The Other Side of Zero — Class 6 Maths (Ganita Prakash)
"Until now, numbers have only gone up from zero. But there is a whole world on the other side — the world of negative numbers."
1. About This Chapter
Until Class 6, students have worked with natural numbers (1, 2, 3...) and whole numbers (0, 1, 2, 3...). Chapter 10 of Ganita Prakash shatters the floor: numbers exist below zero too. These are negative integers. The chapter introduces integers — the set of positive numbers, negative numbers, and zero — and shows how they model real situations like temperatures below freezing, floors below ground, losses in business, and depths below sea level.
2. What Are Integers?
Integers include:
- Positive integers: +1, +2, +3, ... (also written as 1, 2, 3...)
- Zero: 0 (neither positive nor negative)
- Negative integers: −1, −2, −3, ...
The set of integers is written as:
3. Why Negative Numbers? — Real-Life Examples
Negative numbers are not just mathematical abstractions — they describe the real world:
| Situation | Positive Means | Negative Means |
|---|---|---|
| Temperature | Above 0°C | Below 0°C (−5°C, −10°C) |
| Money/Bank Balance | Credit/Profit | Debit/Loss/Debt |
| Building Floors | Above ground (1st floor, 2nd...) | Below ground (Basement −1, −2...) |
| Sea Level | Height above sea level | Depth below sea level |
| Sports | Points scored | Penalty points |
4. The Number Line with Integers
The number line extends to the LEFT of zero:
<---|---|---|---|---|---|---|---|---|---|--->
−5 −4 −3 −2 −1 0 1 2 3 4 5
Key observations:
- Zero is at the centre
- Positive integers are to the RIGHT of zero
- Negative integers are to the LEFT of zero
- Numbers increase as you move RIGHT
- Numbers decrease as you move LEFT
5. Comparing Integers
Rule 1: Positive vs Negative
Any positive integer is greater than any negative integer.
Rule 2: Two Positives
The one further right on the number line is greater.
Rule 3: Two Negatives
The one closer to zero (further right) is greater.
Think of temperature: −3°C is warmer than −5°C!
Rule 4: Any number is greater than numbers to its left.
6. Absolute Value — Distance from Zero
The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always non-negative.
The absolute value of a number is written between two vertical bars: |x|.
Key insights:
- |a| is always ≥ 0 (never negative)
- |a| = |−a| (opposites have the same absolute value)
- |a| = 0 only when a = 0
7. Ordering Integers
Arrange in ascending order: 3, −7, 0, −2, 5, −1
On the number line from left to right: −7, −2, −1, 0, 3, 5 ✓
Arrange in descending order: −4, 2, −8, 0, 6 From right to left: 6, 2, 0, −4, −8 ✓
8. Opposites
Every positive integer has an opposite negative integer at the same distance from zero:
| Number | Opposite |
|---|---|
| 3 | −3 |
| 7 | −7 |
| 0 | 0 (opposite of itself) |
| −5 | 5 |
Opposites are reflections across zero on the number line.
9. Representing Real Situations with Integers
| Situation | Integer |
|---|---|
| Temperature 5°C below zero | −5°C |
| Profit of ₹200 | +200 or 200 |
| Loss of ₹150 | −150 |
| 10 m above sea level | +10 m or 10 m |
| 25 m below sea level | −25 m |
| 3rd floor above ground | +3 |
| 2nd basement | −2 |
| Deposit in bank | +₹500 |
| Withdrawal from bank | −₹500 |
10. Addition and Subtraction of Integers (Introduction)
The chapter introduces basic integer operations conceptually through the number line:
- Moving RIGHT on the number line = adding a positive number
- Moving LEFT on the number line = adding a negative number (or subtracting)
Example: 3 + (−5) = −2 Start at 3, move 5 steps left → land at −2.
This is developed more fully in Class 7 — Class 6 focuses on understanding the number system and comparison.
11. Key Concepts Summary
| Concept | Definition | Example |
|---|---|---|
| Integer | Whole number that can be positive, negative, or zero | ..., −3, −2, −1, 0, 1, 2, 3... |
| Negative Integer | Integer less than zero | −1, −5, −100 |
| Positive Integer | Integer greater than zero | 1, 5, 100 |
| Number Line | Visual representation of integers in order | Zero at centre |
| Absolute Value | Distance from zero on the number line | |
| Opposite | Number at equal distance from zero on opposite side | Opposite of 4 is −4 |
12. Important Vocabulary
- Integer: A number from the set {..., −3, −2, −1, 0, 1, 2, 3, ...}
- Negative: Less than zero
- Positive: Greater than zero
- Number Line: A straight line with numbers placed at equal intervals
- Absolute Value: The distance of a number from zero, always non-negative
- Opposite: The number obtained by changing the sign (e.g., opposite of −6 is 6)
13. Worked Examples
Example 1: Compare using >, <, or =
a) −8 ___ 3 → −8 < 3 (negative < positive)
b) −2 ___ −7 → −2 > −7 (−2 is to the right of −7)
c) |−9| ___ 9 → |−9| = 9, so |−9| = 9
Example 2: Write as integers
a) 15°C below zero → −15°C
b) A deposit of ₹750 → +750
c) 30 m below sea level → −30 m
d) 5th floor above ground → +5
Example 3: Arrange in ascending order
−12, 7, 0, −5, 3, −1
Solution: On number line from left to right: −12, −5, −1, 0, 3, 7
Example 4: Find the absolute value
|−15| = 15, |23| = 23, |0| = 0, |−1| = 1
14. Conclusion
The Other Side of Zero expands the mathematical universe of Class 6 students from the familiar land of positive numbers into the complete world of integers. This understanding — that numbers can represent direction (above/below, profit/loss, forward/backward) — is foundational for algebra, coordinate geometry, physics, economics, and virtually every quantitative discipline. Integers complete the number line and prepare students for the rational numbers, equations, and graphs that await in higher classes.
