Brahmagupta, in his Brāhmasphuṭasiddhānta \((628 \ CE)\), gave rules for multiplying and dividing positive and negative numbers.
He used:
- Fortune (dhana) for a positive number.
- Debt (rna) for a negative number.
His rules can be expressed as follows:
- The product or quotient of two fortunes is a fortune.
- The product or quotient of two debts is a fortune.
- The product or quotient of a debt and a fortune is a debt.
- The product or quotient of a fortune and a debt is a debt.
This means:
| Numbers | Product / Quotient |
|
Positive \(\times\) Positive
Positive \(\div\) Positive
|
Positive |
|
Negative \(\times\) Negative
Negative \(\div\) Negative
|
Positive |
|
Positive \(\times\) Negative
Positive \(\div\) Negative
|
Negative |
|
Negative \(\times\) Positive
Negative \(\div\) Positive
|
Negative |
For both multiplication and division:
Same signs \(\rightarrow\) Positive
Different signs \(\rightarrow\) Negative
Important!
The magnitude of the product or quotient is determined by the magnitudes of the numbers; their signs determine whether the result is positive or negative.
Example:
At \(6 \ a.m.\), the temperature in a hill station was \(-4^\circ C\). During the next \(5\) hours, the temperature decreased by \(2^\circ C\) every hour. By afternoon, it increased by \(9^\circ C\). What was the temperature in the afternoon?
Solution:
Initial temperature \(= -4^\circ C\)
Decrease in \(5\) hours \(= 5 \times (-2) = -10^\circ C\)
Temperature after the decrease \(= (-4)^\circ C + (-10)^\circ C = (-14)^\circ C\)
Increase in the afternoon \(= (-14)^\circ C + 9^\circ C = -5^\circ C\)
Therefore, the afternoon temperature was \(-5^\circ C\).