Multiplication of Integers
Have you ever wondered what happens when you multiply numbers that include negatives? In Class 6, you learned about integers and how to add and subtract them. Now in Class 7, we take the next step: multiplying integers. Multiplication of integers is one of the most important skills you will use throughout mathematics, science, and even in daily life. Whether you are calculating the total loss a shopkeeper makes over several days, figuring out temperature drops, or working out scores in a quiz where negative marking applies, you are multiplying integers.
Multiplication of integers follows specific sign rules that determine whether the answer is positive or negative. Once you master these rules, you will find that multiplying integers is just as easy as multiplying whole numbers, with one extra step of figuring out the sign. In this chapter, we will explore the rules for multiplying integers, understand why they work, look at many solved examples, and practise problems that will make you confident with this topic.
Think of it this way: if you lose Rs. 5 every day for 4 days, your total loss is 4 times Rs. 5 = Rs. 20. In integer language, this is 4 x (-5) = -20. The multiplication tells us the total amount lost. This simple idea is the foundation of everything we will learn here.
What is Multiplication of Integers?
Integers are the set of whole numbers and their negatives: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Multiplication of integers means finding the product of two or more integers. Just like multiplying whole numbers, multiplying integers can be thought of as repeated addition. For example, 3 x (-4) means adding (-4) three times: (-4) + (-4) + (-4) = -12.
When we multiply two integers, we multiply their absolute values (the numbers without the sign) and then determine the sign of the product using specific rules. The absolute value of an integer is its distance from zero on the number line, regardless of direction. For example, the absolute value of -7 is 7, and the absolute value of 7 is also 7. We write this as |-7| = 7.
Sign Rules for Multiplication of Integers:
| First Integer | Second Integer | Product | Example |
|---|---|---|---|
| Positive (+) | Positive (+) | Positive (+) | (+3) x (+4) = +12 |
| Positive (+) | Negative (-) | Negative (-) | (+3) x (-4) = -12 |
| Negative (-) | Positive (+) | Negative (-) | (-3) x (+4) = -12 |
| Negative (-) | Negative (-) | Positive (+) | (-3) x (-4) = +12 |
An easy way to remember: same signs give a positive product, and different signs give a negative product. Think of it like this: two friends (same sign) agree (positive result), but two people who disagree (different signs) create a conflict (negative result).
Also, any integer multiplied by zero gives zero: a x 0 = 0. This is called the zero property of multiplication.
Multiplication of Integers Formula
Sign Rules for Multiplication of Integers:
(+) x (+) = (+)
(+) x (-) = (-)
(-) x (+) = (-)
(-) x (-) = (+)
Quick Rule:
Same signs → Positive product
Different signs → Negative product
Zero Property:
a x 0 = 0 x a = 0 (for any integer a)
Multiplying More Than Two Integers:
When multiplying several integers together, count the number of negative integers:
- If the count of negative integers is even (0, 2, 4, ...), the product is positive.
- If the count of negative integers is odd (1, 3, 5, ...), the product is negative.
Types and Properties
Multiplication of integers can be categorised into different cases based on the signs of the numbers involved:
Case 1: Positive x Positive
This is just like regular multiplication of whole numbers. For example, 5 x 3 = 15. Both numbers are positive, so the product is positive.
Case 2: Positive x Negative
Multiply the absolute values, then make the answer negative. For example, 6 x (-4) = -24. Think of it as: if you lose Rs. 4 each day for 6 days, your total loss is Rs. 24, which we write as -24.
Case 3: Negative x Positive
This is the same as Case 2, just written in a different order. For example, (-4) x 6 = -24. The commutative property tells us that the order does not matter.
Case 4: Negative x Negative
Multiply the absolute values, and the answer is positive. For example, (-5) x (-3) = 15. This might seem surprising at first! Here is one way to think about it: if losing Rs. 5 per day is represented by -5, then going back 3 days in time is -3. Going back in time on a loss means you are undoing the loss, which is a gain. So (-5) x (-3) = +15.
Case 5: Any Integer x Zero
Any integer multiplied by zero is always zero. For example, (-7) x 0 = 0 and 0 x 12 = 0.
Case 6: Multiplying More Than Two Integers
When multiplying three or more integers, multiply them two at a time from left to right, or simply count the negative signs. An even number of negatives gives a positive product; an odd number gives a negative product. For example: (-2) x (-3) x (-4) has 3 negatives (odd), so the product is negative: -(2 x 3 x 4) = -24.
Solved Examples
Example 1: Multiplying Two Positive Integers
Problem: Find the product: 8 x 7
Solution:
Step 1: Both integers are positive.
Step 2: Multiply the numbers: 8 x 7 = 56
Step 3: Same signs (both positive), so the product is positive.
Answer: 8 x 7 = 56
Example 2: Multiplying a Positive and a Negative Integer
Problem: Find the product: 6 x (-9)
Solution:
Step 1: One integer is positive (6) and the other is negative (-9).
Step 2: Multiply the absolute values: 6 x 9 = 54
Step 3: Different signs, so the product is negative.
Answer: 6 x (-9) = -54
Example 3: Multiplying a Negative and a Positive Integer
Problem: Find the product: (-12) x 5
Solution:
Step 1: First integer is negative (-12) and second is positive (5).
Step 2: Multiply the absolute values: 12 x 5 = 60
Step 3: Different signs, so the product is negative.
Answer: (-12) x 5 = -60
Example 4: Multiplying Two Negative Integers
Problem: Find the product: (-7) x (-8)
Solution:
Step 1: Both integers are negative.
Step 2: Multiply the absolute values: 7 x 8 = 56
Step 3: Same signs (both negative), so the product is positive.
Answer: (-7) x (-8) = 56
Example 5: Multiplying an Integer by Zero
Problem: Find the product: (-15) x 0
Solution:
Step 1: One of the integers is 0.
Step 2: Any number multiplied by 0 is 0 (Zero Property).
Answer: (-15) x 0 = 0
Example 6: Multiplying Three Integers
Problem: Find the product: (-3) x 4 x (-5)
Solution:
Step 1: Count the negative integers: there are 2 negatives (-3 and -5).
Step 2: Even number of negatives, so the product will be positive.
Step 3: Multiply the absolute values: 3 x 4 x 5 = 60
Answer: (-3) x 4 x (-5) = 60
Alternatively, step by step: (-3) x 4 = -12, then (-12) x (-5) = 60. Same answer!
Example 7: Multiplying Four Integers
Problem: Find the product: (-2) x (-3) x (-1) x 4
Solution:
Step 1: Count the negative integers: there are 3 negatives (-2, -3, -1).
Step 2: Odd number of negatives, so the product will be negative.
Step 3: Multiply the absolute values: 2 x 3 x 1 x 4 = 24
Answer: (-2) x (-3) x (-1) x 4 = -24
Example 8: Word Problem: Temperature Drop
Problem: The temperature in Shimla drops by 3°C every hour after sunset. What will be the total drop in temperature after 5 hours?
Solution:
Step 1: A drop of 3°C is represented as -3.
Step 2: Total drop in 5 hours = 5 x (-3)
Step 3: Different signs (positive x negative), so the answer is negative: 5 x 3 = 15, so the product is -15.
Answer: The total temperature drop is -15°C (the temperature falls by 15°C).
Example 9: Word Problem: Pocket Money Loss
Problem: Ravi spends Rs. 8 more than his pocket money each day. How much extra has he spent after 6 days? Represent this as an integer multiplication.
Solution:
Step 1: Spending more than pocket money means a loss of Rs. 8 per day, represented as -8.
Step 2: Over 6 days: 6 x (-8)
Step 3: Multiply: 6 x 8 = 48, and different signs give a negative result.
Answer: 6 x (-8) = -48. Ravi has overspent by Rs. 48.
Example 10: Word Problem: Quiz with Negative Marking
Problem: In a quiz, each wrong answer carries -2 marks. Priya answered 7 questions wrong. What is her total score from wrong answers?
Solution:
Step 1: Each wrong answer = -2 marks.
Step 2: Total marks from wrong answers = 7 x (-2)
Step 3: Multiply: 7 x 2 = 14, different signs give negative result.
Answer: 7 x (-2) = -14. Priya lost 14 marks due to wrong answers.
Real-World Applications
Multiplication of integers is used in many real-life situations:
Banking and Finance: When you withdraw money from an ATM, each withdrawal can be seen as a negative number. If you withdraw Rs. 500 three times, the total change in your account is 3 x (-500) = -1500, meaning your account decreases by Rs. 1,500. Banks use integer multiplication to calculate interest charges, penalties, and transaction summaries.
Temperature: Weather scientists use integer multiplication to calculate temperature changes. If the temperature drops by 2°C every hour, then after 6 hours the total drop is 6 x (-2) = -12°C. In cold storage and refrigeration, understanding how quickly temperature changes over time is critical.
Sports and Games: In cricket, if a batsman's scoring rate drops by 3 runs per over for 4 overs, the change is 4 x (-3) = -12 runs. In quiz shows with negative marking, wrong answers are multiplied by negative values. If each wrong answer costs 2 marks and you get 5 wrong, the total penalty is 5 x (-2) = -10 marks.
Business: A shopkeeper who makes a loss of Rs. 50 per day will have a total loss of 7 x (-50) = -350 over a week. Conversely, understanding that (-1) x (-50) = 50 helps in understanding that reversing a loss means a gain. Profit and loss calculations in business regularly use integer multiplication.
Science: In physics, if a car decelerates (slows down) at a rate of -2 m/s each second, then after 5 seconds the total change in speed is 5 x (-2) = -10 m/s. Integer multiplication is essential in understanding forces, charges, and directions. Positive and negative charges in electricity follow similar sign rules.
Elevators and Buildings: If an elevator goes down 3 floors at a time and does this 4 times, it has moved 4 x (-3) = -12 floors, meaning 12 floors below where it started. Building plans and construction calculations use integers to represent above and below ground levels.
Time Zones: Time zone differences can be represented with integers. If a city is 5 hours behind another (represented as -5), and you want to calculate the time difference for 3 such zones, the result is 3 x (-5) = -15 hours. International travel planning uses this concept.
Sea Level and Depth: Oceanographers measure depth below sea level using negative integers. If a submarine descends 50 metres per minute for 8 minutes, the total descent is 8 x (-50) = -400 metres (400 metres below sea level).
Key Points to Remember
- When two integers with the same sign are multiplied, the product is positive: (+) x (+) = (+) and (-) x (-) = (+).
- When two integers with different signs are multiplied, the product is negative: (+) x (-) = (-) and (-) x (+) = (-).
- Any integer multiplied by zero gives zero.
- Any integer multiplied by 1 gives the same integer (multiplicative identity).
- Any integer multiplied by -1 gives the opposite of that integer.
- When multiplying more than two integers, count the negative signs: even count = positive product, odd count = negative product.
- The commutative property holds: a x b = b x a.
- The associative property holds: (a x b) x c = a x (b x c).
- The distributive property holds: a x (b + c) = (a x b) + (a x c).
- Always multiply the absolute values first, then determine the sign of the answer.
Practice Problems
- Find the product: (-9) x 7
- Find the product: (-11) x (-6)
- Find the product: 15 x (-4)
- Find the product: (-2) x (-3) x (-5)
- Find the product: (-1) x (-1) x (-1) x (-1)
- A submarine dives 8 metres deeper every minute. Represent the total depth after 10 minutes as an integer multiplication and find the answer.
- The temperature in a freezer drops by 4°C every hour. What is the total change after 7 hours?
- In a game, a player loses 5 points for each foul. If the player commits 6 fouls, what is the total change in points?
Frequently Asked Questions
Q1. What is the rule for multiplying two negative integers?
When two negative integers are multiplied, the product is always positive. For example, (-4) x (-5) = 20. This is because two negatives cancel each other out. The rule is: same signs give a positive product.
Q2. Why is negative times negative equal to positive?
Think of it this way: if you owe someone Rs. 10 per day (that is -10), and we go back 3 days (-3 days), then you undo 3 days of debt, which is a gain of Rs. 30. So (-10) x (-3) = +30. Mathematically, it follows from the pattern: (-1) x 3 = -3, (-1) x 2 = -2, (-1) x 1 = -1, (-1) x 0 = 0, (-1) x (-1) = +1. Each time the second number decreases by 1, the product increases by 1.
Q3. What happens when you multiply any integer by zero?
Any integer multiplied by zero is always zero. This is called the Zero Property of Multiplication. For example, (-100) x 0 = 0 and 0 x 57 = 0. It does not matter how large or small the integer is; the product with zero is always zero.
Q4. How do you multiply three or more integers?
When multiplying three or more integers, multiply the absolute values of all the numbers. Then count how many of the integers are negative. If the count is even (0, 2, 4, ...), the product is positive. If the count is odd (1, 3, 5, ...), the product is negative. For example, (-2) x 3 x (-4) has 2 negatives (even), so the product is +(2 x 3 x 4) = +24.
Q5. Is the commutative property valid for multiplication of integers?
Yes, the commutative property holds for multiplication of integers. This means a x b = b x a for any two integers a and b. For example, (-5) x 3 = -15 and 3 x (-5) = -15. The order of multiplication does not change the product.
Q6. What is the difference between multiplying and adding integers?
In addition, you add the values considering their signs (same signs: add and keep the sign; different signs: subtract and keep the sign of the larger absolute value). In multiplication, you multiply the absolute values and then apply the sign rule: same signs give positive, different signs give negative. The operations are fundamentally different.
Q7. What is the product of an integer and -1?
Multiplying any integer by -1 gives the additive inverse (opposite) of that integer. For example, (-1) x 7 = -7 and (-1) x (-9) = 9. This property is useful for changing the sign of a number.
Q8. Can you give a real-life example of multiplying integers?
Sure! Suppose a shop loses Rs. 200 every day. After 5 days, the total loss is 5 x (-200) = -1000, meaning the shop has lost Rs. 1,000. Another example: if you go down 2 floors in a lift, 4 times, you have gone 4 x (-2) = -8, which means 8 floors below your starting position.
