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Why negative times negative = positive?

An Abstract Algebra text book has a sentence on its 1st chapter about natural numbers that i cannot get around easily. The sentence reads "Why minus multiplied by minus needs to be plus is something you might reflect on now." I cannot answer the question, why minus multiplied by minus needs to be plus? thanks

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marked as duplicate by J. M., Arturo Magidin, Qiaochu Yuan Apr 27 '12 at 16:40

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Which "Abstract Algebra" book? (It is not an idle question: I'm unsure at what level to answer this question... numbers, basic algebra, abelian groups...) –  Arturo Magidin Apr 27 '12 at 15:56
    
the book is: A first course in abtract algebra, rings, groups, and fields. –  marcel Apr 27 '12 at 15:57

2 Answers 2

In a ring, $-a$ is the unique element that, when added to $a$, is equal to $0$. And $a$ is the unique element that, when added to $-a$, is equal to $0$, because additive inverses are unique.

In order to show that $-(-a)$ must equal $a$, we need to show that when we add it to $-a$, we get $0$; because then both $a$ and $-(-a)$ have the property "when you add me to $-a$ you get zero", and there's supposed to be only one element with that property. Indeed, by the very definition of the symbol $-$, we have that $-(-a) + (-a) = 0$; so $-(-a)=a$. This is an instance of "minus multiplied by minus is plus".

In order to show that $(-a)(-b)$ must equal $ab$, we need to show that $(-a)(-b)$ has the property that, when added to $-(ab)$, we get $0$; for the same reason as above. First, note that $(-a)x = -(ax)$ for any $x$: $$(-a)x + ax = \Bigl((-a)+a\Bigr)x = 0x = 0,$$ so $(-a)x$ and $-(ax)$ both have the property "when you add me to $ax$ you get $0$", so they must be equal. Similarly, $x(-b) = -(xb)$ for any $x$.

And now we have: $$\begin{align*} (-a)(-b) &= -\Bigl(a(-b)\Bigr) &\text{(since }(-a)x=-(ax)\text{ for any }x)\\ &= -\Bigl( -(ab)\Bigr) &\text{(since }x(-b)=-(xb)\text{ for any }x)\\ &= -(-(ab))\\ &= ab &\text{(since }-(-y) = y\text{ for any }y). \end{align*}$$

So if we want the basic properties of rings to hold, we need $(-a)(-b)$ to be the same thing as $ab$.

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Thank you Arturo! –  marcel Apr 27 '12 at 16:11
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Arturo's answers are certainly not unhelpful. :) –  J. M. Apr 27 '12 at 16:24

Hint $\rm\ xy,\ (-x)(-y)\ $ are both inverses of $\rm\ x(-y)\ $ so they're equal by uniqueness of inverses.

As I often emphasize, uniqueness theorems provide powerful tools for proving equalities.

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