Prove X-x = 0

i have been wondering about something and i can think of a way to prove myself wrong so... here it is:

if X= the square root of 1

the square root of 1 = 1 or -1

the square root of 1 = the square root of 1 these are true yes?

therefore 1 = -1

X-X = 2 because 1--1=2
X-X = -2 because -1-1=2
X-X = 0 because 1-1=0

DaveC426913
Gold Member

i have been wondering about something and i can think of a way to prove myself wrong so... here it is:

if X= the square root of 1

the square root of 1 = 1 or -1

the square root of 1 = the square root of 1 these are true yes?

therefore 1 = -1
You cannot conclude that 1 = -1 from your initial arguments. It does not follow.

arildno
Homework Helper
Gold Member
Dearly Missed

Suppose you have the equation:
$$x^{2}=1$$

This can be rewritten as:
$$(x-1)*(x+1)=0$$
and x can then either be 1 or -1.

1=-1

X=X
the square root of 1 = the square root of 1
the square root of 1 = 1
the square root of 1 = -1
therefore x can be 1 or -1
therefore x-x=2 or -2 or 0 im trying to find an explanation as to why its wrong.. i know its wrong but i dont know why... it seems logical and illogical at the same time

DaveC426913
Gold Member

X can be 1 OR -1. Not BOTH at the same time - or not one then the other within the same equation.

Walk it through.

root(x) can be 1 or -1.
Now pick one.
Now put it into x-x=0. Paradox goes away.

JasonRox
Homework Helper
Gold Member

Suppose you have the equation:
$$x^{2}=1$$

This can be rewritten as:
$$(x-1)*(x+1)=0$$
and x can then either be 1 or -1.

1=-1
Above!

your not explaining your telling, sorry for not taking your word as gospel but feckin prove it

DaveC426913
Gold Member

your not explaining your telling, sorry for not taking your word as gospel but feckin prove it
Are you seriously copping a 'tude?

You're the one who doesn't understand. It has been proven.
Ths onus is on you to lay down math that is valid. It is not valid.
The fact that you don't follow it does not give you cause to be rude.

I'll try again.

X can be 1 or -1.
That's an OR; it is not an AND. X cannot be 1 AND -1.

When you write your equation x-x=0
and then substitute for x, you write (1)-(-1)=0. You cannot do this.
X has one value.

i just want an explanation ... it doesnt make sense and im sorry ive been thinking about it for ages and you just say no with no evidence

why cant they be used independently?

surely 1 equation cant give 2 answers
X must have 2 values in this case?

Your equation is x2=1. x=1 and x=-1 are the two possible real numbers that satisfy this equation. This means that there are two possible cases, one or the other is true.

Case 1: x=1; does this satisfy the equation? Yes, 12=1. Then x-x=1-1=0.

Case 2: x=-1; does this satisfy the equation? Yes, (-1)2=1. Then x-x=(-1)-(-1)=0.

The two cases do not overlap, one is true or the other one is true, if you consider the original equation with x being 1 sometimes and -1 at other times, perhaps (x)(x)=(1)(-1)=-1 and this doesn't satisfy the equation.

surely 1 equation cant give 2 answers
X must have 2 values in this case?
Of course an equation can give two answers, an equation can give as many answers as there are numbers.

hmm im still not convinced

Then you need to meditate on it, because I don't think any more explanation will help.

when you say

Of course an equation can give two answers, an equation can give as many answers as there are numbers.

what do you mean? 1+1 = 2 and only 2?
2+2=2 and only 2

the square root of one equals 1 and -1

X = the square root of 1 theres your overlap

I mean exactly what I say. Your equation is a perfect example. x2=1, x=1 satisfies it and x=-1 satisfies it, there are two answers. A cubic equation might have a third solution, a quartic equation might have 4 etc. Consider sinx=0, there are an infinite number of values of x that satisfy it, any multiple of Pi.

if theres an infinite number of values then doesnt that basically mean its meaningless? actually dont explain that i learn too slow... look im sorry i just dont understand why there is no overlap but its fine i can live with it... cheers tho anyway

It is not meaningless, its meaning is that x=nPi where n is any integer.

haha i get it... ok thanks very much... sorry for being a bit dense ha cheers

ahh i ve lost it again

haha joke...

x=nPi that has a variable though ... x= tsro1 doesnt

tsro=the square root of btw

i have been wondering about something and i can think of a way to prove myself wrong so... here it is:

if X= the square root of 1

the square root of 1 = 1 or -1

the square root of 1 = the square root of 1 these are true yes?

therefore 1 = -1

X-X = 2 because 1--1=2
X-X = -2 because -1-1=2
X-X = 0 because 1-1=0
Square root of a number is positive as defined by the principle root. (Square root of 1) =/= -1...However, functionally, X2 = 1 could have X = 1 or X = -1...so it is stated that X = (+ or -)*square root of 1.

One way that might convince you is in the bold statement. Saying the square root of 1 = (1 or -1) is a logical condition. In the case of and, you have both conditions satisfied when at least one is true...but in the case of or, you only have one condition satisfied (the one that is true). Thus, if X = -1, then you cannot say that X = 1.

You can also refer to this statement:
...exponentiation is not a function; it's what is sometimes called a "multi-valued function": to each input value, there can be more than one output values.

Last edited:
DaveC426913
Gold Member

tsro=the square root of btw
Just use 'root'. It has the same number of letters and has the advantage of not having to be explained.

The error you're making is thinking that the square root of 1 is 1 and -1.
This is not true.
The square root of 1 is 1 or -1.

X is 1 or -1. You must pick one before using it in an equation. You keep waffling on which one you want it to be, using a different one at different points in the same equation.