Division of Integers
After learning how to multiply integers, the next natural step is to divide them. Division of integers is closely related to multiplication, just as division of whole numbers is related to their multiplication. If you know that 3 x 4 = 12, then you also know that 12 / 4 = 3 and 12 / 3 = 4. The same relationship holds for integers, but now we need to handle positive and negative signs.
Division of integers appears everywhere in daily life. Imagine you and your friends share a pizza bill equally, or a company divides its losses among its partners, or you need to figure out how many floors an elevator went down per trip. All of these situations involve dividing integers. In competitive exams and in NCERT Class 7 Maths, this topic is essential.
The great news is that the sign rules for division are exactly the same as for multiplication. If you have already learned the multiplication sign rules, you are already halfway there! In this chapter, we will understand the sign rules for dividing integers, see how division and multiplication are connected, work through many solved examples, and practise problems to build confidence.
One important thing to note: unlike multiplication, division of integers does not always give an integer result. For example, 7 / 2 = 3.5, which is not an integer. We say that integers are not closed under division. We will explore what this means and other properties in detail.
What is Division of Integers?
Division of integers means splitting an integer into equal parts or finding how many times one integer fits into another. If a and b are two integers (where b is not zero), then a / b means dividing a by b.
Division is the inverse (opposite) operation of multiplication. If a x b = c, then c / b = a and c / a = b (provided a and b are not zero).
Sign Rules for Division of Integers:
| Dividend | Divisor | Quotient | Example |
|---|---|---|---|
| Positive (+) | Positive (+) | Positive (+) | (+12) / (+3) = +4 |
| Positive (+) | Negative (-) | Negative (-) | (+12) / (-3) = -4 |
| Negative (-) | Positive (+) | Negative (-) | (-12) / (+3) = -4 |
| Negative (-) | Negative (-) | Positive (+) | (-12) / (-3) = +4 |
The sign rules are identical to multiplication: same signs give a positive quotient, and different signs give a negative quotient.
Important Rules:
- Division by zero is not defined. You cannot divide any number by 0. For example, 5 / 0 has no answer.
- Zero divided by any non-zero integer is zero. For example, 0 / (-7) = 0.
- Any integer divided by 1 gives the same integer. For example, (-9) / 1 = -9.
- Any non-zero integer divided by itself gives 1. For example, (-5) / (-5) = 1.
Division of Integers Formula
Sign Rules for Division of Integers:
(+) / (+) = (+)
(+) / (-) = (-)
(-) / (+) = (-)
(-) / (-) = (+)
Quick Rule:
Same signs → Positive quotient
Different signs → Negative quotient
Special Cases:
0 / a = 0 (where a is not zero)
a / 0 = Not Defined
a / 1 = a
a / a = 1 (where a is not zero)
Relationship with Multiplication:
If a x b = c, then c / a = b and c / b = a (for non-zero a and b).
Types and Properties
Let us look at the different cases of dividing integers, with simple explanations:
Case 1: Positive / Positive
This is just like dividing whole numbers. For example, 20 / 4 = 5. Both numbers are positive, so the quotient is positive. Think of sharing 20 chocolates among 4 friends; each friend gets 5 chocolates.
Case 2: Negative / Positive
Divide the absolute values, then make the answer negative. For example, (-20) / 4 = -5. Think of it as: a total debt of Rs. 20 shared equally among 4 people means each person owes Rs. 5, written as -5.
Case 3: Positive / Negative
Divide the absolute values, then make the answer negative. For example, 20 / (-4) = -5. The signs are different, so the result is negative.
Case 4: Negative / Negative
Divide the absolute values, and the answer is positive. For example, (-20) / (-4) = 5. Both signs are the same (negative), so the quotient is positive. Think of it as: if a total debt of Rs. 20 was created by 4 equal losses, each loss was Rs. 5 (positive, because we are measuring the size of each loss).
Case 5: Zero / Any Non-Zero Integer
Zero divided by any non-zero integer is always zero. For example, 0 / (-8) = 0. If you have nothing to share, each person gets nothing.
Case 6: Any Integer / Zero
Division by zero is undefined. You cannot divide by zero. If someone asks "What is 5 / 0?", the answer is "not defined" or "undefined". This is a fundamental rule in mathematics.
Non-Closure Property:
Unlike addition and multiplication, dividing two integers does not always give an integer. For example, (-7) / 2 = -3.5, which is not an integer. This means integers are not closed under division. Other examples include: 5 / 3 = 1.666... and (-10) / 4 = -2.5. This is an important difference between division and the other three operations.
Successive Division:
When dividing a series of integers, perform the operations from left to right. For example, (-24) / 4 / (-2) = first (-24) / 4 = -6, then (-6) / (-2) = 3. Be careful with grouping, as the associative property does not hold for division.
Solved Examples
Example 1: Dividing a Positive by a Positive Integer
Problem: Find: 36 / 9
Solution:
Step 1: Both numbers are positive.
Step 2: Divide: 36 / 9 = 4
Step 3: Same signs (both positive), so the quotient is positive.
Answer: 36 / 9 = 4
Example 2: Dividing a Negative by a Positive Integer
Problem: Find: (-42) / 7
Solution:
Step 1: Dividend is negative (-42), divisor is positive (7).
Step 2: Divide the absolute values: 42 / 7 = 6
Step 3: Different signs, so the quotient is negative.
Answer: (-42) / 7 = -6
Example 3: Dividing a Positive by a Negative Integer
Problem: Find: 56 / (-8)
Solution:
Step 1: Dividend is positive (56), divisor is negative (-8).
Step 2: Divide the absolute values: 56 / 8 = 7
Step 3: Different signs, so the quotient is negative.
Answer: 56 / (-8) = -7
Example 4: Dividing Two Negative Integers
Problem: Find: (-63) / (-9)
Solution:
Step 1: Both numbers are negative.
Step 2: Divide the absolute values: 63 / 9 = 7
Step 3: Same signs (both negative), so the quotient is positive.
Answer: (-63) / (-9) = 7
Example 5: Dividing Zero by an Integer
Problem: Find: 0 / (-13)
Solution:
Step 1: The dividend is 0.
Step 2: Zero divided by any non-zero integer is zero.
Answer: 0 / (-13) = 0
Example 6: Division by Zero
Problem: Find: (-25) / 0
Solution:
Step 1: The divisor is 0.
Step 2: Division by zero is not defined.
Answer: (-25) / 0 is not defined.
Example 7: Using Division as Inverse of Multiplication
Problem: If (-8) x (?) = 72, find the missing integer.
Solution:
Step 1: We need to find (?) = 72 / (-8)
Step 2: Divide absolute values: 72 / 8 = 9
Step 3: Different signs (positive dividend, negative divisor), so the quotient is negative.
Answer: 72 / (-8) = -9. The missing integer is -9.
Check: (-8) x (-9) = 72 ✓
Example 8: Word Problem: Sharing Losses
Problem: A group of 5 friends invested in a business and made a total loss of Rs. 2,500. If the loss is shared equally, how much does each friend lose?
Solution:
Step 1: Total loss = -2500 (negative because it is a loss).
Step 2: Number of friends = 5.
Step 3: Each friend's share = (-2500) / 5 = -500
Answer: Each friend loses Rs. 500.
Example 9: Word Problem: Temperature Change
Problem: The temperature in a city dropped by 18°C over 6 hours. If the drop was the same each hour, what was the hourly temperature change?
Solution:
Step 1: Total temperature change = -18°C (a drop is negative).
Step 2: Time = 6 hours.
Step 3: Hourly change = (-18) / 6 = -3
Answer: The temperature dropped by 3°C each hour (hourly change = -3°C).
Example 10: Word Problem: Elevator Floors
Problem: An elevator went down a total of 24 floors in 4 equal trips. How many floors did it go down in each trip?
Solution:
Step 1: Total floors moved = -24 (down is negative).
Step 2: Number of trips = 4.
Step 3: Floors per trip = (-24) / 4 = -6
Answer: The elevator went down 6 floors in each trip (i.e., -6 floors per trip).
Real-World Applications
Division of integers is useful in many real-life situations:
Sharing Expenses and Losses: When friends split a restaurant bill or share a business loss, they divide the total amount among themselves. If 4 friends share a loss of Rs. 800, each person's share is (-800) / 4 = -200, meaning each person loses Rs. 200. This concept is fundamental in business partnerships where profits and losses are distributed.
Average Calculations: Finding the average of a set of numbers (which may include negatives) involves division. For example, the average of -10, -20, and 30 is (-10 + (-20) + 30) / 3 = 0 / 3 = 0. Averages of temperature changes, altitude variations, and financial data often involve dividing integers.
Speed and Distance: If a car travels -120 km (120 km in the reverse direction) in 3 hours, the speed is (-120) / 3 = -40 km/h, meaning 40 km/h in the reverse direction. In physics, velocity can be positive or negative depending on direction, and dividing total displacement by time gives the average velocity.
Science Experiments: Scientists divide total changes by the number of observations. If a chemical reaction causes a temperature drop of 15°C over 5 experiments, the average change per experiment is (-15) / 5 = -3°C. Water level changes, population declines, and chemical concentration reductions all use integer division.
Gaming and Quizzes: In a quiz where a team loses 30 points total over 6 rounds, the average loss per round is (-30) / 6 = -5 points per round. This helps in analysing performance and identifying which rounds were worse.
Stock Market and Finance: If a stock loses Rs. 100 in value over 4 days, the average daily change is (-100) / 4 = -25, meaning the stock lost Rs. 25 per day on average. Financial analysts use this to track trends and make predictions.
Geography and Altitude: If a hiker descends 600 metres over 4 hours, the average descent per hour is (-600) / 4 = -150 metres per hour. Submarine depths, mountain elevations, and sea-level measurements all involve dividing positive and negative integers.
Agriculture: If the groundwater level in a region drops by 24 cm over 6 months, the average monthly drop is (-24) / 6 = -4 cm per month. Farmers and water management teams use such calculations for planning.
Key Points to Remember
- Same sign division gives a positive quotient: (+) / (+) = (+) and (-) / (-) = (+).
- Different sign division gives a negative quotient: (+) / (-) = (-) and (-) / (+) = (-).
- Division by zero is not defined for any integer.
- Zero divided by any non-zero integer is zero: 0 / a = 0.
- Any integer divided by 1 gives the same integer: a / 1 = a.
- Any non-zero integer divided by itself gives 1: a / a = 1.
- Division is the inverse of multiplication: if a x b = c, then c / b = a.
- Integers are not closed under division because dividing two integers may not give an integer (e.g., 7 / 2 = 3.5).
- The commutative property does not hold for division: a / b is not always equal to b / a.
- The associative property does not hold for division: (a / b) / c is not always equal to a / (b / c).
- Always divide the absolute values first, then apply the sign rule.
Practice Problems
- Find: (-48) / 8
- Find: 72 / (-12)
- Find: (-81) / (-9)
- Find: 0 / (-25)
- Is (-7) / 0 defined? Explain why.
- If (-6) x (?) = -54, find the missing integer using division.
- A submarine dived 60 metres in 5 equal descents. How many metres did it dive each time? Write your answer as an integer.
- The total score change in a game for a team over 8 rounds was -32 points. What was the average score change per round?
Frequently Asked Questions
Q1. What are the sign rules for dividing integers?
The sign rules for dividing integers are the same as for multiplication. When you divide two integers with the same sign (both positive or both negative), the quotient is positive. When you divide two integers with different signs (one positive and one negative), the quotient is negative. For example, (-20) / (-4) = +5 (same signs) and 20 / (-4) = -5 (different signs).
Q2. Why can we not divide by zero?
Division by zero is not defined because there is no number that, when multiplied by 0, gives a non-zero result. If we try 5 / 0, we are asking: what number times 0 equals 5? There is no such number because anything times 0 is 0. For 0 / 0, every number would work, making the answer undefined. So mathematicians say division by zero is simply not allowed.
Q3. Is zero divided by a negative number equal to zero?
Yes. Zero divided by any non-zero integer (positive or negative) is always zero. For example, 0 / (-5) = 0, 0 / 3 = 0, and 0 / (-100) = 0. This is because 0 shared among any number of groups gives 0 to each group.
Q4. Are integers closed under division?
No, integers are not closed under division. This means that dividing one integer by another does not always give an integer. For example, (-7) / 2 = -3.5, which is not an integer. This is different from addition, subtraction, and multiplication, where the result of operating on two integers is always an integer.
Q5. Does the commutative property work for division of integers?
No, the commutative property does not hold for division. This means a / b is not always equal to b / a. For example, 12 / (-3) = -4, but (-3) / 12 = -0.25. Since -4 is not equal to -0.25, the order of division matters.
Q6. How is division of integers related to multiplication?
Division is the inverse (opposite) of multiplication. If you know that (-6) x 4 = -24, then you can say (-24) / 4 = -6 and (-24) / (-6) = 4. To check a division answer, multiply the quotient by the divisor; if you get the dividend, your answer is correct.
Q7. What is (-1) divided by (-1)?
(-1) / (-1) = 1. Since both the dividend and divisor have the same sign (both negative), the quotient is positive. And 1 / 1 = 1, so the answer is +1.
Q8. Can you give a real-life example of dividing integers?
Suppose a group of 4 students has to pay off a total debt of Rs. 1,200. Each student's share is (-1200) / 4 = -300. This means each student owes Rs. 300. Another example: if the temperature dropped 12°C over 4 hours uniformly, the drop per hour was (-12) / 4 = -3°C.
