Mathematics: Class 6 — Chapter 10: The Other Side of Zero

10. Introduction: More and More Numbers!

In our study of mathematics, we progress through different sets of numbers to represent various real-world scenarios:

Counting Numbers: $1, 2, 3, 4, \dots$
Zero ($0$): Represents nothing, comes before $1$, and has a rich history in India and globally.
Fractions: Numbers existing between whole numbers, such as $\frac{1}{2}$, $\frac{3}{2}$, and $\frac{13}{6}$.

While the standard number line starting at $0$ and extending to the right is geometrically a number ray, we need numbers to the left of $0$ to complete it into a true number line. These numbers are less than $0$ and are called negative numbers.

10.1 Bela's Building of Fun

Using a multi-storied building with floors above and below the ground level, we can model numbers on both sides of zero:

The ground floor is designated as Floor 0.
Floors above the ground are represented by positive numbers (e.g., $+1, +2, +3$).
Floors below the ground are represented by negative numbers (e.g., $-1, -2, -3$).
Zero ($0$) is neither positive nor negative.

Addition to Keep Track of Movement

We can track movement between floors using the expression:

$$\text{Starting Floor} + \text{Movement} = \text{Target Floor}$$

Going up is represented by a positive movement (e.g., $+2$).
Going down is represented by a negative movement (e.g., $-3$).
Example: Starting at Floor $+1$ and moving $+2$ floors leads to: $$(+1) + (+2) = +3\text{ (Book Store)}$$
Example: Starting at Floor $+2$ and moving $-3$ floors leads to: $$(+2) + (-3) = -1\text{ (Toy Store)}$$

Back to Zero & Additive Inverses

If you are at Floor $+3$ and want to return to Floor $0$, you must move $-3$ floors. This is represented as:

$$(+3) + (-3) = 0$$

We call $-3$ the additive inverse of $+3$, and $+3$ the additive inverse of $-3$. Every number has an additive inverse which, when added to it, results in zero.

Comparing Numbers Using Floors

We can determine which number is larger by looking at their relative positions on the vertical floor plan:

A floor that is lower is represented by a smaller number. Thus, $+3 < +4$ and $-4 < -3$.
All negative numbers are less than $0$.
All positive numbers are greater than $0$.

Subtraction to Find Movement

To find the movement needed to go from a starting floor to a target floor, we use subtraction:

$$\text{Target Floor} - \text{Starting Floor} = \text{Movement Needed}$$

Example: To go from Starting Floor $+2$ to Target Floor $+5$: $$(+5) - (+2) = +3\text{ (Move 3 floors up)}$$
Example: To go from Starting Floor $+3$ to Target Floor $-1$: $$(-1) - (+3) = -4\text{ (Move 4 floors down)}$$
Example: To go from Starting Floor $-2$ to Target Floor $+2$: $$(+2) - (-2) = +4\text{ (Move 4 floors up)}$$

10.2 The Token Model

The token model is a visual way to perform operations on integers using colored tokens:

Positive Token: Represented as green with a plus sign ($+$).
Negative Token: Represented as red with a minus sign ($-$).
Zero Pair: One positive token and one negative token cancel each other out because $(+1) + (-1) = 0$.

Addition Using Tokens

To add integers, combine the tokens and remove all zero pairs:

Example: Add $+5$ and $-8$. Combine $5$ positive tokens and $8$ negative tokens. Cancel out $5$ zero pairs. You are left with $3$ negative tokens: $$(+5) + (-8) = -3$$

Subtraction Using Tokens

To subtract, take away the specified tokens. If there are not enough tokens, add zero pairs first:

Example: $(+5) - (+4) = +1$ (Take away $4$ positive tokens from $5$ positive tokens).
Example: $(+5) - (+6)$. Start with $5$ positive tokens. Since we cannot take away $6$, add $1$ zero pair (one positive and one negative). Now take away $6$ positive tokens, leaving $1$ negative token: $$(+5) - (+6) = -1$$
Example: $+4 - (-6)$. Start with $4$ positive tokens. To take away $6$ negative tokens, add $6$ zero pairs. Remove the $6$ negative tokens, leaving $10$ positive tokens: $$(+4) - (-6) = +10$$

10.3 Integers in Other Places

Integers are widely used to represent opposite physical quantities in various fields:

Credits and Debits (Banking)

Credits: Positive numbers representing money deposited into an account.
Debits: Negative numbers representing money withdrawn or spent.
An account balance can become negative if debits exceed credits (overdraft).

Geographical Elevations

Sea Level: Represented as $0\text{ m}$.
Heights above sea level (mountains, plateaus) are positive numbers.
Depths below sea level (ocean trenches, deep mines) are negative numbers.

Temperature

Measured in degrees Celsius ($^\circ\text{C}$).
The freezing point of water is $0^\circ\text{C}$.
Temperatures below freezing are represented as negative numbers (e.g., $-4^\circ\text{C}$).

10.4 Explorations with Integers

Hollow Integer Grids

A hollow integer grid is a $3 \times 3$ grid with the center cell left empty. It has a special property called the border sum, where the sum of the numbers in the top row, bottom row, leftmost column, and rightmost column are all equal.

10.5 A Pinch of History

The concept of negative numbers and zero has deep historical roots in Asia:

China: In *The Nine Chapters on Mathematical Art* (completed by the 1st or 2nd century CE), positive and negative numbers were represented using red and black rods.
India:
- Kautilya's *Arthashastra* (c. 300 BCE) detailed the concepts of credit and debit.
- The *Bakshali Manuscript* (c. 300 CE) used a special symbol placed after a number to denote it as negative.
- Brahmagupta, in his treatise *Brahma-sphuta-siddhanta* (628 CE), gave the first systematic rules for arithmetic operations involving positive numbers, negative numbers, and zero.

Brahmagupta's Rules for Addition

The sum of two positives is positive (e.g., $2 + 3 = 5$).
The sum of two negatives is negative (e.g., $(-2) + (-3) = -5$).
To add a positive and a negative, subtract the smaller value from the greater value (ignoring signs) and keep the sign of the greater value (e.g., $-5 + 3 = -2$).
The sum of a number and its inverse is zero (e.g., $2 + (-2) = 0$).
The sum of any number and zero is the same number (e.g., $-2 + 0 = -2$).

Brahmagupta's Rules for Subtraction

Subtracting a smaller positive from a larger positive yields a positive result (e.g., $3 - 2 = 1$).
Subtracting a larger positive from a smaller positive yields a negative result (e.g., $2 - 3 = -1$).
Subtracting a negative number is the same as adding its positive counterpart (e.g., $2 - (-3) = 2 + 3 = 5$).
Subtracting a number from itself gives zero (e.g., $2 - 2 = 0$).
Subtracting zero from a number leaves it unchanged (e.g., $-2 - 0 = -2$). Subtracting a number from zero gives its additive inverse (e.g., $0 - (-2) = 2$).