Multiplying integers

The token model

Green tokens stand for +1 and red tokens for −1; a green and a red together form a zero pair, worth 0. To multiply, start with an empty bag. In a × b, the multiplier a tells you how many times, and the multiplicand b tells you which group of tokens:

• If the multiplier is positive, put the group into the bag that many times.
• If the multiplier is negative, take the group out of the bag that many times.

The four cases

• 4 × 2: put 2 positives in, 4 times: 8 positives, so +8.
• 4 × (−2): put 2 negatives in, 4 times: 8 negatives, so −8.
• (−4) × 2: take 2 positives out, 4 times. The bag is empty, so each time first put in 2 zero pairs, then take out the 2 positives. 8 negatives are left: −8.
• (−4) × (−2): take 2 negatives out, 4 times, again from zero pairs. 8 positives are left: +8.

Taking out positives does the same as putting in negatives, so (−4) × 2 can also be seen as adding 2 negatives 4 times. And a number gives the same answer however its tokens look: 2 red tokens, or 4 red and 2 green, both stand for −2, and 4 times either is −8.

Patterns that continue below zero

Write 3 × 3 = 9, 2 × 3 = 6, 1 × 3 = 3, 0 × 3 = 0. Each time the multiplier drops by 1, the product drops by 3, so the next ones are (−1) × 3 = −3 and (−2) × 3 = −6. With −3 as the multiplicand, each drop in the multiplier makes the product rise by 3: 0 × (−3) = 0, then (−1) × (−3) = 3 and (−2) × (−3) = 6. The patterns agree with the tokens.

The sign rules

The magnitude of the product never depends on the signs: it is the product of the magnitudes. The sign follows these rules:

Signs of the two numbers	Sign of the product
both positive, or both negative	positive
one positive, one negative	negative

Two special cases are worth remembering: 1 × a = a, and (−1) × a = −a, the additive inverse of a. In 628 CE, Brahmagupta stated the same rules in words: the product of two fortunes, or of two debts, is a fortune; the product of a fortune and a debt is a debt. It was the first time anyone wrote them down.

Worked examples

• Negative marking: an exam gives +5 for a correct answer and −2 for a wrong one. Mala has 30 correct and 20 wrong: 30 × 5 + 20 × (−2) = 150 + (−40) = 110 marks. The highest possible score is 50 × 5 = 250 and the lowest, with all 50 wrong, is 50 × (−2) = −100.
• A lift in a mine: moving down 3 m a minute is −3 per minute, so in 60 minutes the change is 60 × (−3) = −180, that is 180 m below ground level.
• Without calculating: if 123 × 456 = 56088, then (−123) × 456 = −56088 and (−123) × (−456) = 56088, because only the sign changes.

Why it matters

Multiplying integers lets you handle repeated losses, falls in temperature, debts and movements below a starting point in one step. Next, learn dividing integers and the properties of integer multiplication. Answers to the textbook questions are in the NCERT solutions for Part 2 Chapter 2; see all of Part 2 Chapter 2 notes.