Chapter 10 — Operations with Integers

Chapter 10 'Operations with Integers' covers multiplication and division of integers, using the token model and number-line model to explain sign rules, and proves that integer multiplication is commutative, associative, and distributive over addition.

When both the multiplier and multiplicand are positive, the product is positive. When both are negative, the product is positive. When one is positive and the other is negative, the product is negative.

The token model explains it: for (−4) × (−2), you remove 2 negative (red) tokens from an empty bag 4 times. To do that you first add zero pairs; after removing the negatives, 8 positive (green) tokens remain, giving +8. The pattern of multiplications also confirms this — when the multiplicand is negative, every unit decrease in the multiplier increases the product, so crossing zero gives a positive result.

You start with an empty bag. A positive multiplier means you place tokens into the bag that many times; a negative multiplier means you remove tokens that many times. Green tokens represent +1 and red tokens represent −1. A zero pair (one green + one red) adds nothing and is used when you need to remove tokens that are not yet in the bag.

Division follows the same sign rules as multiplication. For positive integers a and b (b ≠ 0): a ÷ (−b) = −(a ÷ b), (−a) ÷ b = −(a ÷ b), and (−a) ÷ (−b) = a ÷ b. The quotient is positive when dividend and divisor have the same sign, and negative when they have opposite signs.

Yes. For any two integers a and b, a × b = b × a. The magnitude of the product depends only on the magnitudes of the two numbers, and the sign rules are symmetric — swapping the multiplier and multiplicand does not change whether the product is positive or negative.

Yes. For any three integers a, b, and c, a × (b × c) = (a × b) × c. The product remains the same when three or more integers are multiplied in any order or grouping.

For any integers a, b, and c: a × (b + c) = (a × b) + (a × c). This property holds for integers just as it does for positive integers, and can be visualised using a rectangular arrangement of green and red tokens.

−1 × a = −a for all integers a. Multiplying any integer by −1 gives its additive inverse — the same magnitude but opposite sign.

Brahmagupta, in his Brāhmasphuṭasiddhānta written in 628 CE, articulated the first explicit rules. He used 'fortune' (dhana) for positive values and 'debt' (ṛṇa) for negative values, stating: the product of two fortunes is a fortune, the product of two debts is a fortune, and the product of a fortune and a debt is a debt.

Rightward movement is represented as a positive integer and leftward movement as a negative integer. If a coin starts at 0 and moves a units then b units (each positive or negative), its final position is P = a + b. For example, a first strike of +5 and a second of −7 gives a final position of 5 + (−7) = −2, meaning 2 units to the left of 0.

Yes, the PDF is free to view and download on cbseprepmaster.com — no sign-up or account required.