# Question #91ebb

Sep 28, 2016

Maths

#### Explanation:

Negative numbers multiplied by negative numbers become positive.

Oct 13, 2016

This is a particular case of a more general property:

$\left(- a\right) \cdot b = - a b$

#### Explanation:

The definition of the opposite $- x$ of a number $x$ is the number that added to $x$ gives $0$. This is whatever the number $x$ is.

In particular $- a b$ is the number that added to $a b$ gives $0$.

a). Now consider $\left(- a\right) \cdot b$, and add:

$\left(- a\right) \cdot b + a b = \left(\left(- a\right) + a\right) \cdot b$, by the distributive property.

But $\left(- a\right) + a = 0$ by definition, so we have $0 \cdot b$, and then $0 \cdot b = 0$.

So we see that $\left(- a\right) \cdot b + a b = 0$, and then $\left(- a\right) \cdot b = - a b$

b). Also, since $1 + \left(- 1\right) = 0$ by definition, this tells us that $1$ is the opposite of $- 1$, that is $1 = - \left(- 1\right)$

Now, to the question itself:

$\left(- 1\right) \cdot \left(- 1\right) = - \left(1 \cdot \left(- 1\right)\right)$ because of the proof in a). And again

$\left(- 1\right) \cdot \left(- 1\right) = - \left(1 \cdot \left(- 1\right)\right) = - \left(- \left(1 \cdot 1\right)\right)$. Because of the proof in b), we now have

$\left(- 1\right) \cdot \left(- 1\right) = - \left(- 1\right) = 1$

QED

Remark: I have assumed that we know $0 \cdot b = 0$. This can also be proven