Integers (The Other Side of Zero)

Chapter at a Glance

This chapter introduces integers, extending the number ray starting at 0 to the left to form a complete number line. Using the models of "Bela's Building of Fun" (elevators moving above and below ground floor) and a deep "Mineshaft," it teaches addition, subtraction, and comparison of positive and negative numbers. Students learn to use two-color token sets (zero pairs) and unmarked number lines. The chapter links integers to credits/debits, sea level, and temperature, and presents historical rules codified by Brahmagupta in 628 CE.

Key Definitions & Terminology

Negative Numbers: Numbers less than zero, written with a minus sign ($-$) in front (e.g. $-1, -2, -3$). They lie to the left of zero on the horizontal number line.
Positive Numbers: Numbers greater than zero, sometimes written with a plus sign ($+$) (e.g. $+1, +2$).
Integers: The complete set of numbers consisting of positive integers ($1, 2, 3, \dots$), negative integers ($\dots, -3, -2, -1$), and zero ($0$).
Zero: A number representing nothing, which is neither positive nor negative.
Additive Inverse (or Inverse): The number that when added to a given number gives zero. For any integer $a$, its inverse is $-a$ (e.g. $+4$ and $-4$ are inverses). Zero is its own inverse.
Zero Pair: A pair consisting of one positive token ($+1$) and one negative token ($-1$) that cancel each other out to have a net value of zero.
Credits & Debits: In accounting, credits represent money added (positive integers) and debits represent money spent or owed (negative integers).
Sea Level: The reference elevation ($0\text{ m}$) from which geographical heights (positive) and depths (negative) are measured.

Formulas, Rules & Properties

Starting Position + Movement = Target Position:
Used to model integer addition. E.g., starting at floor $+1$ and pressing $+2$ lands on $+3$ ($(+1) + (+2) = +3$).
Target Position - Starting Position = Movement Needed:
Used to model integer subtraction. E.g., to go to target $-1$ from start $-2$ requires pressing $+1$ ($(-1) - (-2) = +1$).
Integer Order Property:
Integers increase as you move right and decrease as you move left on the number line:
$$\dots < -3 < -2 < -1 < 0 < 1 < 2 < 3 < \dots$$
Brahmagupta's Rules for Addition (628 CE):
Sum of two positives: Positive (e.g. $2 + 3 = 5$).
Sum of two negatives: Negative. Add their values without signs and attach a minus sign (e.g. $(-2) + (-3) = -5$).
Sum of a positive and a negative: Subtract the smaller number (without sign) from the larger (without sign), and attach the sign of the larger number (e.g. $-5 + 3 = -2$).
Sum of opposites: A number plus its inverse is zero (e.g. $2 + (-2) = 0$).
Brahmagupta's Rules for Subtraction (628 CE):
Subtracting a negative number is equivalent to adding its positive counterpart:
$$a - (-b) = a + b$$
Subtracting a number from itself gives zero ($a - a = 0$).
Subtracting a number from zero gives its inverse ($0 - a = -a$).

Core Concepts & Topics

Bela's Building of Fun:
Ground floor is Floor $0$ (Welcome Hall).
Floors above are positive: $+1$ (Food Court), $+2$ (Art Centre), $+3$ (Book Store).
Floors below are negative: $-1$ (Toy Store), $-2$ (Video Games), $-3$ (Cinema Centre), $-5$ (Sports Centre).
The Token Model:
Positive integers are represented by green tokens, and negative integers by red tokens.
Pair up one green and one red token to form a zero pair and remove them.
For subtraction (e.g. $+4 - (-6)$): Start with 4 positive tokens. Since there are no negative tokens to take away, add 6 zero pairs (which does not change the net value). Then, remove the 6 negative tokens, leaving 10 positive tokens ($+4 - (-6) = +10$).
Unmarked Number Line (UNL):
A diagram showing only 0 to evaluate large additions or subtractions visually by drawing arrows for relative movements on paper or mentally.
Historical Development of Integers:
Negative numbers were used in ancient China (Nine Chapters on Mathematical Art, 1st-2nd century CE) using red and black rods, and in India (Kautilya's Arthashastra, c. 300 BCE) for accounting.
The Bakhshali Manuscript (c. 300 CE) wrote negative numbers with a special symbol placed after the digit.
Brahmagupta (628 CE) formally codified the arithmetic operations on positive numbers, negative numbers, and zero, establishing the foundations of modern algebra.
No Year Zero Rule:
In chronological time, there is no year 0. The year directly before 1 CE is 1 BCE.
Hollow Integer Grid Puzzles:
Placing numbers in a hollow square layout such that the sum of the numbers in the top row, bottom row, left column, and right column all equal a specified "border sum."
Magic Grid Elimination Game:
Circling numbers and striking out their rows and columns in a grid. The sum of the circled numbers always equals a constant final sum, depending on the grid's algebraic pattern.

Worked Examples

Elevator Movements:
Question: You start at Floor $+2$ and press $-3$ in the lift. Where do you land?
Solution: $(+2) + (-3) = \mathbf{-1}$ (Toy Store).
Question: If you are at Floor $-2$ and want to reach the Sports Centre at $-5$, what button must you press?
Solution: $(-5) - (-2) = \mathbf{-3}$ (press down three times).
Comparing Integers:
$-5 < -3$ (since Floor $-5$ is lower than Floor $-3$, or $-5$ lies to the left of $-3$).
$+6 > -6$ (all positive numbers are greater than all negative numbers).
$-10 > -12$ ($-10$ lies to the right of $-12$ on the number line).
Unmarked Number Line (UNL) Calculations:
$-125 + (-30) = \mathbf{-155}$ (moving 30 units left from $-125$).
$+105 - (-55) = 105 + 55 = \mathbf{160}$ (subtracting a negative is adding a positive).
$-99 - (-200) = -99 + 200 = \mathbf{101}$.
Credit & Debit Account Balance:
Question: An account starts at ₹0. Find the balance after credits of ₹30, ₹40, and ₹50, and debits of ₹40, ₹50, and ₹60.
Solution: $(+30 + 40 + 50) - (40 + 50 + 60) = 120 - 150 = \mathbf{-₹30}$ (a negative balance).
Geographical Elevation Sequence:
Question: Sequence A ($+1500\text{ m}$), B ($-500\text{ m}$), C ($+300\text{ m}$), D ($-1200\text{ m}$), E ($+1200\text{ m}$), F ($-200\text{ m}$), G ($+100\text{ m}$) in decreasing order of heights.
Solution: A, E, C, G, F, B, D. (Note that $-200 > -500 > -1200$).
No Year Zero Chronology:
Question: What year was it 2200 years ago from the year 2026 CE?
Solution: $2026 - 2200 = -174$. Since there is no year 0, we adjust by 1 year. The year was 175 BCE.
Question: What will be the year 320 years after 680 BCE?
Solution: $680\text{ BCE} - 320 = \mathbf{360\text{ BCE}}$.
Dice Sums Puzzle:
Question: Two dice have faces labeled ${-1, 2, -3, 4, -5, 6}$. What sums between the minimum $-10$ and maximum $12$ are impossible to get by adding two rolled numbers?
Solution: The impossible sums are: $-9, -7, -5, 0, 2, 7, 9, 11$.

Practical Activities & Experiments

Classroom Elevator Simulation: Creating a vertical line of floors on a strip of paper, moving a clip marker to model lift movements and write equations.
Two-Color Token Addition/Subtraction: Placing green (positive) and red (negative) counters on desks, pairing them to remove zero pairs, and demonstrating subtraction by introducing zero pairs.
Integers Snakes & Ladders: Playing a board game with pawns starting at 0. Players roll one positive and one negative die, calculate the sum/difference, and move towards $+50$ (positive outcome) or $-50$ (negative outcome) to win.
Thermometer Ice-Water Experiment: Using a laboratory thermometer in a beaker of crushed ice to observe the temperature dropping to and below $0^\circ\text{ C}$.

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