Class 7 — Mathematics (NCERT)
Integers: Multiplication and Division, Properties
Class 7
- ✓By the end of this lesson students will be able to multiply integers using the correct sign rules.
- ✓By the end of this lesson students will be able to divide integers using the correct sign rules.
- ✓By the end of this lesson students will be able to identify and apply the properties of multiplication of integers (Closure, Commutative, Associative, Distributive, Multiplicative Identity, Property of Zero).
- ✓By the end of this lesson students will be able to solve problems involving multiplication and division of integers, including those requiring the use of properties.
Key concepts
When multiplying integers, we follow specific rules for determining the sign of the product:\n1. The product of two positive integers is a positive integer. (e.g., 3 × 4 = 12)\n2. The product of two negative integers is a positive integer. (e.g., (-3) × (-4) = 12)\n3. The product of a positive integer and a negative integer is a negative integer. (e.g., 3 × (-4) = -12 or (-3) × 4 = -12)\n4. The product of any integer and zero is zero. (e.g., a × 0 = 0)
When dividing integers, we follow specific rules for determining the sign of the quotient:\n1. The quotient of two positive integers is a positive integer. (e.g., 12 ÷ 4 = 3)\n2. The quotient of two negative integers is a positive integer. (e.g., (-12) ÷ (-4) = 3)\n3. The quotient of a positive integer and a negative integer is a negative integer. (e.g., 12 ÷ (-4) = -3 or (-12) ÷ 4 = -3)\n4. Zero divided by any non-zero integer is zero. (e.g., 0 ÷ a = 0, where a ≠ 0)\n5. Division by zero is undefined. (e.g., a ÷ 0 is undefined)
Integers exhibit certain properties under multiplication:\n1. Closure Property: For any two integers 'a' and 'b', the product 'a × b' is always an integer. (e.g., 5 × (-3) = -15, which is an integer).\n2. Commutative Property: For any two integers 'a' and 'b', the order of multiplication does not change the product. That is, a × b = b × a. (e.g., (-2) × 5 = -10 and 5 × (-2) = -10).\n3. Associative Property: For any three integers 'a', 'b', and 'c', the grouping of integers does not affect the product. That is, (a × b) × c = a × (b × c). (e.g., (2 × (-3)) × 4 = (-6) × 4 = -24 and 2 × ((-3) × 4) = 2 × (-12) = -24).\n4. Distributive Property: For any three integers 'a', 'b', and 'c', multiplication distributes over addition and subtraction. That is, a × (b + c) = (a × b) + (a × c) and a × (b - c) = (a × b) - (a × c). (e.g., 5 × (2 + (-3)) = 5 × (-1) = -5 and (5 × 2) + (5 × (-3)) = 10 + (-15) = -5).\n5. Multiplicative Identity: For any integer 'a', multiplying by 1 does not change the integer. That is, a × 1 = 1 × a = a. Here, 1 is called the multiplicative identity for integers.\n6. Property of Zero: For any integer 'a', multiplying by 0 always results in 0. That is, a × 0 = 0 × a = 0.
Key facts to remember
- 1The product of two integers with the same sign is always positive.
- 2The product of two integers with different signs is always negative.
- 3The quotient of two integers with the same sign is always positive.
- 4The quotient of two integers with different signs is always negative.
- 5Multiplication by zero always results in zero (a × 0 = 0).
- 6Division by zero is undefined.
- 71 is the multiplicative identity for integers (a × 1 = a).
- 8The set of integers is closed under multiplication.
Worked examples
Example 1
Find the product: (-12) × 9 × (-5)
Answer
540
When multiplying more than two integers, count the number of negative signs. If it's even, the product is positive. If it's odd, the product is negative.
Example 2
Evaluate: (-144) ÷ (-12)
Answer
12
Example 3
Using a suitable property, evaluate: 25 × (-48) + (-48) × (-15)
Answer
-480
The distributive property helps simplify calculations by factoring out common terms.
Common mistakes
- ✗Incorrectly applying the sign rules for multiplication and division, especially when dealing with multiple negative signs.
- ✗Confusing the rules for multiplication/division with those for addition/subtraction of integers.
- ✗Attempting to divide by zero, which is undefined.
- ✗Errors in applying the distributive property, particularly when negative signs are involved in the terms being distributed.
Exam tips
- ★Always remember the sign rules: 'Same signs, positive; Different signs, negative' for both multiplication and division.
- ★When solving problems with multiple operations, follow the BODMAS rule (Brackets, Of, Division, Multiplication, Addition, Subtraction).
- ★Practice identifying and applying the properties of integers, as they can significantly simplify calculations.
- ★Double-check your calculations, especially when working with negative numbers, to avoid careless errors.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.