.
• Adding a negative integer means moving to the left on the number line.
• Subtracting a positive integer is equivalent to adding its negative counterpart (a - b = a + (-b)).
• Subtracting a negative integer is equivalent to adding its positive counterpart (a - (-b) = a + b).
• The product or quotient of two integers with the same sign is positive.
• The product or quotient of two integers with different signs is negative.
• Division by zero is undefined.

Essential Definitions

Integers

A set of numbers that includes all whole numbers (0, 1, 2, ...) and their negative counterparts (..., -2, -1). Represented by Z.

Absolute Value

The distance of an integer from zero on the number line, always a non-negative value. Denoted by |a|. For example, |-5| = 5 and |5| = 5.

Additive Inverse

For any integer 'a', its additive inverse is '-a' such that when added together, they result in zero (a + (-a) = 0). For example, the additive inverse of 7 is -7.

Additive Identity

The integer 0 is the additive identity because for any integer 'a', a + 0 = a.

Multiplicative Identity

The integer 1 is the multiplicative identity because for any integer 'a', a × 1 = a.

Commutative Property

For any two integers 'a' and 'b', an operation is commutative if changing the order of operands does not change the result (a + b = b + a; a × b = b × a).

Associative Property

For any three integers 'a', 'b', and 'c', an operation is associative if changing the grouping of operands does not change the result ((a + b) + c = a + (b + c); (a × b) × c = a × (b × c)).

Distributive Property

Multiplication distributes over addition for integers: a × (b + c) = (a × b) + (a × c).

Operations on Integers and Their Properties

Understanding how to perform operations (addition, subtraction, multiplication, and division) with integers is fundamental. Beyond just calculating, knowing the properties of these operations helps simplify calculations and solve complex problems more efficiently.

Addition of Integers

When adding integers:

• If two positive integers are added, their sum is a positive integer (e.g., 5 + 3 = 8).
• If two negative integers are added, their sum is a negative integer (e.g., -5 + (-3) = -8).
• If a positive and a negative integer are added, find the difference between their absolute values and keep the sign of the integer with the larger absolute value (e.g., -5 + 8 = 3; 5 + (-8) = -3).

Properties of Addition:

• Closure Property: The sum of any two integers is always an integer. (a + b is an integer).
• Commutative Property: The order of addition does not matter. (a + b = b + a).
• Associative Property: The grouping of integers for addition does not affect the sum. ((a + b) + c = a + (b + c)).
• Additive Identity: Adding zero to any integer leaves the integer unchanged. (a + 0 = a).
• Additive Inverse: For every integer 'a', there exists an integer '-a' such that their sum is zero. (a + (-a) = 0).

Subtraction of Integers

Subtraction of integers can be thought of as adding the additive inverse of the second integer to the first. For example, a - b = a + (-b).

Properties of Subtraction:

• Closure Property: The difference between any two integers is always an integer. (a - b is an integer).
• Not Commutative: a - b ≠ b - a (e.g., 5 - 3 ≠ 3 - 5).
• Not Associative: (a - b) - c ≠ a - (b - c) (e.g., (5 - 3) - 2 = 0, but 5 - (3 - 2) = 4).

Multiplication of Integers

When multiplying integers:

• Positive × Positive = Positive (e.g., 3 × 4 = 12)
• Negative × Negative = Positive (e.g., (-3) × (-4) = 12)
• Positive × Negative = Negative (e.g., 3 × (-4) = -12)
• Negative × Positive = Negative (e.g., (-3) × 4 = -12)

Properties of Multiplication:

• Closure Property: The product of any two integers is always an integer. (a × b is an integer).
• Commutative Property: The order of multiplication does not matter. (a × b = b × a).
• Associative Property: The grouping of integers for multiplication does not affect the product. ((a × b) × c = a × (b × c)).
• Distributive Property: Multiplication distributes over addition. (a × (b + c) = (a × b) + (a × c)).
• Multiplicative Identity: Multiplying any integer by one leaves the integer unchanged. (a × 1 = a).
• Property of Zero: Multiplying any integer by zero always results in zero. (a × 0 = 0).

Division of Integers

When dividing integers:

• The quotient of two integers with the same sign (both positive or both negative) is positive (e.g., 12 ÷ 3 = 4; (-12) ÷ (-3) = 4).
• The quotient of two integers with different signs (one positive, one negative) is negative (e.g., (-12) ÷ 3 = -4; 12 ÷ (-3) = -4).

Properties of Division:

• Not Closed: Division of integers is not always an integer (e.g., 5 ÷ 2 = 2.5, which is not an integer).
• Not Commutative: a ÷ b ≠ b ÷ a.
• Not Associative: (a ÷ b) ÷ c ≠ a ÷ (b ÷ c).
• Division by Zero is Undefined.

Property Checklist for Integer Operations

Worked Examples

• 1. Simplify: -15 + (-7) - (-10) -15 + (-7) - (-10) = -15 - 7 + 10 (Subtracting a negative is adding a positive) = -22 + 10 = -12
• 2. Evaluate: (-8) × 5 + (-20) ÷ (-4) Following BODMAS/PEMDAS: (-8) × 5 = -40 (-20) ÷ (-4) = 5 So, -40 + 5 = -35
• 3. Using Distributive Property: -6 × (7 + (-3)) -6 × (7 + (-3)) = (-6 × 7) + (-6 × -3) = -42 + 18 = -24

Exam Tip: Avoiding Common Mistakes

Many students lose marks in integer problems due to sign errors. Always double-check the signs when adding, subtracting, multiplying, or dividing. Remember:

• Two negatives make a positive only in multiplication and division (e.g., -5 × -2 = 10, -10 ÷ -2 = 5). For addition/subtraction, -5 - 2 = -7.
• Pay close attention to the Order of Operations (BODMAS/PEMDAS): Brackets, Orders (powers/roots), Division/Multiplication (from left to right), Addition/Subtraction (from left to right).
• Don't confuse additive inverse (sum is 0) with multiplicative inverse (product is 1, usually a fraction, not typically an integer). For Class 7, focus on additive inverse primarily.

Practice Questions with Solutions

• Q: What is the additive inverse of 25? And what property explains a + 0 = a? A: The additive inverse of 25 is -25. The property explaining a + 0 = a is the Additive Identity.
• Q: Calculate: (-12) × (-3) + (-50) ÷ 10. A: (-12) × (-3) = 36. (-50) ÷ 10 = -5. So, 36 + (-5) = 31.
• Q: Is the statement 'The difference of two integers is always an integer' true or false? A: True. This is the Closure Property under subtraction.
• Q: If a × b = b × a, which property is being demonstrated? A: The Commutative Property of multiplication.

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