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I was browsing through the non-mandatory tasks on my college's website and I stumbled upon one which cracks my head quite a bit. It goes as follows:

Prove why multiplying two numbers with different signs gives us a negative number while multiplying two of the same sign - positive.

Thing is... I don't even know where to start. How is one supposed to prove thing like that? I mean - it seems so natural, uncontested and taken as granted from one's early years of life that I can't even figure how one could prove it. Is there some proof of such thing I could read or the question itself is somehow tricky?

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    That's all that was in the task description. Keeping in mind we didn't cover a lot o$f$ material, though, I'd say they refer to real numbers.2012-11-07

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Use the distributive property: $-a\cdot b = (0-a)\cdot b = 0\cdot b - a\cdot b.$

(Here it is assumed $a > 0$ and $b > 0$, although it is not a necessary condition -- it just allows us to consider one case without loss of generality).

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    That seems what I was asking. Thank you a lot!2012-11-07
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There's the algebraic point of view, but there's also the following. If I'm not mistaken, negative numbers were introduced in Italy in the middle ages to represent debts. If you have $\$30$ and you owe $\$20$; your net worth is $\$10$: $\$30+(-\$20)=\$10$. If you have $\$30$ and you owe $\$50$; your net worth is $\$30+(-\$50)=-\$20.

So you gain 5$ debts of $\$7$ each; this changes your net worth by $5\cdot(-\$7)=-\$35$.

Then suppose $5$ of your debts of $\$7$ each are forgiven. To your total number of debts, $-5$ is added; your net worth changes by $-5\cdot(-\$7)=+\$35$.

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To complement the other answers, here is a proof that $(-1)(-1) = 1$.

We know $(-1) \cdot 1 = -1$, since this is the definition of $1$ (its use in multiplication does not change the value of the other number).

We know also that $(-1) \cdot 0 = 0$.

Now, $ \begin{align*} 0 &= (-1) \cdot 0\\ &= (-1) (1 + -1)\\ &= (-1)(1) + (-1)(-1)\\ &= (-1) + (-1)(-1). \end{align*} $ Adding $1$ to both sides gives $ 1 + 0 = 1 + (-1) + (-1)(-1), $ which simplifies to $ 1 = (-1)(-1). $

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    Oh my, didn't notice it. Great one, then. Thank you a lot :)2012-11-07
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Use the fact that $(-1)(-1)=1$, $(1)(1)=1$, commutativity and assocativity to write $(-a)(-b)=(-1)a(-1)b=(-1)(-1)ab$ $(a,b>0)$.