Class 7 Maths Part 2 Chapter 2: Operations with Integers Notes

What is this chapter about?

In Class 6 you added and subtracted integers. Operations with Integers goes further: it gives meaning to multiplying and dividing negative numbers, using a bag of tokens and number patterns, finds the sign rules, and checks which properties of multiplication still hold. Along the way there are puzzles, a carrom coin, exam marks with negative marking, a lift in a mine, a magic grid and pattern machines.

A quick recap

Rakesh thinks of two numbers whose sum is 25 and difference is 11: by trying pairs you find 18 and 7. For a difference of −11, swap them: 7 and 18. Integers also describe movement. If a carrom coin moves right for positive numbers and left for negative ones, two strikes a and b leave it at P = a + b: a strike of 5 and then −7 leaves it at −2.

With tokens, a green token is +1, a red token is −1, and a zero pair (one of each) is 0. To find 7 − 18, put in 11 zero pairs so that 18 positives can be removed; 11 negatives are left, so 7 − 18 = −11. Subtracting a number is the same as adding its additive inverse: 7 − 18 = 7 + (−18) and 4 − (−12) = 4 + 12.

Multiplying integers with tokens

Start with an empty bag. In a × b, the multiplier a says how many times, and the multiplicand b says which group of tokens. A positive multiplier puts groups in; a negative multiplier takes groups out.

So 4 × (−2) puts 2 negatives in 4 times: −8. For (−4) × 2 the bag is empty, so put in 2 zero pairs each time and take out the 2 positives, 4 times; 8 negatives remain: −8. For (−4) × (−2), take out 2 negatives 4 times from zero pairs; 8 positives remain: +8. Read more in multiplying integers.

Patterns and sign rules

When the multiplicand is 3, each step down in the multiplier lowers the product by 3, and the pattern runs on past zero: (−1) × 3 = −3, (−2) × 3 = −6. When the multiplicand is −3, each step down raises the product by 3: (−1) × (−3) = 3, (−2) × (−3) = 6. So the size of the product is always the product of the sizes, and:

• Same signs (both positive or both negative): the product is positive.
• Different signs: the product is negative.

Also 1 × a = a, and (−1) × a = −a, the additive inverse of a. In 628 CE, Brahmagupta’s Brahmasphutasiddhanta gave these rules for the first time, calling positive numbers fortunes (dhana) and negative numbers debts (rina): the product or quotient of two fortunes, or of two debts, is a fortune, and of a fortune and a debt is a debt.

Integers at work

In an exam with +5 for a correct answer and −2 for a wrong one, Mala’s 30 correct and 20 wrong answers give 30 × 5 + 20 × (−2) = 150 + (−40) = 110 marks. A lift in a mine that goes down 3 m every minute changes its position by −3 each minute: from ground level it is at 60 × (−3) = −180 after an hour, and from 15 m above ground it is at 15 + 45 × (−3) = −120 after 45 minutes.

The magic grid

Circle a number in a 4 × 4 grid, strike out its row and column, and repeat until no number is left; then multiply the circled numbers. Whatever you choose, the product is −30240. The secret is that the grid is a multiplication table: each circled set uses every row number (2, −7, 3, 5) and every column number (4, −2, 6, −3) exactly once.

Dividing integers

Division undoes multiplication. (−100) ÷ 25 asks what times 25 gives −100; since 25 × (−4) = −100, the answer is −4. In the same way (−100) ÷ (−4) = 25 and 50 ÷ (−25) = −2. Divide the magnitudes and use the same sign rules: same signs give a positive quotient, different signs a negative one. See dividing integers.

Properties of multiplication

Multiplication of integers is commutative (3 × (−4) = (−4) × 3) and associative ((5 × (−3)) × 4 = 5 × ((−3) × 4) = −60), so several integers can be multiplied in any order. Multiplying −1 again and again gives 1, −1, 1, −1: a product with an odd number of negative factors is negative. Multiplication is also distributive over addition: 4 × (2 + (−3)) = 4 × 2 + 4 × (−3). Read properties of integer multiplication.

The chapter ends with pattern machines that take three numbers and give a result, such as a + b − c, for you to decode. Answers to every question are in the NCERT solutions for Part 2 Chapter 2.