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!
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
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.
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.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.