1. Chapter at a Glance
In Class 6 you added and subtracted integers using lifts, number lines and tokens. Now we ask: what is 4 × (−3)? What about (−4) × (−3)? Why is a negative times a negative positive? This chapter builds the rules for multiplying and dividing integers through patterns and reasoning, and studies the properties of these operations so that you can calculate quickly and correctly.
- Quick revision: adding and subtracting integers.
- Multiplying integers: patterns and sign rules.
- Dividing integers and the sign of the quotient.
- Properties: closure, commutative, associative, distributive, identity.
- Using properties for smart calculation.
- Brahmagupta's rules for fortunes and debts.
2. Quick Revision: Adding and Subtracting
- Same signs: add the sizes, keep the sign. (−7) + (−5) = −12.
- Different signs: subtract the sizes, take the sign of the bigger size. (−9) + 4 = −5.
- Subtracting = adding the opposite: 6 − (−8) = 6 + 8 = 14.
3. Multiplying Integers
Positive × negative
3 × (−4) means three lots of −4: (−4) + (−4) + (−4) = −12. So a positive times a negative is negative. Since multiplication can be done in either order, (−4) × 3 = −12 too.
Negative × negative — a pattern
Watch the pattern as the first number goes down by 1:
| Product | Value |
|---|---|
| 3 × (−4) | −12 |
| 2 × (−4) | −8 |
| 1 × (−4) | −4 |
| 0 × (−4) | 0 |
| (−1) × (−4) | 4 |
| (−2) × (−4) | 8 |
Each step adds 4 to the answer. Continuing the pattern past zero gives positive answers. So a negative times a negative is positive.
- (+) × (+) = (+)
- (+) × (−) = (−)
- (−) × (+) = (−)
- (−) × (−) = (+)
For a product of many integers: if the number of negative factors is even, the product is positive; if it is odd, the product is negative.
Find (−5) × 6 × (−2) × (−1).
Solution: There are three negative factors (odd), so the answer is negative. Size: 5 × 6 × 2 × 1 = 60. Answer = −60.
4. Dividing Integers
Division is the reverse of multiplication. Since (−4) × 3 = −12, we get −12 ÷ 3 = −4 and −12 ÷ (−4) = 3. The sign rules are the same as for multiplication:
- Dividing by zero is not defined.
- 0 divided by any non-zero integer is 0.
- Division of integers does not always give an integer: (−7) ÷ 2 = −3.5.
The temperature fell by 18 °C evenly over 6 hours. What was the change per hour?
Solution: Change = −18 ÷ 6 = −3 °C per hour.
5. Properties of Operations on Integers
| Property | Addition | Multiplication | Subtraction / Division |
|---|---|---|---|
| Closure (result is an integer) | Yes | Yes | Subtraction yes; division no |
| Commutative (order) | a + b = b + a | a × b = b × a | No |
| Associative (grouping) | Yes | Yes | No |
| Identity | 0 (a + 0 = a) | 1 (a × 1 = a) | — |
| Inverse | −a (a + (−a) = 0) | — | — |
Distributive property: a × (b + c) = a × b + a × c also holds for integers. Example: (−6) × (10 − 2) = −60 + 12 = −48, and directly (−6) × 8 = −48 ✔.
Multiplication by −1 gives the additive inverse: (−1) × a = −a. And a × 0 = 0 for every integer.
Find (−25) × 37 × 4.
Solution: Regroup: (−25 × 4) × 37 = −100 × 37 = −3700.
6. Brahmagupta's Rules
In Brāhmasphuṭasiddhānta (628 CE), Brahmagupta described positive numbers as "fortunes" and negative numbers as "debts" and stated rules that match ours exactly: the product or quotient of two fortunes is a fortune; of two debts is a fortune; of a debt and a fortune is a debt. These rules made India the first place where negative numbers were treated as full members of arithmetic.
7. Solved Problems (NCERT Style)
In a quiz, +4 for a correct answer and −2 for a wrong one. Rohan answered 15 correctly and 10 wrongly. Find his score.
Solution: 15 × 4 + 10 × (−2) = 60 − 20 = 40.
A company earns ₹8 profit per bag of white cement and loses ₹5 per bag of grey cement. It sells 3,000 white and 5,000 grey bags. Profit or loss?
Solution: 3000 × 8 + 5000 × (−5) = 24,000 − 25,000 = −1,000. A loss of ₹1,000.
An elevator descends into a mine shaft at 6 m per minute. Where is it after 45 minutes if it started 10 m above ground?
Solution: Change = 45 × (−6) = −270 m. Position = 10 − 270 = −260 m (260 m below ground).
Find 53 × (−18) + (−18) × (−3).
Solution: (−18) × (53 − 3) = (−18) × 50 = −900.
8. Master Formula Table
| Rule | Example |
|---|---|
| (+)(+) = + | 4 × 5 = 20 |
| (+)(−) = − | 4 × (−5) = −20 |
| (−)(−) = + | (−4) × (−5) = 20 |
| Even number of negatives → + | (−1)⁴ = 1 |
| Odd number of negatives → − | (−1)³ = −1 |
| (−a) ÷ (−b) = a ÷ b | (−24) ÷ (−6) = 4 |
| a × 0 = 0; a ÷ 0 undefined | (−9) × 0 = 0 |
| a(b + c) = ab + ac | (−3)(7 + 3) = −30 |
9. Master Summary Table
| Topic | Key idea | Exam focus |
|---|---|---|
| Multiplication | Sign rules via patterns | Products of several integers |
| Division | Same sign rules | Rates of change |
| Properties | Commutative, associative, distributive | Smart calculation |
| Zero and one | Identities; ÷ 0 undefined | True/false statements |
| Applications | Profit/loss, depth, scores | Word problems |
10. Mind Map
- Pattern reasoning
- (−)(−) = +
- Count negatives
- Reverse of ×
- Same sign rules
- ÷ 0 undefined
- Closure
- Commutative, associative
- Distributive
- Scores, profit/loss
- Temperature, depth
11. Tips and Tricks
- Decide the sign first by counting negative factors, then multiply the sizes.
- Pair numbers that make 10, 100 or 1000 (like 25 × 4) before multiplying.
- Take out a common factor using the distributive property.
- Applying the multiplication sign rule to addition: (−3) + (−4) is −7, not +7.
- Writing (−2)⁴ as −16 (it is +16); but −2⁴ means −(2⁴) = −16.
- Dividing by zero.
12. Important One-liners
- A positive times a negative is negative.
- A negative times a negative is positive.
- The product is negative when there is an odd number of negative factors.
- Division follows the same sign rules as multiplication.
- Division by zero is not defined.
- Integers are closed under addition, subtraction and multiplication but not division.
- 1 is the multiplicative identity; 0 is the additive identity.
- (−1) × a = −a.
- Brahmagupta gave the sign rules in 628 CE.
13. Exam-pattern Questions with Detailed Answers
Find (−9) × (−7).
- −63
- 63
- −16
- 16
Show answer
Negative × negative = positive: 63.
Answer: (B) 63
Find (−84) ÷ 12.
Show answer
Different signs → negative: −7.
Answer: −7
What is the sign of (−1) × (−1) × … (25 factors)?
Show answer
25 negative factors — an odd number — so the product is −1.
Answer: Negative (−1)
Find (−12) × 7 × (−5).
Show answer
Two negatives → positive. 12 × 7 × 5 = 420.
Answer: 420
Fill in: ___ ÷ (−8) = 9.
Show answer
Multiply back: 9 × (−8) = −72.
Answer: −72
Which statement is false?
- (−5) × 0 = 0
- (−6) ÷ (−2) = 3
- (−3) − (−3) = −6
- (−4) × 1 = −4
Show answer
(−3) − (−3) = −3 + 3 = 0, not −6.
Answer: (C)
Use the distributive property to find (−37) × 99.
Show answer
(−37) × (100 − 1) = −3700 + 37 = −3663.
Answer: −3663
A submarine dives from sea level at 4 m per second. Where is it after 1 minute?
Show answer
60 × (−4) = −240 m.
Answer: 240 m below sea level (−240 m)
Find two integers whose product is −36 and sum is 5.
Show answer
Product negative → one positive, one negative. Factor pairs of 36: (9, 4) works since 9 + (−4) = 5.
Answer: 9 and −4
Evaluate [(−8) × (−3) − 30] ÷ (−2).
Show answer
(−8) × (−3) = 24; 24 − 30 = −6; (−6) ÷ (−2) = 3.
Answer: 3
Show by an example that division of integers is not associative.
Show answer
(−24 ÷ 4) ÷ 2 = (−6) ÷ 2 = −3, but −24 ÷ (4 ÷ 2) = −24 ÷ 2 = −12. The results differ.
Answer: −3 ≠ −12
In a test, +3 for each correct answer and −1 for each wrong one (no blanks). A student attempted 30 questions and scored 50. How many were correct?
Show answer
Let c be correct; wrong = 30 − c. 3c − (30 − c) = 50 → 4c = 80 → c = 20.
Answer: 20 correct (10 wrong)
14. Quick Revision Summary
- Like signs → positive; unlike signs → negative (for × and ÷).
- Many factors: count negatives — even → positive, odd → negative.
- ÷ 0 is undefined; 0 ÷ a = 0.
- Addition and multiplication are commutative and associative; × distributes over + and −.
- Brahmagupta's debts and fortunes gave these rules first.