Addition of Integers
Now that you know what integers are and how to represent them on a number line, it is time to learn how to add them. Adding positive numbers is something you have been doing since Class 1 — but adding negative numbers is a new and exciting skill. What does it mean to add a negative number? Think of it this way: if you have Rs. 50 (that is +50) and you earn Rs. 30 more (that is +30), your total is +80. Simple. But what if you have Rs. 50 and you spend Rs. 30? That spending is like adding a negative: 50 + (-30) = 20. Addition of integers follows clear rules that you can master with practice. The key idea is to use the number line as your guide — adding a positive number means moving to the right, and adding a negative number means moving to the left. Once you understand this visual approach, you can then learn the rules for adding integers without needing to draw a number line every time. This topic is the foundation for all arithmetic with integers — subtraction, multiplication and division all build on addition. Let us master it step by step.
What is Addition of Integers?
The addition of integers is the process of finding the sum of two or more integers. Since integers include both positive and negative numbers, there are different cases to consider.
Adding Integers on a Number Line:
The number line is the best way to visualise integer addition:
Rule 1: Start at the first number on the number line.
Rule 2: If you are adding a positive number, move to the right (towards bigger numbers).
Rule 3: If you are adding a negative number, move to the left (towards smaller numbers).
Rules for Adding Integers (Without Number Line):
Case 1: Adding Two Positive Integers
Simply add the numbers as usual. The result is positive.
Example: (+5) + (+3) = +8
Case 2: Adding Two Negative Integers
Add the absolute values (the numbers without signs) and put a negative sign on the result.
Rule: (-a) + (-b) = -(a + b)
Example: (-5) + (-3) = -(5 + 3) = -8
Think of it as: you owe Rs. 5 and you owe Rs. 3 more — now you owe Rs. 8 total.
Case 3: Adding a Positive and a Negative Integer
Find the difference of their absolute values (subtract the smaller from the larger) and use the sign of the number with the larger absolute value.
Rule: If |a| > |b|, then (+a) + (-b) = +(a - b)
Rule: If |a| < |b|, then (+a) + (-b) = -(b - a)
Example: (+8) + (-3) = +(8 - 3) = +5 (since 8 > 3, result is positive)
Example: (+3) + (-8) = -(8 - 3) = -5 (since 8 > 3 and the bigger number 8 is negative, result is negative)
Additive Identity: Adding zero to any integer gives the same integer: a + 0 = a. Zero is called the additive identity.
Additive Inverse: For every integer a, there exists an integer -a such that a + (-a) = 0. The number -a is called the additive inverse of a. For example, the additive inverse of +7 is -7 because 7 + (-7) = 0.
Types and Properties
Let us look at the properties and different approaches to adding integers:
1. Properties of Integer Addition
Closure Property: The sum of any two integers is always an integer. For example, (-3) + 5 = 2 (an integer). You will never get a fraction or decimal by adding integers.
Commutative Property: The order of addition does not matter: a + b = b + a.
Example: (-3) + 7 = 4, and 7 + (-3) = 4. Same result.
Associative Property: When adding three or more integers, the grouping does not matter: (a + b) + c = a + (b + c).
Example: [(-2) + 3] + (-4) = 1 + (-4) = -3, and (-2) + [3 + (-4)] = (-2) + (-1) = -3. Same result.
Additive Identity: a + 0 = 0 + a = a. Adding zero changes nothing.
Additive Inverse: a + (-a) = 0. Every integer cancels out with its opposite.
2. The Number Line Method (Visual Approach)
This is the most intuitive method for beginners:
- To add (+3): start at your position and take 3 steps to the right.
- To add (-4): start at your position and take 4 steps to the left.
Example: (-2) + (+5): Start at -2. Move 5 steps right: -2 → -1 → 0 → 1 → 2 → 3. Answer: +3.
Example: 3 + (-7): Start at 3. Move 7 steps left: 3 → 2 → 1 → 0 → -1 → -2 → -3 → -4. Answer: -4.
3. The Absolute Value Method (Quick Mental Math)
Same signs: Add the absolute values, keep the common sign.
(-4) + (-6) → add 4 + 6 = 10 → both negative → answer is -10.
(+4) + (+6) → add 4 + 6 = 10 → both positive → answer is +10.
Different signs: Subtract the absolute values (smaller from larger), take the sign of the larger absolute value.
(+9) + (-4) → subtract 9 - 4 = 5 → 9 is bigger and positive → answer is +5.
(+4) + (-9) → subtract 9 - 4 = 5 → 9 is bigger and negative → answer is -5.
4. The Money Analogy
Think of positive numbers as money you have and negative numbers as money you owe.
- (+50) + (+30) = you have Rs. 50 and earn Rs. 30 more = you have Rs. 80. Answer: +80.
- (-50) + (-30) = you owe Rs. 50 and borrow Rs. 30 more = you owe Rs. 80. Answer: -80.
- (+50) + (-30) = you have Rs. 50 and spend Rs. 30 = you have Rs. 20 left. Answer: +20.
- (+30) + (-50) = you have Rs. 30 but spend Rs. 50 = you owe Rs. 20. Answer: -20.
Solved Examples
Example 1: Example 1: Adding two positive integers
Problem: Find (+12) + (+8).
Solution:
Both numbers are positive. Simply add them.
(+12) + (+8) = +20
Number line method: Start at 12. Move 8 steps to the right. Land on 20.
Answer: +20
Example 2: Example 2: Adding two negative integers
Problem: Find (-7) + (-5).
Solution:
Both numbers are negative. Add the absolute values: 7 + 5 = 12. Since both are negative, the result is negative.
(-7) + (-5) = -12
Number line method: Start at -7. Move 5 steps to the left (because adding a negative means going left). -7 → -8 → -9 → -10 → -11 → -12.
Money analogy: You owe Rs. 7 and borrow Rs. 5 more. Now you owe Rs. 12.
Answer: -12
Example 3: Example 3: Adding a positive and a negative integer (positive result)
Problem: Find (+15) + (-6).
Solution:
Different signs. Subtract the absolute values: 15 - 6 = 9. Since 15 (the positive number) has the larger absolute value, the result is positive.
(+15) + (-6) = +9
Money analogy: You have Rs. 15 and spend Rs. 6. You have Rs. 9 left.
Answer: +9
Example 4: Example 4: Adding a positive and a negative integer (negative result)
Problem: Find (+4) + (-11).
Solution:
Different signs. Subtract the absolute values: 11 - 4 = 7. Since 11 (the negative number) has the larger absolute value, the result is negative.
(+4) + (-11) = -7
Number line method: Start at 4. Move 11 steps to the left. 4 → 3 → 2 → 1 → 0 → -1 → -2 → -3 → -4 → -5 → -6 → -7.
Money analogy: You have Rs. 4 but need to pay Rs. 11. You pay Rs. 4 (all you have) and still owe Rs. 7.
Answer: -7
Example 5: Example 5: Adding an integer and its additive inverse
Problem: Find (+25) + (-25).
Solution:
These are additive inverses — they are opposites. An integer plus its opposite always equals zero.
(+25) + (-25) = 0
Money analogy: You have Rs. 25 and spend exactly Rs. 25. You have Rs. 0 left.
Answer: 0
Example 6: Example 6: Adding zero to an integer
Problem: Find (a) (-13) + 0, (b) 0 + (+42).
Solution:
Adding zero to any number gives the same number (zero is the additive identity).
(a) (-13) + 0 = -13
(b) 0 + (+42) = +42
Answer: (a) -13, (b) +42
Example 7: Example 7: Adding more than two integers
Problem: Find (-3) + (+7) + (-5) + (+2).
Solution:
Method 1 — Add step by step (left to right):
(-3) + (+7) = +4
(+4) + (-5) = -1
(-1) + (+2) = +1
Method 2 — Group positives and negatives separately:
Positive integers: +7 + (+2) = +9
Negative integers: -3 + (-5) = -8
Now add: +9 + (-8) = +1
Both methods give the same answer: +1.
Example 8: Example 8: Temperature word problem
Problem: The temperature at 6 AM in Shimla was -3°C. By noon, it increased by 8 degrees. By 6 PM, it dropped by 5 degrees from the noon temperature. What was the temperature at 6 PM?
Solution:
Temperature at 6 AM = -3°C.
Temperature at noon = -3 + 8 = +5°C.
Temperature at 6 PM = +5 + (-5) = 0°C.
Answer: The temperature at 6 PM was 0°C.
Example 9: Example 9: Bank balance problem
Problem: Ravi's bank balance is Rs. 2,500. He withdraws Rs. 1,800, deposits Rs. 3,200, and then withdraws Rs. 4,500. What is his final balance?
Solution:
Starting balance: +2500
Withdraw Rs. 1800: +2500 + (-1800) = +700
Deposit Rs. 3200: +700 + (+3200) = +3900
Withdraw Rs. 4500: +3900 + (-4500) = -600
Answer: His final balance is -Rs. 600, meaning he has an overdraft of Rs. 600 (he owes the bank Rs. 600).
Example 10: Example 10: Verifying properties of addition
Problem: Verify the commutative and associative properties for: a = -4, b = 7, c = -2.
Solution:
Commutative Property: a + b = b + a
LHS: (-4) + 7 = 3
RHS: 7 + (-4) = 3
LHS = RHS ✓ Verified.
Associative Property: (a + b) + c = a + (b + c)
LHS: [(-4) + 7] + (-2) = 3 + (-2) = 1
RHS: (-4) + [7 + (-2)] = (-4) + 5 = 1
LHS = RHS ✓ Verified.
Real-World Applications
Addition of integers has many practical applications:
Temperature Changes: Meteorologists track temperature changes throughout the day using integer addition. If the morning temperature is -5°C and it rises by 12 degrees, the new temperature is -5 + 12 = 7°C. Temperature changes over days, weeks and months are all tracked using integer addition.
Banking and Finance: Every bank transaction is integer addition — deposits are positive, withdrawals are negative. Your bank balance is the running sum of all transactions. Credit cards track spending (negative) and payments (positive). Accountants use integer addition constantly to calculate profit and loss.
Altitude and Depth: When a submarine at -200 metres (below sea level) rises 75 metres, its new depth is -200 + 75 = -125 metres. Mountain climbers track their altitude changes the same way — each ascent is positive and each descent is negative.
Sports Scoring: In golf, scores relative to par use integer addition. If a golfer scores -2 on the first round and -3 on the second, the total is (-2) + (-3) = -5 (5 under par). In football, goal difference is calculated by adding goals scored (positive) and goals conceded (negative).
Gaming: Video games use integer addition constantly — gaining health points (+20), taking damage (-15), earning coins (+100), paying for items (-75). Your score is the sum of all positive and negative changes.
Elevators: An elevator at floor -2 (Basement 2) going up 5 floors reaches floor -2 + 5 = floor 3. Going down 3 floors from floor 1 means 1 + (-3) = floor -2 (Basement 2).
Key Points to Remember
- Adding two positive integers: add the values, result is positive. (+5) + (+3) = +8.
- Adding two negative integers: add the absolute values, result is negative. (-5) + (-3) = -8.
- Adding integers with different signs: subtract the absolute values, take the sign of the number with larger absolute value.
- On the number line: adding positive means move right, adding negative means move left.
- Additive identity: a + 0 = a. Zero does not change the value.
- Additive inverse: a + (-a) = 0. Every integer and its opposite sum to zero.
- Commutative property: a + b = b + a (order does not matter).
- Associative property: (a + b) + c = a + (b + c) (grouping does not matter).
- Closure property: the sum of two integers is always an integer.
- When adding multiple integers, group all positives and all negatives separately for easier calculation.
Practice Problems
- Find: (a) (+9) + (+6), (b) (-4) + (-8), (c) (+12) + (-5), (d) (-7) + (+3).
- Use a number line to find (-5) + (+8). Show each step.
- Find the sum: (-12) + (+12). What property does this demonstrate?
- Calculate: (-6) + (+4) + (-3) + (+8) + (-1).
- The temperature was -7°C. It rose by 4 degrees, then dropped by 6 degrees. What is the final temperature?
- Verify the commutative property for: a = -8, b = 5.
- A submarine is at -150 metres. It goes up 60 metres, then down 40 metres. What is its final depth?
- Find two integers whose sum is -3. Can you find three different pairs?
Frequently Asked Questions
Q1. How do you add two negative numbers?
Add the absolute values (ignore the minus signs and add) and then put a negative sign on the result. For example, (-4) + (-6): add 4 + 6 = 10, then make it negative: -10. Think of it as two debts adding up — if you owe Rs. 4 and owe Rs. 6, you owe Rs. 10 total.
Q2. How do you add a positive and a negative number?
Subtract the smaller absolute value from the larger one, then use the sign of the number with the larger absolute value. For example, (+8) + (-3): subtract 8 - 3 = 5; since 8 is bigger and positive, the answer is +5. For (+3) + (-8): subtract 8 - 3 = 5; since 8 is bigger and negative, the answer is -5.
Q3. Why does adding a negative number make the result smaller?
Adding a negative number is like taking away. If you have 10 and add -3, you are essentially removing 3 from 10, giving you 7. On the number line, adding a negative number means moving to the left, which always gives a smaller number. Think of it as losing money — adding a negative amount to your balance reduces it.
Q4. What is the additive inverse?
The additive inverse of a number is the number you add to it to get zero. The additive inverse of +7 is -7 (because 7 + (-7) = 0). The additive inverse of -12 is +12 (because -12 + 12 = 0). The additive inverse of 0 is 0 (because 0 + 0 = 0). Every integer has exactly one additive inverse, which is its opposite.
Q5. Does the order of adding integers matter?
No, the order does not matter — this is the commutative property of addition. For example, (-5) + 3 = -2 and 3 + (-5) = -2. You get the same answer regardless of the order. Similarly, when adding three or more integers, the grouping does not matter (associative property). This flexibility makes it easier to rearrange numbers for quicker calculation.
Q6. How do you add many integers at once?
The easiest method is to separate the positive and negative integers into two groups. Add all the positive integers together to get a positive sum. Add all the negative integers together to get a negative sum. Then add the positive sum and the negative sum. For example: (-3) + 7 + (-5) + 2 → Positive: 7 + 2 = 9. Negative: -3 + (-5) = -8. Total: 9 + (-8) = 1.
Q7. Is (-5) + 3 the same as 3 + (-5)?
Yes! By the commutative property of addition, the order does not affect the result. Both equal -2. You can verify: (-5) + 3 → subtract 5 - 3 = 2, bigger number (5) is negative, so answer is -2. And 3 + (-5) → subtract 5 - 3 = 2, bigger number (5) is negative, so answer is -2.
Q8. Can the sum of two negative numbers ever be positive?
No, never. When you add two negative numbers, you add their absolute values and put a negative sign. Since both absolute values are positive, their sum is positive, and then the negative sign makes the result negative. For example, (-3) + (-5) = -8, (-100) + (-1) = -101. The sum of two negative numbers is always negative.
Q9. What is the sum of all integers from -5 to 5?
The integers from -5 to 5 are: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. Notice that -5 and 5 cancel out (sum to 0), -4 and 4 cancel out, -3 and 3 cancel out, -2 and 2 cancel out, -1 and 1 cancel out. That leaves just 0. So the sum is 0. In general, the sum of consecutive integers from -n to +n is always 0.
Q10. How is integer addition used in everyday life?
Integer addition is used whenever we track changes that can be positive or negative. Bank transactions (deposits and withdrawals), temperature changes (rising and falling), altitude changes (ascending and descending), game scores (gaining and losing points), and even your phone battery (charging and draining) all involve adding positive and negative numbers.
