Question #91ebb

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Answer 1 · 2016-09-28T07:25:32

Maths

Negative numbers multiplied by negative numbers become positive.

Answer 2 · 2016-10-13T17:41:07

This is a particular case of a more general property:

$(-a) * b = -ab$

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 $-ab$ is the number that added to $ab$ gives $0$ .

a). Now consider $(-a)*b$ , and add:

$(-a) * b + ab = ((-a) + a) * b$ , by the distributive property.

But $(-a)+ a=0$ by definition, so we have $0*b$ , and then $0*b=0$ .

So we see that $(-a) * b + ab = 0$ , and then $(-a) * b =- ab$

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

Now, to the question itself:

$(-1) * (-1) = - (1*(-1))$ because of the proof in a). And again

$(-1) * (-1) =- (1*(-1))=-(-(1*1))$ . Because of the proof in b), we now have

$(-1) * (-1) =-(-1) = 1$

QED

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