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Math Help - Proof that 1 = 2

  1. #16
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    Quote Originally Posted by TheAbstractionist View Post
    Here is one.

    \cos^2x-\sin^2x\ =\ \cos2x (identity)

    \cos^2x\ =\ \cos2x+\sin^2x (rearramging)

    \cos x\ =\ \sqrt{\cos2x+\sin^2x} (taking square root)

    \cos\frac{3\pi}4\ =\ \sqrt{\cos\frac{3\pi}2+\sin^2\frac{3\pi}4} (substituing x=\frac{3\pi}4)

    -\frac1{\sqrt2}\ =\ \sqrt{0+\frac12}=\frac1{\sqrt2} (evaluating the substitution)

    -\frac12\ =\ \frac12 (multiplying both sides by \frac1{\sqrt2})

    1\ =\ 2 (adding \frac32 to both sides)
    Quite late , but this proof is incorrect simply due to the fact that in the second quadrant (\frac{\pi}{2} \leq \theta \leq \pi), the result is negative (meaning cosx = -\sqrt{cos2x+sin^2x} ), but you used the positive result.
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  2. #17
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    Quote Originally Posted by paulrb View Post
    Are there any 1=2 "proofs" that do not involve dividing by zero? I am trying to think of other mathematical rules that can be broken to derive such a proof.
    Irrational number - Wikipedia, the free encyclopedia

    I saw this the other day. Look under the "History" section. Hippasus' proof that an odd number is even. It's not exactly the same as a 1=2 "proof," but is at the very least relevant to what you are touching upon here.

    It's worth mentioning that--if I've understood correctly--a major difference is that Hippasus's proof is valid. No fallacies in logic can be pointed out as in the examples here. I therefore speculate that it is conceivable that a valid 1=2 proof can exist, and that it would merely prove, as Hippasus' proof does, that irrational numbers exist.

    My speculation is probably totally hare-brained though, as I am not a real mathematician.
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  3. #18
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    Quote Originally Posted by rainer View Post
    Irrational number - Wikipedia, the free encyclopedia

    I saw this the other day. Look under the "History" section. Hippasus' proof that an odd number is even. It's not exactly the same as a 1=2 "proof," but is at the very least relevant to what you are touching upon here.

    It's worth mentioning that--if I've understood correctly--a major difference is that Hippasus's proof is valid. No fallacies in logic can be pointed out as in the examples here. I therefore speculate that it is conceivable that a valid 1=2 proof can exist, and that it would merely prove, as Hippasus' proof does, that irrational numbers exist.

    My speculation is probably totally hare-brained though, as I am not a real mathematician.
    Actually, in his proof he assumes (in contradiction), that there are no irrational numbers. He then reaches a cotradiction - that the number b, which he specified, must be both even and odd, however no such number exists. therefore, the claim that there are no irratioal numbers is false.
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  4. #19
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    Quote Originally Posted by TheAbstractionist View Post
    Here is one.
    \cos^2x-\sin^2x\ =\ \cos2x (identity)

    \cos^2x\ =\ \cos2x+\sin^2x (rearramging)

    \cos x\ =\ \sqrt{\cos2x+\sin^2x} (taking square root)

    \cos\frac{3\pi}4\ =\ \sqrt{\cos\frac{3\pi}2+\sin^2\frac{3\pi}4} (substituing x=\frac{3\pi}4)

    -\frac1{\sqrt2}\ =\ \sqrt{0+\frac12}=\frac1{\sqrt2} (evaluating the substitution)

    -\frac12\ =\ \frac12 (multiplying both sides by \frac1{\sqrt2})

    1\ =\ 2 (adding \frac32 to both sides)

    you forgot that square rootting something is '+' OR '-'.
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  5. #20
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    Exclamation Another Proof that 1 = 2

    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
    where there are x x's.

    Now, we differentiate both sides:
    1 + 1 + 1 + ... + 1 = 2x

    There are x 1's, so we can sum all the 1's to get
    x = 2x

    And dividing by x we get
    1=2
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  6. #21
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    Re: Another Proof that 1 = 2

    Nice example of what happens if the variable gets partly treated as a constant.
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  7. #22
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    Re: Another Proof that 1 = 2

    Quote Originally Posted by mathbyte View Post
    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 ?
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  8. #23
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    Re: Another Proof that 1 = 2

    Quote Originally Posted by agentmulder View Post
    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).
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  9. #24
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    Re: Another Proof that 1 = 2

    Quote Originally Posted by Archie Meade View Post
    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
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  10. #25
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    Re: Another Proof that 1 = 2

    let's look at the equation:

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

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

    now, let's differentiate that same equation:

    \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.
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  11. #26
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    Re: Proof that 1 = 2

    -1 = 1
    0 = 1 + 1
    0 = 2
    so it will zero not 1 =2 .
    word problem help
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  12. #27
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    Re: Proof that 1 = 2

    Go away with your commercials...
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  13. #28
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    Re: Proof that 1 = 2

    Quote Originally Posted by TheAbstractionist View Post
    Here is one.

    \cos^2x-\sin^2x\ =\ \cos2x (identity)

    \cos^2x\ =\ \cos2x+\sin^2x (rearramging)

    \cos x\ =\ \sqrt{\cos2x+\sin^2x} (taking square root)

    \cos\frac{3\pi}4\ =\ \sqrt{\cos\frac{3\pi}2+\sin^2\frac{3\pi}4} (substituing x=\frac{3\pi}4)

    -\frac1{\sqrt2}\ =\ \sqrt{0+\frac12}=\frac1{\sqrt2} (evaluating the substitution)

    -\frac12\ =\ \frac12 (multiplying both sides by \frac1{\sqrt2})

    1\ =\ 2 (adding \frac32 to both sides)
    you made a mistake here bro:
    \cos x\ =\ \sqrt{\cos2x+\sin^2x} (taking square root)
    there should absolute function over cosx. since sqrt(x^2)=abs(x). Its the most common mistake that students makes while learning Pre-calculus.But we soon realize the importance of absolute function when we study limits.
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  14. #29
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    Re: Proof that 1 = 2

    \cos^2x-\sin^2x\ =\ \cos2x (identity)

    \cos^2x\ =\ \cos2x+\sin^2x (rearranging)

    |\cos x}|\ =\ \sqrt{\cos2x+\sin^2x} (taking square root)

    And thus there is no contradiction when x\ =\ \frac{3\pi}4. This absolute value sign must be included if we are taking the positive square root on the right hand side of the equation.
    Last edited by Stephen347; November 27th 2012 at 11:05 AM.
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  15. #30
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    Re: Another Proof that 1 = 2

    Quote Originally Posted by Archie Meade View Post
    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 xx(x^2) is not a continuous function therefore not differentiable.
    If x is not a natural number then x^2 = x+x+...+x is not a valid equality.
    Last edited by ChessTal; July 20th 2013 at 03:02 PM.
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