Multiplication of Two Negative integers
When you multiply a negative number by another negative number, the result is a positive number. This rule is not obvious and proving it is not straightforward.
However, here is a clever way to prove the rule by starting with an equation and factoring out terms.
Goal
Prove that the product of two negative numbers or terms is positive:
(−a)(−b) = ab
where a and b can be:
- Numbers (i.e. a = 5, b = 1/2)
- Constants
- Variables
- Expressions [i.e. a = (y2 + 6), b = (h − w + z)]
Proof
A clever way to prove that (−a)(−b) = ab is to consider the equation:
x = ab + (−a)(b) + (−a)(−b)
You want use this equation to show that x = ab and x = (−a)(−b).
Factor out −a
First, factor out −a from the expression (−a)(b) + (−a)(−b):
x = ab +(−a)(b) + (−a)(−b)
Thus
x = ab + (−a)[b + (−b)]
Since b + (−b) = 0
x = ab + (−a)(0)
Thus
x = ab
Factor out b
Now, with the original equation, factor out b from the expression ab + (−a)(b):
x = ab + (−a)(b) + (−a)(−b)
x = b[a + (−a)] + (−a)(−b)
x = b(0) + (−a)(−b)
Thus
x = (−a)(−b)
Result
Since x = ab and x = (−a)(−b):
(−a)(−b) = ab
This can be extended to any even amount of negative numbers by factoring out in steps:
(−a)(−b)(−c)(−d) = ab(−c)(−d) = abcd
Summary
This clever method proves that (−a)(−b) = ab.
The fact that the product of two negative numbers, terms, or expressions is positive can be extended to any even number of negative items.
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