Re: Another Proof that 1 = 2

Nice example of what happens if the variable gets partly treated as a constant.

Re: Another Proof that 1 = 2

Quote:

Originally Posted by

**mathbyte** I'm surprised no one brought this one up, since it's been quite a number of years since I first heard of it.

Here is a 1=2 proof using differentiation:

Start with:

x^2 = x^2

x^2 is x*x, so we can represent one side by addition:

x + x + x + ... + x = x^2

Theres got to be something fishy about that representation

x + x + x + ... + x = x^2

if x = -1, how many x do you have on the left of your equation? How about if x = 1/3 ?

Re: Another Proof that 1 = 2

Quote:

Originally Posted by

**agentmulder** Theres got to be something fishy about that representation

x + x + x + ... + x = x^2

if x = -1, how many x do you have on the left of your equation? How about if x = 1/3 ?

Even if "x" is natural,

the number of x's summed is itself a function of x

and must be taken into account when calculating the derivative (product rule).

Re: Another Proof that 1 = 2

Quote:

Originally Posted by

**Archie Meade** Even if "x" is natural,

the number of x's summed is itself a function of x

and must be taken into account when calculating the derivative (product rule).

I agree. To me it looks fishy even before the derivative is taken, x^2 = x^2 holds for any x, not so sure about the other representation.

Yes, x*x = x^2 doesn't pose any problems i can think of

Re: Another Proof that 1 = 2

let's look at the equation:

$\displaystyle \sum_{i=1}^kx = x^2$

when we solve for k, we get k = x.

now, let's differentiate that same equation:

$\displaystyle \sum_{i=1}^k 1 = 2x$.

when we solve for k, we get k = x/2.

evidently, x = x/2, so 2x = x, so x = 0.

therefore, the flaw in the proof is the very last step...we have divided by 0.

Re: Another Proof that 1 = 2

Quote:

Originally Posted by

**Archie Meade** Even if "x" is natural,

the number of x's summed is itself a function of x

and must be taken into account when calculating the derivative (product rule).

More simply: If x is natural x·x(x^2) is not a continuous function therefore not differentiable.(Nod)

If x is not a natural number then x^2 = x+x+...+x is not a valid equality.