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I was given the assignment to find a way to explain why the product of two negative numbers is positive. If a middle school or high school student were to ask why is it true, I must show some logic explanation as to how we can get a positive number.

$$(-a)\times(-b)=(a)\times(b)$$

I wanted to talk about a video of a guy running backwards but the video itself playing backwards. When I hit play, it actually shows the guy running forward. But I can not find a logic way to explain it.

Any ideas or a way to explain this to a middle school student? I have to convince the student that this is a fact.

Cesar Agama
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  • The backward of backward is forward. – giobrach Sep 25 '16 at 05:55
  • An alternative approach that anticipates some concepts of complex analysis would be to visualize multiplication by $-1$ as a rotation of the number line by 180 degrees around the origin. If you do that twice, you have rotated 360 degrees and you're back where you started. – giobrach Sep 25 '16 at 05:59
  • You might find this video from Mathologer interesting : https://youtu.be/ij-EK-MZv2Q ultimately, it comes down to assigning a meaning to negative times negative, and the only one that is consistent with the usual properties of multiplication (especially distributivity) is to give it a positive value. – Joel Cohen Sep 25 '16 at 07:42
  • I always viewed it as a negative flipping things backwards. – fleablood Sep 26 '16 at 02:31

2 Answers2

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Proof of $\quad -(-x)=x$

$\forall x\; \exists y\quad x+y=y+x=0$ (From Z.F.C.)

Then we can say, $\quad x=-y\quad$ and likewise $\quad y=-x$

$-x=y\quad $ then ,$-(-x)=-(y)=-y\quad$ and we know $-y=x$

$-(-x)=-(y)=-y=x\quad \Box\quad$ (Note:This $y$ is unique.)

$\quad$

$\forall a$, $-a=(-1)a$,

Because ,$\quad 0=0a=(1+(-1))a=1a+(-1)a=a+(-1)a \quad \Box \quad$(this $(-1)a$ is unique.)

Then if we write $a=-1$;

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

Then;

$(-a)(-b)=(-1)a(-1)b=(-1)(-1)ab=ab\quad\Box$

Micheal Brain Hurts
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Strategy: rewrite $ab$ several times using a few laws of arithmetic (each in two variants, equivalent because $xy=yx$):

(1a) $x=x+0$, (1b) $0+x=x$; (2a) $0=x0$, (2b) $0x=0$; (3a) $x\left( y+z\right)=xy+xz$, (3b) $\left( x+y\right)z=xz+yz$.

We'll also repeatedly recall $u-v$ is an abbreviation for $u+\left( -v\right)$.

Now for the rewrite: $$ab=ab+\left( -a\right)\left( b-b\right)=\left(a -a\right)b + \left( -a\right)\left( -b\right)=\left( -a\right)\left( -b\right).$$The first $=$ sign uses (1) and (2a); similarly, the third $=$ sign uses (1b) and (2b). The second $=$ sign is more complicated: it expands a product so we have three terms, and factorises the leftmost two of them, viz. (3a) and (3b).

J.G.
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