Basic Question about 2x = 0

If I have 2x = 0, can someone explain what's going on or what rule doesn't allow for these two cases to exist...or can they and why...

Case 1:

2x = 0

(2x)/2 = 0/2

x = 0

O.K.

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Case 2:

2x = 0

x + x = 0

x = -x

x = (-1)x

x/x = ((-1)x)/x

1 = -1

???

...so if anybody could clear this up, it would be great! It's funny that after all of these years of doing math that I could stumble on something so basic that REALLY confuses me:) Thanks!

Re: Basic Question about 2x = 0

Quote:

Originally Posted by

**Alexander4444** x = (-1)x

x/x = ((-1)x)/x

1 = -1 ???

You correctly proved that $\displaystyle x=0$ so, the step $\displaystyle x/x = ((-1)x)/x$ is not valid because $\displaystyle 1/x$ does not exist.

Re: Basic Question about 2x = 0

Case 1 :

You can do so only if the field in which you are has characteristic not equal to 2(Group Theory).

For example if:

x^2=x in a field say F s.t xbelongs to F

then, -x=(-x)^2=x^2 s.t xbelongs to F

i.e. x=-x s.t xbelongs to F

or 2x=0 s.t xbelongs to F

This is a boolean Field. Here u cannot divide by a 2 which is equivalent to dividing by 0 in our number system.

Case 2:

2x = 0

x + x = 0

x = -x

x = (-1)x

x/x = ((-1)x)/x

1 = -1

???

For this one would say u cannot divide by a number whose multiplicative inverse is not defined eg the '0' here

Also 1=-1 in a boolean field

Re: Basic Question about 2x = 0

Thanks for the clarification. That was very insightful!

Re: Basic Question about 2x = 0

If you had **not** known in advance that a=0 and did the problem by saying that 2x= 0 gives x+ x= 0, x= -x, then the next step would be to say "**If x is not 0** then -x/x= x/x, so -1= 1". Since that is obviously not true, you conclude that x= 0.

Re: Basic Question about 2x = 0

Re: Basic Question about 2x = 0

in mod2 algebra truly -1=1

x may or maynot be 0

Re: Basic Question about 2x = 0

Quote:

Originally Posted by

**Learner248** in mod2 algebra truly -1=1 x may or maynot be 0

That is right, but this thread is included in Pre-Algebra forum so I suppose the OP meant $\displaystyle x\in\mathbb{R}$.

Re: Basic Question about 2x = 0

Quote:

That is right, but this thread is included in Pre-Algebra forum so I suppose the OP meant x is in the set of the real numbers.

Yes, I did. But still, bring on the advanced:) Much better to be thorough than to conform to the contextual limits of this portion of the forum. Unfortunately, I didn't know that this COULD go outside the scope of pre-algebra/algebra. Now I do.