Class 6 The Other Side of Zero
Explore numbers on both sides of zero using interactive number lines, Bela's Building of Fun lift simulator, zero-pair token models, real-world banking and altitude applications, and Brahmagupta's ancient mathematical rules.
Quick Answer • What is The Other Side of Zero in Class 6 Mathematics?
In NCERT Class 6 Mathematics (Ganita Prakash), The Other Side of Zero introduces integers—completing the number ray into a true two-way number line:
Numbers greater than zero written with a '+' sign (or no sign): $+1, +2, +3, \dots$ lying to the right of 0.
Numbers less than zero written with a '–' sign: $-1, -2, -3, \dots$ lying to the left of 0.
Zero ($0$) is neither positive nor negative. It acts as the reference origin separating both sides.
Every number $a$ has an opposite $-a$ such that $a + (-a) = 0$. (e.g. $+5$ and $-5$).
Subtracting a negative number is the same as adding its positive inverse: $a - (-b) = a + b$.
On a number line, numbers farther to the right are always greater (e.g., -2 > -5).
Chapter 10 Table of Contents • Learning Roadmap
Bela's Building of Fun & The Vertical Lift Model
Interactive Lift Movement Simulator (Bela's Building of Fun)
Choose a Starting Floor and Target Floor to calculate lift button presses and mathematical expressions:
(+0) + (+3) = +3
Starting Floor + Movement = Target Floor(+3) – (+0) = +3
Target Floor – Starting Floor = Movement NeededInteractive Horizontal Number Line & Additive Inverses (pp. 252–254)
Selected Number: -33 units
+3
(-3) + (+3) = 0
The Token Model & Zero Pairs (+ and –)
Interactive Token Addition & Zero Pair Eliminator
Adjust the number of positive (green) and negative (red) tokens to visualize zero pair cancellation:
Subtracting with Tokens: When You Don't Have Enough Tokens! (pp. 257–258)
To subtract a quantity when you don't have enough tokens of that type, add zero pairs (which equal 0), and then take away the required tokens!
Start with 5 positives. To take away 6 positives, add 1 zero pair (+, –). Now take away 6 positives → –1 remains!
Start with 4 positives. To take away 6 negatives, add 6 zero pairs. Now take away 6 negatives → 4 + 6 = +10 remains!
Integers in Real Life: Banking, Altitude & Temperature
Bank Account Balances
Deposits are credits (+, positive) and payments are debits (–, negative). If debits exceed credits, your account balance becomes negative (overdraft).
Sea Level as Zero Reference
Altitudes above sea level are positive (+8848 m Mount Everest) and oceanic depths below sea level are negative (–10,994 m Challenger Deep).
Freezing Point as 0°C
Temperatures above water's freezing point are positive (+40°C heat wave), and winter cold temperatures in places like Leh, Ladakh drop below zero (–2°C, –4°C).
Integer Explorations: Hollow Grids & Magic Elimination
Hollow Integer Grid & Border Sums (Page 263)
In a hollow integer grid, the numbers along each of the 4 outer borders (top row, bottom row, left column, right column) add up to the same constant sum:
Top: 4+(–1)+(–3) = 0 • Bottom: (–1)+(–1)+2 = 0 • Border Sum = 0
The Amazing Magic Elimination Grid (Page 264)
Circle any number, strike out its entire row and column, and repeat until 4 numbers are circled. The sum is ALWAYS the exact same constant!
Try choosing completely different cells—no matter which unstruck cells you choose, the sum will always equal –1!
Brahmagupta's Rules for Integers (628 CE)
Brahmagupta's 10 Classical Rules for Addition & Subtraction (628 CE)
- Pos + Pos: Sum is positive ($2 + 3 = 5$).
- Neg + Neg: Add numbers, keep minus sign ($(-2) + (-3) = -5$).
- Pos + Neg: Subtract smaller from larger, take sign of larger ($(-5) + 3 = -2$).
- Inverse: $a + (-a) = 0$.
- Zero: $a + 0 = a$.
- Large Pos – Small Pos: Result is positive ($3 - 2 = 1$).
- Small Pos – Large Pos: Result is negative ($2 - 3 = -1$).
- Subtracting Negative: $a - (-b) = a + b$.
- Number from Itself: $a - a = 0$ and $(-2) - (-2) = 0$.
- Zero Subtraction: $a - 0 = a$ and $0 - (-a) = a$.
Interactive Integer Operations Calculator (Brahmagupta's Method)
Enter any two integers and select operation to see step-by-step evaluation:
Class 6 The Other Side of Zero Practice Lab
Which of the following numbers is neither positive nor negative?
In Bela's Building of Fun, you start at Floor +2 (Art Centre) and press '–3' in the lift. Where do you reach?
What is the additive inverse of –543?
Which statement correctly compares –5 and –2?
Evaluate the expression: (+40) + (–50).
Evaluate the subtraction: (+8) – (–7).
In the Token Model, what is the net value of 5 green positive tokens and 8 red negative tokens?
To evaluate (–3) – (+5) using the token model, how many zero pairs must you add to take away 5 positive tokens?
You open a bank account with ₹100, deposit a credit of ₹60, pay an electric bill debit of ₹30, and make a business purchase debit of ₹150. What is your final bank balance?
What is the highest point above sea level on Earth and its elevation?
What is the lowest known point on Earth with respect to sea level?
In Leh, Ladakh, temperature at 2:00 PM is 14°C and at 2:00 AM it drops to –4°C. What is the total temperature drop?
In the 3×3 Hollow Integer Grid, what is the 'border sum' if the numbers are 5, –3, –5 (top row) and (–8), (–2), 7 (bottom row)?
Two dice have faces numbered {–1, 2, –3, 4, –5, 6}. Which of the following sums is IMPOSSIBLE to roll?
What year was it 320 years after 680 BCE?
Complete the sequence: (–40), (–34), (–28), (–22), ___, ___, ___.
A string of 100 tokens repeats the pattern (+ + + – –) consisting of 3 positive and 2 negative tokens. What is the total value of the 100 tokens?
Which ancient Indian treatise first systematically gave complete arithmetic rules for positive numbers, negative numbers, and zero on an equal footing in 628 CE?
What is always the result of: (Negative Integer) – (Positive Integer)?
In the Integers Snakes and Ladders game (Page 271), players start at 0. What are the two winning target scores?
Advanced Integer Challenges & Proofs
Prove why a – (–b) = a + b holds for all integers using both the Lift Movement model (Target Floor – Starting Floor = Movement) and the Additive Inverse definition.
In the 4×4 'Amazing Grid of Numbers' (Page 264), prove why circling any 4 numbers such that no two share a row or column always yields the exact same constant sum (e.g. –1).
Two dice have faces {–1, 2, –3, 4, –5, 6}. Prove why the sum 0 is impossible to roll, and identify all 8 impossible integer sums between the minimum sum –10 and maximum sum +12.
A long string of 100 tokens follows the repeating 5-token block pattern: (+, +, +, –, –). Determine the total value of the string, and find the value if the string had 103 tokens instead.
Given the six integer cards: (+1), (+7), (+18), (–5), (–2), (–9), construct an arithmetic expression using addition and subtraction that evaluates to exactly –30.
Determine and justify whether each expression is always positive, always negative, or can be either: (a) Positive – Negative; (b) Positive + Negative; (c) Negative + Negative; (d) Negative – Negative; (e) Negative – Positive; (f) Negative + Positive.
In a 3×3 hollow integer grid where corner cells are shared by rows and columns, explain why the 4 corners contribute to two border lines simultaneously, and find a solution for border sum = +4 with given cells (–10), 9, (–5).
Explain why Brahmagupta's 628 CE rules for addition and subtraction of zero, positive, and negative numbers satisfied the formal mathematical properties of an abelian group under addition.
Show how an unmarked number line (UNL) with only 0 as an anchor can be used to solve the missing addend problem: (–100) – (+250) = ? by setting up 250 + ? = –100.
In the bank account problem (Page 260, Question 2), a person starts with ₹0, incurs 8 successive doubling debits of ₹1, ₹2, ₹4, ₹8, ₹16, ₹32, ₹64, ₹128, and then deposits a single credit of ₹256. Prove using geometric progression why the final balance is exactly +₹1.
Common Integer Mistakes to Avoid
Although 5 > 2, for negative numbers –5 lies to the left of –2 on the number line, so –5 < –2.
Zero is neither positive nor negative. It is the neutral origin point between both sets.
While +5 can be written simply as 5, dropping the minus sign from –5 fundamentally changes the number to +5!
Subtracting a negative number is the same as adding a positive: 7 – (–3) = 7 + 3 = 10, not 4.
Distance from zero is always non-negative (distance of –5 is 5 units), but the number itself is –5.
The historical calendar jumps directly from 1 BCE to 1 CE with no Year 0 in between.
Key Takeaways • Ganita Prakash Chapter 10 Summary
Integers include negative numbers (..., –3, –2, –1), zero (0), and positive numbers (1, 2, 3, ...).
... < –3 < –2 < –1 < 0 < +1 < +2 < +3 < ... Numbers farther right are always greater.
Every number has an opposite: $a + (-a) = 0$. Subtracting an integer is adding its additive inverse.
Integers accurately model building lifts, mineshafts, banking credits/debits, temperatures, and altitudes.
Frequently Asked Questions (Class 6 The Other Side of Zero)
Signed Binary Arithmetic, 2's Complement & Computer Memory
In computer science, negative integers are represented in computer memory using Two's Complement binary representation. Zero pairs and additive inverses directly power CPU Arithmetic Logic Units (ALUs) to perform subtraction purely using addition circuitry!