Multiplication and Division of Integers

The Story

"What is (−3) × 4?" asked Ms. Rao.

Kabir thought of it as repeated addition. "−3, added 4 times: −3 + (−3) + (−3) + (−3) = −12."

"Exactly right," said Ms. Rao. "A positive times a negative — or a negative times a positive — always gives a negative answer."

She then asked: "What about (−3) × (−4)?"

Kabir hesitated. "Two negatives... multiplied?"

Anaya jumped in. "Think of it like a game rule: 'the opposite of the opposite.' (−3) × (−4) means 'the opposite of 3 groups of −4' — which flips it back to positive. (−3) × (−4) = 12."

"Here's the pattern that makes it stick," said Ms. Rao, and she wrote a short table: 3 × (−4) = −12, then 2 × (−4) = −8, then 1 × (−4) = −4, then 0 × (−4) = 0. "Each answer went up by 4 as the first number dropped by 1. Follow the pattern one more step: −1 × (−4) should be −4 + 4 = ... 4."

Kabir saw it. "The pattern itself proves negative times negative has to be positive — it's not just a memorised rule."

"Same logic applies to division," said Ms. Rao. "Division and multiplication follow the exact same sign rules. Same signs give a positive result; different signs give a negative result."

She gave a final round: (−20) ÷ (−4). "Same signs — both negative — so the answer is positive." Kabir calculated: 20 ÷ 4 = 5. "So (−20) ÷ (−4) = 5."

Anaya summarised the whole rule in one line: "Same signs, positive answer. Different signs, negative answer. Works for both multiplying and dividing."