Basic Question about 2x = 0

• July 16th 2011, 01:49 AM
Alexander4444
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.

----------------------------------------------

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!
• July 16th 2011, 01:57 AM
FernandoRevilla
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 $x=0$ so, the step $x/x = ((-1)x)/x$ is not valid because $1/x$ does not exist.
• July 16th 2011, 04:11 AM
Learner248
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
• July 16th 2011, 06:54 AM
Alexander4444
Re: Basic Question about 2x = 0
Thanks for the clarification. That was very insightful!
• July 16th 2011, 05:52 PM
HallsofIvy
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.
• July 16th 2011, 10:59 PM
Alexander4444
Re: Basic Question about 2x = 0
That makes great sense.
• July 17th 2011, 07:15 PM
Learner248
Re: Basic Question about 2x = 0
in mod2 algebra truly -1=1
x may or maynot be 0
• July 17th 2011, 10:41 PM
FernandoRevilla
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 $x\in\mathbb{R}$.
• July 17th 2011, 11:48 PM
Alexander4444
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.