Notes For All Chapters – Maths Class 6 Ganita Prakash
📘 Ganita Prakash • Grade 6
🔢 The Other Side of Zero
Discover the world of Integers — numbers that go beyond zero into the negative side of the number line!
Chapter 10
Integers
Positive & Negative Numbers
Number Line
Addition & Subtraction
Integers
Positive & Negative Numbers
Number Line
Addition & Subtraction
📑 Contents
- Introduction — More and More Numbers!
- Bela’s Building of Fun — Positive & Negative Numbers
- Numbering the Floors — Floor System
- Addition of Integers
- Back to Zero — Additive Inverse
- Comparing Integers
- Subtraction of Integers
- The Number Line
- The Token Model
- Integers in Real Life
- Brahmagupta’s Rules
- Chapter Summary
- Practice Questions
Introduction — More and More Numbers!
We started our journey with counting numbers: 1, 2, 3, 4, … Then we learned about 0 (zero), which represents nothing and comes before 1. We also discovered fractions like 1/2, 3/2, 13/6 that live between whole numbers.
The Big Question!
We know numbers go to the right of 0: 0, 1, 2, 3, 4 … But are there numbers to the LEFT of 0? Are there numbers that are LESS than zero? That’s exactly what this chapter is about!
We know numbers go to the right of 0: 0, 1, 2, 3, 4 … But are there numbers to the LEFT of 0? Are there numbers that are LESS than zero? That’s exactly what this chapter is about!
📏 From Number Ray to Number Line
The number line we knew earlier was actually a number ray — it started at 0 and went only to the right. Now we’ll complete it into a true number line that goes in BOTH directions!
0
−1
−2
−3
−4
1
2
3
4
← Negative numbers
Positive numbers →
Key Fact
Numbers to the LEFT of 0 are called Negative Numbers. Numbers to the RIGHT of 0 are called Positive Numbers. Zero itself is NEITHER positive NOR negative.
Numbers to the LEFT of 0 are called Negative Numbers. Numbers to the RIGHT of 0 are called Positive Numbers. Zero itself is NEITHER positive NOR negative.
Bela’s Building of Fun
Imagine a special building with floors both above and below the ground. This building has a lift with only two buttons: + (go up) and − (go down).
🏪 Above Ground Floors
Welcome Hall (Ground), Food Court, Art Centre, Books, Sports, Ice Cream, Space — numbered +1, +2, +3 …
Welcome Hall (Ground), Food Court, Art Centre, Books, Sports, Ice Cream, Space — numbered +1, +2, +3 …
🎮 Below Ground Floors
Toys, Video Games, Cinema, Shoot House, Dinosaur Land — numbered −1, −2, −3 …
Toys, Video Games, Cinema, Shoot House, Dinosaur Land — numbered −1, −2, −3 …
🔘 How the Lift Works
| Button Press | Written As | Meaning |
|---|---|---|
| + pressed once | +1 | Go up 1 floor |
| + pressed three times (+++) | +3 | Go up 3 floors |
| − pressed once | −1 | Go down 1 floor |
| − pressed four times (−−−−) | −4 | Go down 4 floors |
Easy Trick!
A number with + in front → Positive Number (above ground). A number with − in front → Negative Number (below ground). 0 → Ground Floor!
A number with + in front → Positive Number (above ground). A number with − in front → Negative Number (below ground). 0 → Ground Floor!
🏗️ Floor Numbering System
| Floor Name | Floor Number | Button from Ground |
|---|---|---|
| Space Zone | +6 | Press + six times |
| Sports Centre | +5 | Press + five times |
| Book Store | +3 | Press + three times |
| Art Centre | +2 | Press + twice |
| Food Court | +1 | Press + once |
| Welcome Hall | 0 | Ground Floor! |
| Toy Store | −1 | Press − once |
| Video Games | −2 | Press − twice |
| Cinema | −3 | Press − three times |
| Dinosaur Land | −5 | Press − five times |
Addition of Integers
In the Building of Fun, addition can be understood in two ways:
📍 Method 1: Starting Floor + Movement = Target Floor
Starting Floor + Movement = Target Floor
Example: You are at Food Court (Floor +1). You press +2 in the lift.
Starting Floor = +1 (Food Court)
Movement = +2 (press + twice)
Target Floor = (+1) + (+2) = +3 (Book Store ✓)
Movement = +2 (press + twice)
Target Floor = (+1) + (+2) = +3 (Book Store ✓)
🔗 Method 2: Combining Button Presses
Gurmit was in the Toy Store (Floor −1) and accidentally pressed +2, then realised and pressed −3. Where did he end up?
Accidental press: +2
Correction press: −3
Result = (+2) + (−3) = −1
(He went 1 floor below where he started!)
Correction press: −3
Result = (+2) + (−3) = −1
(He went 1 floor below where he started!)
Brahmagupta’s Rules for Addition
- (+) + (+) = (+): Two positives add to a positive. E.g., (+3)+(+4) = +7
- (−) + (−) = (−): Two negatives add to a negative. E.g., (−3)+(−4) = −7
- (+) + (−): Subtract smaller from larger, keep sign of larger. E.g., (−5)+(+3) = −2
- n + 0 = n: Adding zero doesn’t change the number.
✏️ Practice Examples
(+1) + (+4) = ?
Start at +1, go up 4 = +5
Start at +1, go up 4 = +5
(+4) + (−3) = ?
Start at +4, go down 3 = +1
Start at +4, go down 3 = +1
(−1) + (+2) = ?
Start at −1, go up 2 = +1
Start at −1, go up 2 = +1
0 + (−2) = ?
Start at 0, go down 2 = −2
Start at 0, go down 2 = −2
Back to Zero — Additive Inverse
Basant was on the ground floor (Floor 0) and accidentally pressed +3. To come back to zero, he pressed −3. These two actions cancelled each other out!
(+3) + (−3) = 0 | (−5) + (+5) = 0
Definition: Additive Inverse
For every number, there is another number called its Additive Inverse (or simply Inverse). When you add a number and its inverse, you always get zero.
For every number, there is another number called its Additive Inverse (or simply Inverse). When you add a number and its inverse, you always get zero.
- The inverse of +4 is −4
- The inverse of −7 is +7
- The inverse of 0 is 0
- The inverse of −543 is +543
🤯
Think About It!
The additive inverse of any number n is −n. So the inverse of the inverse brings you back! The inverse of (−4) is +4, and the inverse of +4 is −4 again!
The additive inverse of any number n is −n. So the inverse of the inverse brings you back! The inverse of (−4) is +4, and the inverse of +4 is −4 again!
🎯 Inverse Pairs
| Number | Its Additive Inverse | Sum = ? |
|---|---|---|
| +9 | −9 | 0 ✓ |
| −7 | +7 | 0 ✓ |
| +8 | −8 | 0 ✓ |
| −5 | +5 | 0 ✓ |
| 0 | 0 | 0 ✓ |
Comparing Integers
Using Bela’s Building, we can easily compare any two integers. The floor that is lower has a smaller number. The floor that is higher has a larger number.
Lower floor = Smaller number | Higher floor = Larger number
📌 Key Comparison Rules
- All negative numbers are less than 0 (they are all below ground!)
- All positive numbers are greater than 0 (they are all above ground!)
- All negative numbers are less than all positive numbers
- The more negative a number, the smaller it is: −10 < −3
Common Mistake!
Many students think −10 is bigger than −3 because 10 > 3. But on the number line, −10 is further to the LEFT, so −10 is actually SMALLER than −3.
Think of it like floors: Floor −10 is much deeper underground than Floor −3!
Many students think −10 is bigger than −3 because 10 > 3. But on the number line, −10 is further to the LEFT, so −10 is actually SMALLER than −3.
Think of it like floors: Floor −10 is much deeper underground than Floor −3!
✏️ Comparison Examples
−2 vs +5
−2 < +5 (negative is always less than positive)
−2 < +5 (negative is always less than positive)
−5 vs −3
−5 < −3 (Floor −5 is lower than Floor −3)
−5 < −3 (Floor −5 is lower than Floor −3)
0 vs −4
0 > −4 (zero is above all negative floors)
0 > −4 (zero is above all negative floors)
−25 vs −7
−25 < −7 (more negative = smaller)
−25 < −7 (more negative = smaller)
🌍 Order of Integers
… < −4 < −3 < −2 < −1 < 0 < +1 < +2 < +3 < +4 < …
Subtraction of Integers
In integers, subtraction means: “What button do I press to get from the starting floor to the target floor?”
Target Floor − Starting Floor = Movement Needed
🎯 How to Think About Subtraction
Floor Example
You are at Art Centre (+2) and want to reach Sports Centre (+5). How many floors up? Answer: +3 floors. So: (+5) − (+2) = +3
You are at Art Centre (+2) and want to reach Sports Centre (+5). How many floors up? Answer: +3 floors. So: (+5) − (+2) = +3
📝 Worked Examples
| Target Floor | Starting Floor | Button Press | Expression |
|---|---|---|---|
| −1 | −2 | +1 (go up 1) | (−1) − (−2) = +1 |
| −1 | +3 | −4 (go down 4) | (−1) − (+3) = −4 |
| +2 | −2 | +4 (go up 4) | (+2) − (−2) = +4 |
| +5 | +2 | +3 (go up 3) | (+5) − (+2) = +3 |
The Golden Rule of Subtraction!
Subtracting a negative number = Adding the corresponding positive number!
Subtracting a negative number = Adding the corresponding positive number!
(+8) − (−2) = (+8) + (+2) = +10
(−3) − (−5) = (−3) + (+5) = +2
This is one of Brahmagupta’s key rules!
🔄 Subtraction ↔ Addition Conversion
(+7) − (+5) = (+7) + (−5) = +2
(−3) − (+8) = (−3) + (−8) = −11
(+8) − (−2) = (+8) + (+2) = +10 ← subtracting negative!
(+6) − (−9) = (+6) + (+9) = +15 ← subtracting negative!
(−3) − (+8) = (−3) + (−8) = −11
(+8) − (−2) = (+8) + (+2) = +10 ← subtracting negative!
(+6) − (−9) = (+6) + (+9) = +15 ← subtracting negative!
Quick Shortcut:
When you see two minus signs together (like − (−)), they become a PLUS!
When you see + and − together (like + (−)), they become a MINUS!
When you see two minus signs together (like − (−)), they become a PLUS!
When you see + and − together (like + (−)), they become a MINUS!
The Number Line
If we rotate the “infinite lift” sideways by 90°, it becomes the Number Line! To the right is the positive direction; to the left is the negative direction.
−5
−4
−3
−2
−1
0
1
2
3
← Negative
Positive →
🚶 Walking on the Number Line
- Walking to the right = positive movement (+)
- Walking to the left = negative movement (−)
- Smaller numbers are always to the left
- Larger numbers are always to the right
📐 Unmarked Number Line (UNL)
For large numbers, we use an Unmarked Number Line — just a line with 0 marked, and we visualise the positions.
Example: 85 + (−60) = ?
Draw a UNL. Place 0, mark +85 to the right. Move left by 60. You land at +25.
So 85 + (−60) = 25
Draw a UNL. Place 0, mark +85 to the right. Move left by 60. You land at +25.
So 85 + (−60) = 25
The Token Model
Another great way to understand integer addition and subtraction is using tokens:
🟢 Positive Token (+)
Represents +1. Each time the lift goes up, a positive token goes in the pocket.
Represents +1. Each time the lift goes up, a positive token goes in the pocket.
🔴 Negative Token (−)
Represents −1. Each time the lift goes down, a negative token goes in the pocket.
Represents −1. Each time the lift goes down, a negative token goes in the pocket.
Zero Pair!
One positive token + One negative token = Zero Pair. They cancel each other out! Always remove all zero pairs first to find the answer.
One positive token + One negative token = Zero Pair. They cancel each other out! Always remove all zero pairs first to find the answer.
🧮 Token Addition: (+5) + (−3) = ?
Start with: 5 positive tokens (+)(+)(+)(+)(+)
Add: 3 negative tokens (−)(−)(−)Form zero pairs: (+)(−) (+)(−) (+)(−) ← 3 zero pairs removed!
Remaining: (+)(+) ← 2 positive tokens
Add: 3 negative tokens (−)(−)(−)Form zero pairs: (+)(−) (+)(−) (+)(−) ← 3 zero pairs removed!
Remaining: (+)(+) ← 2 positive tokens
Answer: (+5) + (−3) = +2 ✓
➖ Token Subtraction: (+5) − (+6) = ?
Start with: 5 positive tokens
Need to remove: 6 positive tokens (not enough!)→ Add 1 zero pair: now have 6 positive + 1 negative
→ Remove 6 positive tokens
→ Left with: 1 negative token
Need to remove: 6 positive tokens (not enough!)→ Add 1 zero pair: now have 6 positive + 1 negative
→ Remove 6 positive tokens
→ Left with: 1 negative token
Answer: (+5) − (+6) = −1 ✓
Token Trick:
When you don’t have enough tokens to subtract, add zero pairs! Adding a zero pair doesn’t change the value (it adds 0), but it gives you more tokens to work with.
When you don’t have enough tokens to subtract, add zero pairs! Adding a zero pair doesn’t change the value (it adds 0), but it gives you more tokens to work with.
Integers in Real Life
🏦 Credits and Debits (Banking)
Banks use integers all the time! Money coming IN is a Credit (+) and money going OUT is a Debit (−). Your balance = sum of all credits and debits.
Bank Account Story
Start with ₹100. Then Credit ₹60 → Balance = ₹160. Then Debit ₹30 → Balance = ₹130. Then Debit ₹150 → Balance = −₹20 (negative balance = you owe the bank!). Then Credit ₹200 → Balance = ₹180.
Start with ₹100. Then Credit ₹60 → Balance = ₹160. Then Debit ₹30 → Balance = ₹130. Then Debit ₹150 → Balance = −₹20 (negative balance = you owe the bank!). Then Credit ₹200 → Balance = ₹180.
| Day | Transaction | Amount | Balance |
|---|---|---|---|
| Day 1 | Initial deposit | +₹100 | ₹100 |
| Day 2 | Credit (job) | +₹60 | ₹160 |
| Day 3 | Debit (electric bill) | −₹30 | ₹130 |
| Day 4 | Debit (purchase) | −₹150 | −₹20 |
| Day 5 | Credit (business) | +₹200 | ₹180 |
🌋 Geography — Above and Below Sea Level
We measure geographical heights from sea level = 0 m. Heights above are positive, depths below are negative.
🏔️ Above Sea Level (Positive)
Mount Everest: +8,849 m
A plateau: +500 m
A hill: +200 m
Mount Everest: +8,849 m
A plateau: +500 m
A hill: +200 m
🌊 Below Sea Level (Negative)
Mariana Trench: −11,034 m
Dead Sea shore: −430 m
Ocean depths: −1000 m
Mariana Trench: −11,034 m
Dead Sea shore: −430 m
Ocean depths: −1000 m
🌡️ Temperature
We measure temperature in Celsius (°C). Temperatures above 0°C are positive; temperatures below 0°C (below freezing!) are negative.
| Temperature | Meaning | Real Example |
|---|---|---|
| +40°C | Very hot summer | Delhi in May 🥵 |
| +15°C | Cool and pleasant | Leh, 11 am in Nov |
| 0°C | Freezing point of water | Water turns to ice |
| −2°C | Below freezing | Leh, late evening in Nov |
| −4°C | Very cold | Leh, 2 am in Nov 🥶 |
🏔️
Did You Know?
In India, places like Leh (Ladakh), Shimla (Himachal Pradesh), Dras (Kashmir), and Sonamarg experience temperatures below 0°C in winter because they are at high altitudes where air is thinner and colder!
In India, places like Leh (Ladakh), Shimla (Himachal Pradesh), Dras (Kashmir), and Sonamarg experience temperatures below 0°C in winter because they are at high altitudes where air is thinner and colder!
Brahmagupta’s Rules & History
A Pinch of History!
Negative numbers were first used in Asia. In China, The Nine Chapters on Mathematical Art (1st–2nd century CE) used red and black rods for positive and negative numbers — just like our red and black tokens! India’s Kautilya wrote about credits and debits in his Arthaśhāstra (~300 BCE). The Bakśhālī Manuscript (~300 CE) used a special symbol for negative numbers.
Negative numbers were first used in Asia. In China, The Nine Chapters on Mathematical Art (1st–2nd century CE) used red and black rods for positive and negative numbers — just like our red and black tokens! India’s Kautilya wrote about credits and debits in his Arthaśhāstra (~300 BCE). The Bakśhālī Manuscript (~300 CE) used a special symbol for negative numbers.
🌟
Brahmagupta — The Greatest!
In the year 628 CE, Indian mathematician Brahmagupta wrote the Brāhma-sphuṭa-siddhānta. He was the FIRST person in the world to give clear and complete rules for positive numbers, negative numbers, AND zero — all treated as equal and valid numbers!
In the year 628 CE, Indian mathematician Brahmagupta wrote the Brāhma-sphuṭa-siddhānta. He was the FIRST person in the world to give clear and complete rules for positive numbers, negative numbers, AND zero — all treated as equal and valid numbers!
➕ Brahmagupta’s Rules for Addition
| # | Rule | Example |
|---|---|---|
| 1 | (+) + (+) = Positive | 2 + 3 = 5 |
| 2 | (−) + (−) = Negative (add magnitudes) | (−2) + (−3) = −5 |
| 3 | (+) + (−) = Sign of larger, subtract magnitudes | −5 + 3 = −2 |
| 4 | n + (−n) = 0 | 2 + (−2) = 0 |
| 5 | n + 0 = n | −2 + 0 = −2 |
➖ Brahmagupta’s Rules for Subtraction
| # | Rule | Example |
|---|---|---|
| 1 | Larger positive − Smaller positive = Positive | 3 − 2 = 1 |
| 2 | Smaller positive − Larger positive = Negative | 2 − 3 = −1 |
| 3 | Subtracting negative = Adding positive | 2 − (−3) = 2 + 3 = 5 |
| 4 | n − n = 0 | 2 − 2 = 0 |
| 5 | n − 0 = n; 0 − n = −n | −2 − 0 = −2; 0 − (−2) = 2 |
Legacy of Brahmagupta
It took centuries for the rest of the world to accept zero and negative numbers! Arab mathematicians adopted them by the 9th century, and Europe accepted them only by the 13th century. Even in the 18th century, French mathematician Lazare Carnot called negatives “absurd”! But today these numbers are fundamental to all of science and mathematics.
It took centuries for the rest of the world to accept zero and negative numbers! Arab mathematicians adopted them by the 9th century, and Europe accepted them only by the 13th century. Even in the 18th century, French mathematician Lazare Carnot called negatives “absurd”! But today these numbers are fundamental to all of science and mathematics.
What Are Integers?
Integers = … , −4, −3, −2, −1, 0, +1, +2, +3, +4, …
(They go both ways from 0, infinitely!)
(They go both ways from 0, infinitely!)
➕ Positive Integers
1, 2, 3, 4, 5, 6, … (go on forever to the right)
All are greater than 0.
1, 2, 3, 4, 5, 6, … (go on forever to the right)
All are greater than 0.
➖ Negative Integers
−1, −2, −3, −4, −5, … (go on forever to the left)
All are less than 0.
−1, −2, −3, −4, −5, … (go on forever to the left)
All are less than 0.
Remember: Zero (0) is an integer but it is NEITHER positive NOR negative. It is the “neutral” number that separates positive and negative integers.
📌 Real-Life Contexts of Integers
🏢 Building FloorsGround = 0, Above = (+), Below = (−)
⛏️ Mine/ShaftGround = 0 m, Above = (+) m, Below = (−) m
🌡️ TemperatureFreezing = 0°C, Warm = (+)°C, Cold = (−)°C
🏦 Bank AccountZero balance = 0, Credit = (+), Debit = (−)
🌍 GeographySea level = 0 m, Mountains = (+) m, Oceans = (−) m
📅 History (BCE/CE)0 as reference, CE = (+), BCE = (−)
Chapter Summary
Negative Numbers
Numbers less than 0. Written with a − sign. Lie to the LEFT of 0 on the number line.
Numbers less than 0. Written with a − sign. Lie to the LEFT of 0 on the number line.
Integers
All whole numbers: …−3, −2, −1, 0, 1, 2, 3… Positive, negative, and zero.
All whole numbers: …−3, −2, −1, 0, 1, 2, 3… Positive, negative, and zero.
Additive Inverse
Every number n has an inverse −n such that n + (−n) = 0.
Every number n has an inverse −n such that n + (−n) = 0.
Addition Rule
Starting Position + Movement = Target Position
Starting Position + Movement = Target Position
Subtraction Rule
Target − Starting = Movement Needed. Subtracting (−) = Adding (+)!
Target − Starting = Movement Needed. Subtracting (−) = Adding (+)!
Comparison
Integers go: …<−3<−2<−1<0<1<2<3<… Smaller = further left.
Integers go: …<−3<−2<−1<0<1<2<3<… Smaller = further left.
📝 All Key Formulas at a Glance
Starting Position + Movement = Target Position
Target Position − Starting Position = Movement Needed
n + (−n) = 0 | Subtracting (−n) = Adding (+n)
Practice Questions & Answers
Q1. What is the additive inverse of −15?
Ans: +15 (because (−15) + (+15) = 0)
Q2. Evaluate: (−8) + (+12)
Ans: +4 (subtract 8 from 12, keep sign of 12 which is positive)
Q3. Is −10 greater or smaller than −3? Why?
Ans: −10 is SMALLER than −3 because −10 lies further to the left on the number line (Floor −10 is deeper underground than Floor −3).
Q4. Solve: (+4) − (−6)
Ans: (+4) − (−6) = (+4) + (+6) = +10 [Subtracting negative = Adding positive]
Q5. You start at Floor +3 in Bela’s Building and press −7. Which floor do you reach?
Ans: (+3) + (−7) = −4 (Floor −4, which is the Cinema level)
Q6. Compare using < or >: (a) −5 __ +2 (b) −3 __ −8 (c) 0 __ −6
Ans: (a) −5 < +2 (b) −3 > −8 (c) 0 > −6
Q7. Write all integers between −4 and +3 in increasing order.
Ans: −3, −2, −1, 0, 1, 2 (integers strictly between −4 and +3)
Q8. A bank account has credits of ₹500 and ₹200, and debits of ₹300 and ₹700. What is the balance?
Ans: Balance = (+500) + (+200) + (−300) + (−700) = 700 − 1000 = −₹300 (negative balance!)
Q9. Who was the first mathematician to give complete rules for positive numbers, negative numbers and zero?
Ans: Brahmagupta, an Indian mathematician, in his work Brāhma-sphuṭa-siddhānta written in 628 CE.
Q10. Evaluate: (−99) − (−200)
Ans: (−99) − (−200) = (−99) + (+200) = +101
Watch Out for These Common Mistakes!
- −10 is NOT greater than −3 (it is smaller — further left!)
- Zero is NOT a positive number and NOT a negative number
- (−) − (−) does NOT always give a positive answer — it depends on the magnitudes
- The inverse of 0 is 0 (not +0 or −0)
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