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[Algebra] Product of Two Negative Numbers

2021-07-16 13:22 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

Prove why the product of two negative real numbers is a positive real number.

【Solution】

Let $a%2Cb$ be two positive real numbers; subsequently, $-a$ and $-b$ are their respective additive inverses.

A clever way to prove that $(-a)(-b)%3Dab$ is to begin by considering the equation

$x%3Dab%2B(-a)(b)%2B(-a)(-b)$

and then use this equation to show that $x%3Dab%20$ and $x%3D(-a)(-b)$ .

First, factor out $-a$ from the expression $(-a)(b)%2B(-a)(-b)$ :

$x%3Dab%2B(-a)(b)%2B(-a)(-b)$

$x%3Dab%2B(-a)%5Bb%2B(-b)%5D$

Since $b%2B(-b)%3D0$ ,

$x%20%3D%20ab%20%2B%20(-a)(0)$

Thus,

$x%3Dab$

Now, with the original equation, factor out $b$ from the expression $ab%2B(-a)(b)$ :

$x%3Dab%2B(-a)(b)%2B(-a)(-b)$

$x%3Db%5Ba%2B(-a)%5D%2B(-a)(-b)$

$x%3Db(0)%2B(-a)(-b)$

Thus,

$x%3D(-a)(-b)$

Since $x%3Dab%20$ and $x%3D(-a)(-b)$ , we discover that

$(-a)(-b)%20%3D%20ab$

Therefore, the product of two negative numbers is positive.

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