为什么负数的乘积是正数？
代数是处理算术运算及其相关符号的数学分支。这些符号被称为变量，当受到不同约束时可能取不同的值。变量主要表示为 x、y、z、p 或 q，可以通过加法、减法、乘法和除法的不同算术运算进行操作，以计算值。
负数
负数对应于小于零的整数值。负数用于表示损失或不足。负数前面带有一个负号。例如，贷款或信贷用负数表示
Rule: Opposite of an opposite is equivalent to the original value.
For example, −(−3) = 3
为什么负数的乘积是正数？
回答：
Upon multiplication of a negative number by another negative number, the resultant operation is positive in nature.
To Prove: 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
To prove (−a)(−b) = ab, we can consider the equation:
x = ab + (−a)(b) + (−a)(−b)
It can be easily shown that x = ab and x = (−a)(−b).
Factor out −a
First, factoring out −a from the expression (−a)(b) + (−a)(−b):
x = ab +(−a)(b) + (−a)(−b)
Thus, we obtain,
x = ab + (−a)[b + (−b)]
Since, b + (−b) = 0
x = ab + (−a)(0)
Thus,
x = ab
Factor out b
Factoring 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)
Therefore,
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
The method easily proves (−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.
示例问题
问题 1. 求 -3a × -20b 的乘积
解决方案：
Here we have to find the product of -3a × -20b
As we know that when we multiply two negative numbers the answer will be positive
Now multiplying
-3a × -20b = +60ab
Therefore, the solution is positive.
问题 2. 计算 2ab + (−a)(5b) + (−2a)(−3b)？
解决方案：
Here we have to find the product of
= 2ab + (−a)(5b) + (−2a)(−3b)
First solving the brackets
= 2ab + (-a × 5b) + (-2a × -3b)
= 2ab + (-5ab) + 6ab
= 2ab – 5ab + 6ab
= 8ab – 5ab
= 3ab
问题 3. 评估 10ab + (−21a)[5b + (−10b)]？
解决方案：
Here we have to find the product of 10ab + (−21a)[5b + (−10b)]
First solving the brackets
= 10ab + (−21a)× [5b −10b]
= 10ab – 21a × (-5b)
= 10ab + 105ab
= 115ab
问题 4. 求 {-(4x + 5x) × (12x – 16x)} 的乘积？
解决方案：
Here we have to find the product of {-(4x + 5x) × (12x – 16x)}
First solving the brackets
= {(-4x – 5x) × (12x – 16x)}
= (-9x) × (-4x)
= 36x2