Properties of integer multiplication

Multiplying by 1 and by −1

Putting a group of tokens into an empty bag once leaves exactly that group, so 1 × a = a for every integer a, positive or negative. Multiplying by −1 takes the group out once, which leaves its opposite: (−1) × a = −a, the additive inverse of a. For example (−1) × 7 = −7 and (−1) × (−7) = 7.

Order does not matter: commutative

Compare 3 × (−4) = −12 with (−4) × 3 = −12, or (−15) × (−8) = 120 with (−8) × (−15) = 120. Swapping the numbers keeps the magnitude, because the magnitudes are just positive numbers being multiplied, and it keeps the sign, because the two signs are the same pair in either order. So multiplication of integers is commutative: a × b = b × a.

Grouping does not matter: associative

For 5 × (−3) × 4, work it either way: (5 × (−3)) × 4 = (−15) × 4 = −60, and 5 × ((−3) × 4) = 5 × (−12) = −60. Even (5 × 4) × (−3) = 20 × (−3) = −60. Multiplication of integers is associative: (a × b) × c = a × (b × c). Together with the commutative property, this means three or more integers can be multiplied in any order: 25 × (−6) × 12 is −1800 whichever pair you multiply first, and 25 × 12 = 300 first makes it easy.

The sign of a long product

Multiply −1 again and again: (−1) × (−1) = 1, then × (−1) = −1, then 1, then −1. The sign flips with every negative factor. So:

• an even number of negative factors gives a positive product;
• an odd number of negative factors gives a negative product.

For example (−2) × (−1) × (−5) × (−3) has four negative factors, so it is +30, while (−7) × 4 × (−1) has two, so it is +28.

Spreading over addition: distributive

Is 5 × (4 + (−2)) the same as 5 × 4 + 5 × (−2)? The left side is 5 × 2 = 10 and the right side is 20 + (−10) = 10. Tokens show why. Arrange tokens in 4 rows, with 2 columns of positives and 3 columns of negatives:

The whole rectangle is 4 × (2 + (−3)), and its two parts are 4 × 2 and 4 × (−3), so the two must be equal: a × (b + c) = a × b + a × c. It also holds with a negative multiplier: (−2) × (4 + (−3)) = (−2) × 1 = −2, and (−2) × 4 + (−2) × (−3) = −8 + 6 = −2.

Properties make sums easier

The properties help you calculate without long multiplication. Given (−548) × 972 = −532656, the value of (−547) × 972 is −532656 + 972 = −531684, because −547 is one more than −548. They also explain the magic grid in the chapter: its numbers form a multiplication table, and by the commutative and associative properties every set of circled numbers multiplies to the same product, −30240.

Why it matters

These laws let you rearrange and regroup a calculation safely, even with negative numbers, which is the basis of algebra later on. Review multiplying integers and dividing integers. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 2; see all of Part 2 Chapter 2 notes.