Confused by x=--x: Exploring the Mystery

[sinas]

x=--x

wtf =/

(this isn't a homework problem but it was brought up today and I'm curious)

 

[NateTG]

x=--x

wtf =/

(this isn't a homework problem but it was brought up today and I'm curious)

Well, what do you start with?
If you start with
$$-1 \times -1 = 1$$
then
$$--x = -1(-1(x))=(-1 \times -1) x= 1 x =x$$
The first equality is by definition, the second because multiplication is associative, the third because you know [itex]-1 \times -1 =1 [/tex] and the last because 1 is the multiplicative identity.

To see that $$-1 \times -1 =1$$:
$$\frac{-1}{-1}=1=\frac{1}{1}$$
so
$$\frac{-1}{1}=\frac{1}{-1}$$
but
$$1=\frac{-1}{-1}=-1 \times \frac{1}{-1}=-1 \times -1$$

 

[arildno]

It's a bit easier than that.
By definition of zero, x+0=x
By definition of the additive inverse,
x+(-x)=0

--x, that is, (-(-x))
fulfills therefore:
(-x)+(-(-x)=0
Add x on both sides:
x+(-x)+(-(-x))=x, or, since x+(-x)=0, we get:
(-(-x))=x

That is the additive inverse to the additive inverse of x is x itself.