Can X=2 and X=-2 at the same time?

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Discussion Overview

The discussion revolves around the mathematical implications of the equation X = the square root of 1, specifically addressing whether X can simultaneously equal both 1 and -1. Participants explore the logical and mathematical reasoning behind the square root operation and the nature of equations that yield multiple solutions.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • Some participants argue that if X = the square root of 1, then X can be either 1 or -1, leading to the claim that 1 = -1, which is challenged by others.
  • Others clarify that the square root of 1 is defined as the principal root, which is 1, and that the equation X^2 = 1 has two distinct solutions (1 and -1) but does not imply that X can take both values simultaneously in the same context.
  • Some participants express confusion over the concept of overlapping solutions and the implications of using X in equations without choosing a single value.
  • There are discussions about the nature of equations and whether they can yield multiple answers, with some asserting that an equation can have multiple solutions while others emphasize the need to select one value for X when substituting in equations.
  • Participants also discuss the distinction between using the square root function and the broader implications of equations that can have multiple solutions, such as cubic or quartic equations.

Areas of Agreement / Disagreement

Participants generally disagree on the interpretation of the square root and the implications of having multiple solutions. While some acknowledge that X can be either 1 or -1, they maintain that X cannot be both at the same time in a single equation. The discussion remains unresolved regarding the understanding of these concepts.

Contextual Notes

There are limitations in the discussion regarding the definitions of square roots and the assumptions about the nature of equations. The participants do not reach a consensus on the interpretation of overlapping solutions and the implications of choosing values for X.

Who May Find This Useful

This discussion may be of interest to those studying algebra, particularly the properties of square roots and the nature of equations with multiple solutions.

chalky00
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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
 
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chalky00 said:
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.
 


Suppose you have the equation:
[tex]x^{2}=1[/tex]

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

It does not follow that
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 I am trying to find an explanation as to why its wrong.. i know its wrong but i don't know why... it seems logical and illogical at the same time
 


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.
 


arildno said:
Suppose you have the equation:
[tex]x^{2}=1[/tex]

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

It does not follow that
1=-1

Above!
 


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


chalky00 said:
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 doesn't make sense and I am sorry I've been thinking about it for ages and you just say no with no evidence
 
  • #10


why can't they be used independently?
 
  • #11


surely 1 equation can't give 2 answers
X must have 2 values in this case?
 
  • #12


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 can't 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.
 
  • #13


hmm I am still not convinced
 
  • #14


Then you need to meditate on it, because I don't think any more explanation will help.
 
  • #15


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
 
  • #16


X = the square root of 1 there's your overlap
 
  • #17


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.
 
  • #18


if there's an infinite number of values then doesn't that basically mean its meaningless? actually don't explain that i learn too slow... look I am sorry i just don't understand why there is no overlap but its fine i can live with it... cheers tho anyway
 
  • #19


It is not meaningless, its meaning is that x=nPi where n is any integer.
 
  • #20


haha i get it... ok thanks very much... sorry for being a bit dense ha cheers
 
  • #21


ahh i ve lost it again
 
  • #22


haha joke...

x=nPi that has a variable though ... x= tsro1 doesnt
 
  • #23


tsro=the square root of btw
 
  • #24


chalky00 said:
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:
Hurkyl said:
...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:
  • #25


chalky00 said:
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.
 
  • #26


i see
 
  • #27


chalky00 said:
x=nPi that has a variable though ... x= tsro1 doesnt

If the presence/absence of a variable is throwing you off, that's an easy matter to fix. If [itex]x[/itex] is a number such that [itex]x^2 = 1[/itex] then [itex]x = n(1)[/itex] where [itex]n=1[/itex] or [itex]n=-1[/itex]. This is exactly what everyone else has been saying!
 
  • #28


DaveC426913 said:
Just use 'root'. It has the same number of letters and has the advantage of not having to be explained.

haha true
 
  • #29


jgens said:
If the presence/absence of a variable is throwing you off, that's an easy matter to fix. If [itex]x[/itex] is a number such that [itex]x^2 = 1[/itex] then [itex]x = n(1)[/itex] where [itex]n=1[/itex] or [itex]n=-1[/itex]. This is exactly what everyone else has been saying!

ok!
 
  • #30


DaveC426913 said:
Just use 'root'.

Or "sqrt" . . .
 

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