Multiplication of Integers

Class 7 Maths · Chapter 10: Operations with Integers · NCERT Ganita Prakash section P2 2.2

Learn

Tokens in a bag: 4 × 2 means put 2 positives into an empty bag 4 times: 8 positives, so 8. And 4 × (−2) means put 2 negatives in 4 times: 8 negatives, so −8. A positive multiplier means putting in.

4 × (−2) = −8

A negative multiplier means taking out. For (−4) × 2, take 2 positives out of the empty bag 4 times. There are none, so first add zero pairs, then remove the positives: 8 negatives stay, so −8. For (−4) × (−2), remove 2 negatives 4 times: 8 positives stay, so +8!

(−4) × (−2) = 8

Patterns agree: 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0, so (−1) × 3 = −3, (−2) × 3 = −6. And 1 × (−3) = −3, 0 × (−3) = 0, so (−1) × (−3) = 3: each step adds 3. Rule: the size is the product of the sizes; same signs → positive, different signs → negative.

same signs: + · different signs: −

Properties: 1 × a = a and (−1) × a = −a. Swapping does not matter: a × b = b × a. Brahmagupta stated these rules in 628 CE using fortune (dhana, positive) and debt (ṛṇa, negative): "the product of two debts is a fortune." Example: 30 correct answers at +5 and 20 wrong at −2 give 150 + (−40) = 110 marks.

30 × 5 + 20 × (−2) = 110

Explore: multiply with tokens.

1. (−4) × 2: what is left in the bag? Show it.
2. (−3) × (−3): show what is left.

Remember

• Same signs give a positive product; different signs give a negative product.
• (−1) × a = −a, and a × b = b × a for all integers.

Quick check

123 × 456 = 56088. What is (−123) × (−456)?

1. 56088
2. −56088
3. 0
4. −123

Think about it

In a quiz, +4 for right and −2 for wrong. Kabir answered all and scored 40 with 15 right. How many questions were there?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.