Question

If a is a negative number, then a(1/a) is equal to -1. If a is a negative number and b is a negative number, then product ab is equal to a negative number

Answer 1

Both claims are false. In fact, $a$ and $\frac{1}{a}$ are one the multiplicative inverse of the other. This means, by definition of multiplicative inverse, that

$a \cdot \dfrac{1}{a} = 1 \quad\forall a \in \mathbb{R}$

So, it doesn't matter if $a$ is positive or negative: the multiplication of one number and its inverse will always be 1: for example,

$(-2) \cdot \dfrac{1}{-2} = \dfrac{-2}{-2} = 1$

Similarly, when you multiply two number, the sign of the product depends on the sign of the factors as follows:

• $(+) \cdot (+) = (+)$
• $(+) \cdot (-) = (-)$
• $(-) \cdot (+) = (-)$
• $(-) \cdot (-) = (+)$

So, the multiplication of two negative numbers is a positive number.