math teaser :)

Maybe some with find out what is happening, and maybe others will not so easily.  :)

Lets say that somebody gave you the equation X = X + 1 and asked you if there is a solution for X in the real numbers or is this a valid equation.  And you tell that person that you need to work it out because it is not obvious (it is obvious to most people but pretent that you don't think it is so obvious).  First, you make the assumption that there is a solution for X and you check to see if you run into any flaws.  You think of a couple of different methods of trying to solve for X.

Method 1:

X = X + 1

0 = 1             (subtract X from both sides)

We see that X = X + 1 implies that 0 = 1 by using this method.  We know that 0 cannot equal 1 so this must mean that the original equation is not valid.  Or you can say that there is no solution for X.  Lets try to solve this problem in a different way.  :)

Method 2:

X = X + 1

X^2 = (X + 1)^2                 (square both sides)

X^2 = X^2 + 2*X + 1           (expand the right side)

0 = 2*X + 1                       (subtract X^2 from both sides)

X = -1/2                           (isolate X)

Do you know why we found X = -1/2 in this method?  The first method shows no solution, but why did we find a solution for X in this second method?  What logic are we missing?  The two methods don't seem to come to the same conclusion.  I know why, but I wounder what you think.  Maybe this can be a good problem to think about for young math students in high school.  :)

I'm 14 so I think I count

the third row you cannot expand on one side

sorry if it doesn't work right...

No problem shadowslayer.  I know the answer to my question.  I just ask it to chess.com members for fun.  :)

It's too hard for me.

no, I have a different theme and I don't know if I hid it well enough...

I can post the answer to the question.  I might wait for a few more responses though.  :)

Err, difficult for me in English, but I think it's because the squared equation is not equivalent to the initial equation :

X=X+1 => X^2 = (X+1)^2 but the reverse is just not true (A^2 = B^2 doesn't imply A=B but rather abs(A) = abs(B)) 

And guess what ?! abs(-1/2) = abs(-1/2+1)

Is't it wonderful ?

What happes if the left side does become 1, what diffrance will it be?

P.S. Hey I figured it out already people, and posted it! Not too hard to figure out how it's wrong now, huh?

hicetnunc and dethwing got the answer that I was looking for.  When a = b then a^2 = b^2 but the converse is not true.  :)

paul211, I don't know.  I think it is just you that have the misunderstanding.  Maybe it has been awhile since you have been in high school.  :)  I only meant for this to be a high school level question and I am just talking about the group of real numbers with all of the familiar operations (+, -, *, /, squaring, square root...) that are familiar to a high school student.  :)

In the real number set your equation is not valid, as substracting "x" from both sides of the equation is not valid, by the way something one learn in high school not at the university.

You are right.  The equation is not valid.  However, what if I asked you to prove it?  Some equations that are not valid may not be so obvious as this one so some work would be needed to show that it is not valid.  Actually, you can operate on an invalid equation just like as if it was valid.  :)  What you are doing is proving by contradiction (assuming the equation is valid and then showing it can't work).  Mathmaticians do prove some things by contradictions.  I'm sure you would know that.  :)

Prove that X does not equal X + 1.  The contradicting statement is:  X = X + 1

X = X + 1

0 = 1           (subtract X from both sides)

Since we know 0 does not equal 1 in the real numbers then we have proved that X does not equal X + 1.  :)

So if the original equation is by removing "x" from left and right of the equation equal to 0=1, then any subsequent operation is also not valid, what are you missing here???

I'm not missing anything.  lol  You are missing the whole point of this.  Maybe I didn't make it clear enough in my orginal post but the equation X = X + 1 was operated on with two different methods.  The most obvious method would be to subtract X from both sides and you get 0 = 1.  I didn't do any subsequent operations from there so I don't know what you are trying to get at.  After 0 = 1 it shows that X does not equal X + 1.  Ok now forget all that and lets look at a different method on trying to show X does not equal X + 1.  lol  :)  Some things in mathematics can be proven in more than one way.

second method:

X = X + 1

X^2 = (X + 1)^2                 (square both sides)

X^2 = X^2 + 2*X + 1           (expand the right side)

0 = 2*X + 1                       (subtract X^2 from both sides)

X = -1/2                           (isolate X)

The point of this whole thing was to think about why we get X = -1/2 from this second method.  When we square both sides we created an extra solution.  And a couple of people have pointed that out already so maybe my post wasn't too hard for some people.  There was nothing wrong with the steps.  We just need to check our solution with our original equation.  :)  Plug in X = -1/2 into X = X + 1 and you see that it doesn't work out.  This means that squaring both sides created this extra solution and that there is no solution to the original equation.

Do you understand all this?  :)

And another thing is that you stated that I was completely wrong but then later you stated that I didn't define the system.  First off, how am I wrong when it is me asking the question?  :)  Maybe you were talking about the mathematical steps I was providing.  Also, how do you know that I am wrong (assuming you were talking about the steps) if I didn't state the system?  Maybe you checked with every system known.  But then you said I can invent my own system.  So therefore you don't know if I am wrong or not because you don't know the system.  :)

Are you talking about the first part where it shows 0 = 1?  Maybe I didn't explain the question very well.  I will try to reword the question better.  :)

I learnt what "converse" meant

So x = -½. but x=x+1, so x= ½. But x= x+1, so x= 1½.............. but x = x+1, so x= 9999½. This is fun.

x=-1/2 is not a solution to the original equation.  Just plug it in, it doesn't work.

This has something to do perhaps with extraneous stuff when you square or get the square root. 

On the other hand, a false proposition usually yields a false conclusion.

it's too hard for me

x+1 is not equal to x.

i'm not really good in english too.. but i try to give my opinion...

x = x + 1

x - x = 1

0 = 1

what about....

Me = Me + You

Me - Me = You

You = zero...

so.. the equation is true if you are vanish..

0 = 1 ... it means the equation become true if 1 (one) is nothing... 

so the equation become  x = x + 0 .... that is the true equation..

So the answer is that when you square both sides, you are actually changing the initial relationship of both sides of the equation and thus get a different answer between the two methods? I say "change the relationship" instead of "make the equation invalid" since it was invalid to begin with

When you square both sides of an equation you add an answer that is not valid with the original equation. You have to check the answers you get with the original equation to see which is correct.