Operations with Integers
Multiplication and Division of Positive and Negative Numbers
Two Numbers Hidden in a Puzzle
Rakesh challenges you: "I'm thinking of two numbers. Their sum is 25 and their difference is 11. Can you find them?"
You might guess and check: 19 and 6 works! (19 + 6 = 25, and 19 - 6 = 13... no, that's not 11). Then 18 and 7... yes! (18 + 7 = 25, and 18 - 7 = 11).
Now Rakesh adds a twist: "What if I say the sum is 25 but the difference is -11?" By simply swapping the numbers (7 and 18 become 7 and 18 with 7 first), you see how negative numbers represent direction — just like negative differences go backward. This is the heart of integer operations.
Imagine a carrom coin on a number line. When you strike it rightward, it moves in the positive direction (+). When you strike it leftward, it moves in the negative direction (-).
If the coin is struck twice: first 5 units right, then 3 units left, where does it end? Use positive (+5) and negative (-3): the final position is 5 + (-3) = 2 units right of zero.
Multiplication is like multiplying the strikes. Multiplying by a positive number means you're placing tokens into a bag repeatedly. Multiplying by a negative number means you're removing tokens. When you remove negatives (two negatives), you end up adding them back, which is why negative × negative = positive.
What Are Integers?
Integers include all whole numbers (1, 2, 3, ...) and their negatives (-1, -2, -3, ...) plus zero. We write them on a number line: ...−5, −4, −3, −2, −1, 0, +1, +2, +3, +4, +5... The direction matters: +5 means 5 units right, −5 means 5 units left.
Addition & Subtraction Review
Adding integers is like moving on a number line. 5 + 3 = 8 (start at 5, move 3 right). But 5 + (−3) = 2 (start at 5, move 3 left). Subtracting is the same as adding the opposite: 5 − 3 = 5 + (−3) = 2.
Multiplication: Positive × Positive
Think of 4 × 3 using the token model: place 3 positive tokens into an empty bag 4 times. Result: 12 positive tokens. So 4 × 3 = 12. This is familiar multiplication.
Multiplication: Positive × Negative
Now try 4 × (−2): place 2 negative tokens into the bag 4 times. Result: 8 negative tokens. So 4 × (−2) = −8. The key insight: multiplying by a negative number reverses the sign. Positive times negative = negative.
Multiplication: Negative × Positive
What about (−4) × 2? When the multiplier is negative, we remove tokens instead of placing them. Remove 2 positive tokens from an empty bag 4 times. But we start empty! So we add zero pairs (one positive + one negative) and then remove the positives, leaving negatives. Result: −8. So (−4) × 2 = −8.
Multiplication: Negative × Negative
For (−4) × (−2): remove 2 negative tokens from an empty bag 4 times. Again, we start with zero pairs, but this time remove the negatives, leaving positives. Result: 8. So (−4) × (−2) = 8. This is the big insight: negative times negative = positive!
The Pattern in Multiplication
Look at the sequence: 4×3=12, 3×3=9, 2×3=6, 1×3=3, 0×3=0, (−1)×3=−3, (−2)×3=−6... Each time we decrease the multiplier by 1, the product decreases by 3 (the multiplicand). This pattern explains why negatives work the way they do.
Division of Integers
Division is the inverse of multiplication. If 25 × (−4) = (−100), then (−100) ÷ 25 = (−4). The rules: positive ÷ positive = positive, negative ÷ negative = positive, positive ÷ negative = negative, negative ÷ positive = negative. Same sign = positive result, different signs = negative result.
In 628 CE, the mathematician Brahmagupta wrote about integer multiplication using the concepts of "fortune" (positive) and "debt" (negative). His rules:
- "The product of two fortunes is a fortune" (positive × positive = positive)
- "The product of two debts is a fortune" (negative × negative = positive)
- "The product of a debt and a fortune is a debt" (negative × positive = negative)
These ancient rules were revolutionary — they were the first explicit articulation of integer arithmetic, laying groundwork for algebra itself!
Does the order matter? 3 × (−4) = −12 and (−4) × 3 = −12. Yes, order doesn't matter: a × b = b × a. This is commutativity.
What about grouping? (5 × −3) × 4 = (−15) × 4 = −60, and 5 × ((−3) × 4) = 5 × (−12) = −60. Grouping doesn't matter either: (a × b) × c = a × (b × c). This is associativity. These properties hold for integers just as they do for positive numbers!
Does the distributive property work? Test: 5 × (4 + (−2)) = 5 × 2 = 10, and 5 × 4 + 5 × (−2) = 20 + (−10) = 10. Yes! And (−2) × (4 + (−3)) = (−2) × 1 = −2, and (−2) × 4 + (−2) × (−3) = −8 + 6 = −2. The distributive property holds for all integers: a × (b + c) = (a × b) + (a × c). This is crucial for algebra.
An elevator descends into a mining shaft at 3 metres per minute. After 60 minutes from ground level (0), where is it? Method 1: 0 − 180 = −180 (subtract 180 metres). Method 2: 60 × (−3) = −180 (60 minutes times downward speed). Both give −180 metres (180 metres below ground). Integer multiplication elegantly captures both magnitude and direction!
The Mistake: Students often think (−5) × (−3) = −15 (applying "negative times negative equals negative"). Or they might think negative × negative should still be negative because "two negatives don't cancel in real life."
Why It Happens: The token model isn't intuitive at first. Students understand (−5) × 3 = −15 (place negatives repeatedly), but struggle with (−5) × (−3) because "removing negative tokens" is abstract.
The Fix: Visualize the pattern:
5 × 3 = 15
(−5) × 3 = −15 (flip the sign)
(−5) × (−3) = 15 (flip twice, back to positive!)
Or use the token model precisely: removing negatives leaves positives. Practice with concrete sequences: (−1)×(−1)=1, (−1)×(−2)=2, (−2)×(−2)=4. The pattern becomes clear.
Socratic Sandbox — Test Your Understanding
What is (−6) × 7?
Reveal Answer & Explanation
Answer: −42 One number is negative, one is positive. The result is negative. Using tokens: place 7 negatives 6 times = 42 negatives = −42.
What is (−8) × (−5)?
Reveal Answer & Explanation
Answer: 40 Both numbers are negative. The result is positive. Using tokens: remove 5 negatives 8 times from zero pairs leaves 40 positives = 40.
Why does negative × negative = positive make sense in the elevator problem?
Reveal Answer & Explanation
Think of it this way: If the elevator is going down (negative speed), and we reverse time (negative direction in time), then effectively the elevator is going up! Two negations reverse each other, resulting in a positive outcome. Mathematically, (−3 m/min) × (−60 min) = +180 metres moved upward (or 180 metres gained in the backward time direction).
A cement company earns ₹8 profit per white cement bag and ₹5 loss per grey cement bag. Why can we write this as 8 and −5?
Reveal Answer & Explanation
Representing direction with integers: Profit (gain) is positive direction: +8. Loss (reduction) is negative direction: −5. The total profit/loss = 3000(8) + 5000(−5) = 24000 − 25000 = −1000 (a loss of ₹1000). Integers let us combine gains and losses in one calculation.
A temperature drops 5°C every hour for 4 hours. Write an expression and find the total change.
Reveal Answer & Explanation
Expression: 4 × (−5) = −20°C
Explanation: The rate is −5°C per hour (dropping). Over 4 hours, the total change is 4 × (−5) = −20°C (a 20°C drop). If it started at 8°C, it ends at 8 + (−20) = −12°C.
Use the distributive property to simplify (−3) × (5 + (−2)) without multiplying (−3) by (5 + (−2)) directly.
Reveal Answer & Explanation
Solution: (−3) × (5 + (−2)) = (−3) × 5 + (−3) × (−2) = −15 + 6 = −9
Check: (−3) × (5 + (−2)) = (−3) × 3 = −9 ✓
The distributive property splits the multiplication, making it easier to compute: negative times positive = negative (−15), negative times negative = positive (6), then add: −15 + 6 = −9.