Q1.
Can you summarise the rules for integer division looking at the above pattern?
The worked division examples on page 38 that the question refers to.
| We know that… | …therefore |
|---|---|
| 25 × (– 4) = (– 100) | (– 100) ÷ 25 = (– 4) |
| (– 4) × 25 = (– 100) | (– 100) ÷ (– 4) = 25 |
| (– 25) × (– 2) = 50 | 50 ÷ (– 25) = (– 2) |
Answer
Division follows exactly the same sign rule as multiplication.
- Magnitudes: divide as if both numbers were positive.
- Sign: same signs give a positive quotient, different signs a negative quotient.
In symbols, for positive integers a and b with b ≠ 0,
a ÷ (–b) = –(a ÷ b)
(–a) ÷ b = –(a ÷ b)
(–a) ÷ (–b) = a ÷ b
(–a) ÷ b = –(a ÷ b)
(–a) ÷ (–b) = a ÷ b
| Division | Rewritten as a multiplication | Quotient |
|---|---|---|
| (–100) ÷ 25 | 25 × ? = –100, and 25 × (–4) = –100 | –4 |
| (–100) ÷ (–4) | (–4) × ? = –100, and (–4) × 25 = –100 | 25 |
| 50 ÷ (–25) | (–25) × ? = 50, and (–25) × (–2) = 50 | –2 |
Why it happens: a division question is a multiplication question in disguise — “what must I multiply the divisor by to reach the dividend?” Since the answer has to obey the multiplication sign rule, the quotient inherits that very same rule. This is exactly what Brahmagupta wrote in 628 CE: the product or quotient of two debts is a fortune.