Integers — Class 6 Mathematics
'Numbers below zero? Yes! Integers help us describe temperature, depth, debt, and so much more.'
1. Introduction
So far we have worked with whole numbers (0, 1, 2, 3, ...). But what about temperatures below freezing? Or a bank balance with an overdraft? Or a building's basement floors? We need negative numbers.
Integers = {... , −3, −2, −1, 0, 1, 2, 3, ...}
- Positive integers: 1, 2, 3, 4, ... (same as natural numbers)
- Negative integers: −1, −2, −3, −4, ... (read as 'negative one', 'negative two')
- Zero: neither positive nor negative
Why Integers Matter
- Temperature: 5°C above zero (+5°C), 3°C below zero (−3°C)
- Elevation: Sea level = 0, mountain top = +8848 m, valley = −86 m
- Finance: Profit (+), Loss (−), Bank balance
- Sports: Golf scores (under par = negative)
2. Integers on the Number Line
<--|----|----|----|----|----|----|----|----|----|----|-->
-5 -4 -3 -2 -1 0 1 2 3 4 5
Key Observations
- Positive numbers are to the RIGHT of zero
- Negative numbers are to the LEFT of zero
- Zero is in the middle (neither positive nor negative)
- The distance from zero is the same for a number and its opposite
Representing on a Number Line
Example: Show −3 on a number line.
- Draw a line with arrows on both ends
- Mark 0 in the middle
- Mark positive numbers (1, 2, 3, ...) to the right
- Mark negative numbers (−1, −2, −3, ...) to the left
- Locate −3 (three steps LEFT from 0)
3. Ordering Integers
Rules
- On a number line, numbers INCREASE as we move RIGHT
- Numbers DECREASE as we move LEFT
- Every positive integer > 0 > every negative integer
Comparison Examples
| Comparison | True/False | Reason |
|---|---|---|
| 5 > 2 | True | 5 is to the right of 2 |
| −3 < −1 | True | −3 is to the left of −1 |
| 0 > −4 | True | 0 is to the right of −4 |
| −7 > −2 | False | −7 is to the left of −2 |
| −5 < 3 | True | Negative < Positive |
Ordering Integers
Ascending order (smallest to largest): −8, −5, −3, 0, 2, 6 Descending order (largest to smallest): 6, 2, 0, −3, −5, −8
'On the negative side, the number with the LARGER absolute value is actually SMALLER. −10 < −1 even though 10 > 1.'
4. Addition of Integers
Same Sign — Add and Keep the Sign
| Expression | Working | Result |
|---|---|---|
| 5 + 3 | Both positive | 8 |
| −5 + (−3) | Both negative | −8 |
Different Signs — Subtract and Take the Sign of the Larger
| Expression | Working | Result |
|---|---|---|
| 5 + (−3) | 5 − 3 = 2, sign of larger (5) | 2 |
| −5 + 3 | 5 − 3 = 2, sign of larger (−5) | −2 |
Using the Number Line for Addition
- Adding a POSITIVE number → move RIGHT
- Adding a NEGATIVE number → move LEFT
Example: (−4) + 6 Start at −4. Move 6 steps right. Reach 2. Answer: 2.
Example: 3 + (−7) Start at 3. Move 7 steps left. Reach −4. Answer: −4.
Worked Examples
| Problem | Solution |
|---|---|
| (−8) + (−3) | Same sign → −(8+3) = −11 |
| (−12) + 7 | Different signs → −(12−7) = −5 |
| 15 + (−9) | Different signs → +(15−9) = 6 |
| (−20) + 20 | Opposite → 0 |
5. Additive Inverse
For every integer a, there exists an integer −a such that: a + (−a) = 0
−a is called the additive inverse of a (and vice versa).
| Number | Additive Inverse | Sum |
|---|---|---|
| 7 | −7 | 7 + (−7) = 0 |
| −4 | 4 | −4 + 4 = 0 |
| 0 | 0 | 0 + 0 = 0 |
'Every integer and its additive inverse are at the same distance from zero but on opposite sides.'
6. Subtraction of Integers
Rule: To subtract an integer, ADD its additive inverse. a − b = a + (−b)
Steps
- Change subtraction to addition
- Change the sign of the second number
- Use addition rules
Examples
| Problem | Step 1 | Step 2 | Result |
|---|---|---|---|
| 7 − 4 | 7 + (−4) | Different signs | 3 |
| 7 − (−4) | 7 + 4 | Same sign | 11 |
| −7 − 4 | −7 + (−4) | Same sign | −11 |
| −7 − (−4) | −7 + 4 | Different signs | −3 |
Pattern Discovery
| Expression | Result | Pattern |
|---|---|---|
| 3 − 1 | 2 | Decrease by 1 each step |
| 3 − 2 | 1 | |
| 3 − 3 | 0 | |
| 3 − 4 | −1 | We can go below zero! |
| 3 − 5 | −2 |
'Subtraction can give us negative answers — this was impossible with whole numbers but is perfectly valid with integers.'
7. Word Problems with Integers
Problem 1 (Temperature)
In Kullu, the temperature at 6 AM was −4°C. By 2 PM, it rose by 11°C. What was the temperature at 2 PM?
Solution: Initial temperature = −4°C Change = +11°C Final = (−4) + 11 = 7°C Answer: 7°C
Problem 2 (Elevation)
The surface of the Dead Sea is 430 m below sea level. Write this as an integer.
Solution: Sea level = 0 Below sea level = negative Answer: −430 m
Problem 3 (Bank Balance)
Mohan deposited Rs. 500 in his bank account. The balance was Rs. 200 (positive). He then withdrew Rs. 700. What is his new balance?
Solution: Initial balance = +200 (deposit of 500 made it 200, assuming he started at −300... let's work simply) Let initial balance = Rs. 200 Withdrawal of Rs. 700 = −700 New balance = 200 + (−700) = −500 Answer: −Rs. 500 (overdraft of Rs. 500)
Problem 4 (Sports)
In a golf tournament, a player scored: +2 (over par), then −3 (under par), then −1, then +4. What is the final score?
Solution: Total = 2 + (−3) + (−1) + 4 = 2 − 3 − 1 + 4 = 2 + 4 − 3 − 1 = 6 − 4 = 2 (over par)
8. Common Mistakes — Fix Them Now
| # | Mistake | Correction |
|---|---|---|
| 1 | Writing negative 3 as −3 but saying 'minus 3' | Say 'negative 3' not 'minus 3'. Minus is operation, negative is sign. |
| 2 | −5 > −3 (thinking 5 > 3 so −5 > −3) | −5 < −3 because −5 is left of −3 on number line |
| 3 | (−5) + (−3) = −2 | Same sign: add and keep sign → −8, not 2 |
| 4 | 5 − (−3) = 2 | 5 − (−3) = 5 + 3 = 8 (two negatives = positive) |
| 5 | Forgetting additive inverse of 0 | 0 is its own additive inverse |
9. Exam Focus
Marks Blueprint
| Question Type | Marks | Topic |
|---|---|---|
| MCQ | 1 | Integer comparison |
| Fill in blank | 1 | Additive inverse |
| Number line | 2 | Represent and add |
| Short answer | 2 | Addition/subtraction |
| Word problem | 3 | Real-life application |
Quick Self-Test (5 Questions)
Q1: Which is larger: −18 or −5?
<details><summary>Answer</summary>−5 is larger (closer to 0 on number line).</details>Q2: What is the additive inverse of −23?
<details><summary>Answer</summary>23, because (−23) + 23 = 0.</details>Q3: Solve: (−15) + 8
<details><summary>Answer</summary>−7 (different signs: 15−8=7, sign of larger = negative).</details>Q4: Solve: 12 − (−7)
<details><summary>Answer</summary>12 + 7 = 19 (subtracting negative = adding positive).</details>Q5: At 5 PM in Shimla, temperature was 8°C. By midnight, it dropped by 12°C. What was the midnight temperature?
<details><summary>Answer</summary>8 + (−12) = −4°C.</details>10. Chapter Summary
- Integers: {...−3, −2, −1, 0, 1, 2, 3...}
- Number line: negative left, positive right, 0 in middle
- Ordering: larger as you go right; every positive > 0 > every negative
- Addition: same sign → add; different signs → subtract, take sign of larger
- Additive inverse: a + (−a) = 0
- Subtraction: add the additive inverse: a − b = a + (−b)
- Real uses: temperature, elevation, finance, sports
'Integers unlock a whole new world of mathematics. With negatives, we can now solve problems that were impossible with just whole numbers.'
