Both claims are false. In fact, [tex] a [/tex] and [tex] \frac{1}{a} [/tex] are one the multiplicative inverse of the other. This means, by definition of multiplicative inverse, that
[tex] a \cdot \dfrac{1}{a} = 1 \quad\forall a \in \mathbb{R} [/tex]
So, it doesn't matter if [tex] a [/tex] is positive or negative: the multiplication of one number and its inverse will always be 1: for example,
[tex] (-2) \cdot \dfrac{1}{-2} = \dfrac{-2}{-2} = 1 [/tex]
Similarly, when you multiply two number, the sign of the product depends on the sign of the factors as follows:
- [tex] (+) \cdot (+) = (+) [/tex]
- [tex] (+) \cdot (-) = (-) [/tex]
- [tex] (-) \cdot (+) = (-) [/tex]
- [tex] (-) \cdot (-) = (+) [/tex]
So, the multiplication of two negative numbers is a positive number.
