Chapter 2: Operations with Integers
Sign Laws, Token Models, Algebraic Properties, and Brahmagupta's Ancient Rules
📖 Visualizing Integers
Moves capture magnitude & direction: Rightward is (+), Leftward is (−). Final spot: P = a + b.
(+1) + (−1) = 0. Subtracting an integer is the same as adding its additive inverse: a − b = a + (−b).
(+) multiplier = add sets into bag.
(−) multiplier = remove sets from bag.
⚡ Sign Rules for (×) and (÷)
1. Same Signs → Positive (+):
• (+) × (+) = (+) | (+) ÷ (+) = (+)
• (−) × (−) = (+) | (−) ÷ (−) = (+)
2. Opposite Signs → Negative (−):
• (+) × (−) = (−) | (+) ÷ (−) = (−)
• (−) × (+) = (−) | (−) ÷ (+) = (−)
3. Chain Product Sign Rule:
• Even number of (−) signs → Positive (+)
• Odd number of (−) signs → Negative (−)
🧩 Algebraic Properties
Order does not change the product: a × b = b × a.
Grouping does not matter: (a × b) × c = a × (b × c).
Multiplication distributes over addition: a × (b + c) = (a × b) + (a × c).
🏛️ Brahmagupta's Heritage & Real-World Modeling
• Brahmasphutasiddhanta (628 CE): Brahmagupta defined integers using Fortune (Dhana = +) and Debt (Rna = −): "The product of two debts is a fortune; product of a debt and a fortune is a debt."
• Elevator & Depth Models: Above ground = (+), Shaft/Mine descent = (−). Ending Depth = Initial + (Time × Velocity).
• Competitive Exam Marking: Total Marks = (Correct × +marks) + (Incorrect × −penalty).
• Traditional Game Terhüchü: Strategy board game from Assam & Nagaland played on a 16-square diagonal grid using jumping integer paths.
⚠️ Common Integer Traps
Trap 1: Addition vs. Multiplication Signs
(−4) + (−2) = −6 (combining debts), but (−4) × (−2) = +8 (multiplying negatives). Never mix up addition with multiplication!
Trap 2: Division is NOT Commutative
While a × b = b × a, division does not allow swapping: a ÷ b ≠ b ÷ a (e.g. (−8) ÷ 2 = −4, but 2 ÷ (−8) = −1/4).