Same size, sign by a rule
The signs decide the sign of a product, never its size.
Four results, one size
From the token bag: 4 × 2 = 8, 4 × (−2) = −8, (−4) × 2 = −8, (−4) × (−2) = 8.
When we change the signs of 4 and 2, what changes in the answer?
What this lesson covers
The idea
The magnitude of a product of two integers does not depend on their signs; the product is positive when both are positive or both negative, and negative when one is positive and the other negative.
Four results, one size
From the token bag: 4 × 2 = 8, 4 × (−2) = −8, (−4) × 2 = −8, (−4) × (−2) = 8.
When we change the signs of 4 and 2, what changes in the answer?
- Only its sign
- Its size and its sign
- Nothing at all
Find all four products
Each card changes the signs of the same two numbers. Find the value of each: the size and the sign.
The size stays, the sign follows a rule
3 × 4, (−3) × 4, 3 × (−4) and (−3) × (−4) all have the same size, 12. Only the sign differs.
Brahmagupta said it in 628 CE with fortunes (positive) and debts (negative): a fortune times a fortune, or a debt times a debt, is a fortune. A debt times a fortune, or a fortune times a debt, is a debt.
The magnitude of a product of two integers does not depend on their signs. The product is positive when both are positive or both negative, and negative when one is positive and the other negative.
- First number | Second number | Product
- positive | positive | positive
- negative | negative | positive
- positive | negative | negative
- negative | positive | negative
Notes
The magnitude of a product does not depend on the signs. The product is positive for two positives or two negatives, and negative when one is positive and the other negative.
Check yourself
Find (−12) × (−5).
Answer: 60
12 × 5 = 60, and two negatives give a positive: 60.
In an exam, 5 marks are given for each correct answer and 2 negative marks for each wrong one. Mala had 30 correct and 20 wrong answers. What are her total marks? Use 30 × 5 + 20 × (−2).
Answer: 110
30 × 5 = 150 and 20 × (−2) = −40, so the total is 150 + (−40) = 110.
Pooja says (−7) × (−3) = −21, because both numbers have a minus sign. What should it be?
Which product is negative?
- 21: two negatives give a positive — correct. Yes! The size is 7 × 3 = 21 and both numbers are negative, so it is positive.
- −21: it is right. Two negative numbers give a positive product, not a negative one.
- −10: add the numbers. We multiply the sizes: 7 × 3 = 21. Both are negative, so the product is +21.
- (−8) × (−6). Both are negative, so the product is positive: 48.
- (−8) × 6 — correct. Yes! One number is negative and the other is positive.
- 8 × 6. Both are positive, so the product is positive: 48.