Arithmetic with Rational Numbers
“Settle the sign first, worry about the size second.”
The formula
(+)×(+)=(+), (−)×(−)=(+), (+)×(−)=(−)How to read it: same signs multiply to a positive, different signs multiply to a negative
- (+)×(+), (−)×(−)
- — product of two numbers with the same sign → positive
- (+)×(−), (−)×(+)
- — product of two numbers with different signs → negative
- ÷
- — division follows the exact same sign rules as multiplication
The hook
Eighty percent of arithmetic with rationals is one thing — the sign rule. Why does multiplying two negatives give a positive?
In plain words
The rules for adding, subtracting, multiplying, and dividing integers and fractions. The key is to settle the sign first, then compute the sizes.
The intuition
Think 'the opposite of the opposite is the original'. Multiplying by a negative flips direction — flip twice and you're back where you started (positive). That's why (−)×(−)=(+).
How it's built
For × and ÷, settle the sign first: same signs give +, different signs give −. For + and −, remember 'subtracting a negative is adding': a−(−b) = a+b.
Example
Take (−2) × (−6) ÷ 4. Sign: (−)×(−)=(+). Size: 2×6÷4 = 12÷4 = 3. So the answer is +3.
Common mistake
(−2)² and −2² are different. (−2)²=(−2)×(−2)=4, but −2² is 2² with a minus in front, so −4. The parentheses decide whether the sign gets squared too.
Where it's used
Temperature swings, deposits and withdrawals, rising and falling game scores — any real-life calculation mixing gains and losses is arithmetic with rationals.
Where it came from
It took a long time to accept negatives as real numbers, but once the sign rules were settled, equations and all of algebra began to run smoothly.
Prerequisites
Quick check
What is (−2) × 3?
- −6✓
- 6
- −5
- 5
Practice
Compute (−5) + (−3).
Answer: -8
- Same-sign addition: add the sizes, keep the sign
- 5 + 3 = 8, sign is −
- → −8
Takeaway: Negative plus negative heads to a deeper negative.
Compute (−7) − (−2).
Answer: -5
- Subtracting a negative is adding: −7 − (−2) = −7 + 2
- = −5
Takeaway: Subtracting a negative flips to adding.
Compute (−2) × (−6) ÷ 4.
Answer: 3
- Sign: (−)×(−)=(+)
- Size: 2 × 6 ÷ 4 = 12 ÷ 4 = 3
- → +3
Takeaway: Settle the sign first, then compute the size.
Compute (−2)³ and write why it comes out negative.
Answer: undefined
- (−2) × (−2) × (−2)
- = 4 × (−2) = −8
- Multiplying a negative an odd number of times leaves it negative
Takeaway: An even count of negatives is positive; an odd count is negative.