# The Other Side of Zero

## Overview

This chapter introduces the concept of **integers**, which includes positive numbers, negative numbers, and zero. Through intuitive examples like buildings, mines, bank balances, temperatures, and historical context, students learn to add, subtract, and compare integers using various models such as number lines, lifts, and tokens.

## Key Topics Covered

### 1. Introduction to Negative Numbers

* Counting numbers and fractions were known earlier.
* The number line we commonly use is a ray starting from 0.
* Negative numbers complete this line, extending it to the left of 0.
* Question explored: **Are there numbers less than zero?**

### 2. Bela’s Building of Fun (Lift Analogy)

* Ground floor = 0, floors above = positive numbers (+1, +2,...), below = negative numbers (–1, –2,...)
* Movement described using + (up) and – (down) button presses.
* Introduction of **positive** and **negative** numbers with practical floor movement examples.

### 3. Addition of Integers Using Movements

* Expression: `Starting Floor + Movement = Target Floor`
* Examples:

* (+1) + (+2) = +3
* (–1) + (+2) = +1
* Combining button presses: (+2) + (–3) = –1

### 4. Inverse and Zero

* The **inverse** of a number is what brings you back to 0.

* Example: (+3) + (–3) = 0
* Zero is **neither positive nor negative**.
* Inverses are used to reach back to the ground floor.

### 5. Comparing Integers

* Number line comparison: smaller numbers are to the left.
* Examples:

* –4 < –3
* 0 < 4
* –2 > –5

### 6. Subtraction as Movement

* Expression: `Target Floor – Starting Floor = Movement`
* Examples:

* (+5) – (+2) = +3
* (–1) – (–2) = +1
* (+2) – (–2) = +4

### 7. Larger Integers in Mineshafts

* Depths and heights in a mine illustrate integers with larger values.
* Both positive and negative numbers go on infinitely.
* Examples:

* (–90) + (–55) = –145
* (+40) – (–50) = +90

### 8. Infinite Number Line & Visualization

* Imagine a lift or a number line that extends infinitely in both directions.
* Concept: Subtracting a negative is the same as adding its positive.

* (+2000) – (–200) = (+2200)
* –99 – (–200) = +101

### 9. The Token Model

* Positive tokens = red, Negative tokens = green
* A positive and negative token form a “zero pair”
* Use tokens to visualize:

* Addition: (+5) + (–3) = +2
* Subtraction: (+5) – (+6) = –1 (add zero pairs to solve)
* Subtraction involving negatives: (+4) – (–6) = +10

### 10. Real-World Applications of Integers

#### A. Banking: Credits and Debits

* Credits = positive numbers; Debits = negative numbers
* Example:

* Start with `100, add `60 (credit) → 160
* Subtract \`30 (debit) → 130
* Subtract \`150 → –20

#### B. Geography: Heights and Depths

* Heights measured from sea level (0 m)
* Above sea level = positive; Below = negative

#### C. Temperature

* 0°C = freezing point
* Positive = above freezing; Negative = below freezing
* Example: Leh temperatures can be –4°C to 14°C in a day

### 11. Integer Grids and Patterns

* Integer grids with equal sums in rows and columns (called “border sums”)
* Puzzle activities to observe patterns using addition with integers

### 12. Integer Games and Puzzles

* Grids and dice used to generate integers and explore their properties
* Integer snakes and ladders: play using a pair of dice with +ve and –ve numbers

### 13. History of Integers

* Ancient origins in China and India (Jiuzhang Suanshu, Bakhshali Manuscript, Arthaśhāstra)
* **Brahmagupta’s Rules** (628 CE) — formalized integer arithmetic, including zero
* Initially rejected in Europe but now integral to mathematics

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## New Terms and Definitions

| Term | Simple Definition |
| ---------------- | ------------------------------------------------------------------------------ |
| Integer | Whole number (positive, negative, or zero) |
| Negative Number | Number less than 0, written with a minus sign (e.g., –3) |
| Positive Number | Number greater than 0, often written without the plus sign (e.g., 4) |
| Zero | Neither positive nor negative, the middle point on the number line |
| Additive Inverse | A number that when added to a given number gives 0 (e.g., inverse of +5 is –5) |
| Lift Model | Using elevator movement to explain integer addition/subtraction |
| Token Model | Using colored tokens to model positive and negative numbers |
| Credit | Money added to a bank account (positive number) |
| Debit | Money removed from a bank account (negative number) |
| Sea Level | Reference point of 0m height in geography |

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## Practice Questions

### Easy (3)

1. What is (+4) + (–2)?
**Answer**: +2 — Subtract and keep the sign of larger number.

2. What is the inverse of –7?
**Answer**: +7 — The opposite sign makes it the inverse.

3. Is –5 > –8?
**Answer**: Yes — –5 is closer to zero.

### Medium (2)

4. Evaluate: (–2) + (–3)
**Answer**: –5 — Add magnitudes, keep the negative sign.

5. Subtract: (+3) – (–4)
**Answer**: +7 — Subtracting a negative is same as adding.

### Difficult (3)

6. Use a number line to solve: (–6) + (+8)
**Answer**: +2 — Movement from –6 forward by 8 units.

7. Simplify: (–5) – (+7)
**Answer**: –12 — Move 7 units backward from –5.

8. Evaluate: (–10) – (–15)
**Answer**: +5 — Subtraction becomes addition: –10 + 15

### Very Difficult (2)

9. Solve: (+120) – (–130)
**Answer**: +250 — Subtracting negative = adding positive.

10. What is the result of: –150 – (+200)?
**Answer**: –350 — Add magnitudes and keep the negative sign.

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