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Math Help - Proof that 1 = 2

  1. #1
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    Proof that 1 = 2

    This is a silly proof that my thirteen-year old cousin showed me the other day. I'm ashamed to admit that it took me a full ten minutes to figure it out.

    Let a = 1
    Let b = 1

    therefore:
    a = b

    multiply both sides by b:
    ab = b^2

    subtract a^2:
    ab - a^2 = b^2 - a^2

    factorise:
    a(b - a) = (b - a)(b + a)

    divide by (b - a):
    [a(b - a)]/(b -a) = [(b - a)(b + a)]/(b -a)

    cancel:
    a = b + a

    returning to our initial property:
    a = 1
    b = 1

    therefore:
    a = b + a
    1 = 1 + 1
    1 = 2


    What is wrong with the above proof? (It's not difficult, but it's a good one to pull out to confuse people)
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  2. #2
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    Division by zero (b-a).
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  3. #3
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    Well done.

    I can't believe it took me that long to figure it out!
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  4. #4
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    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.
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  5. #5
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by paulrb View Post
    Are there any 1=2 "proofs" that do not involve dividing by zero?
    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)
    Last edited by TheAbstractionist; June 3rd 2009 at 12:52 PM.
    Thanks from topsquark
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by TheAbstractionist View Post

    -1\ =\ 1 (multiplying both sides by \sqrt2)

    1\ =\ 2 (adding 1 to both sides)
    The last step should lead to 0=2...
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  7. #7
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    Quote Originally Posted by Chris L T521 View Post
    The last step should lead to 0=2...
    ^ How?

    The Abstractionist is correct!
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  8. #8
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by blueirony View Post
    ^ How?

    The Abstractionist is correct!
    Read what I quoted. Its correct now since he edited it.
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  9. #9
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by blueirony View Post
    ^ How?

    The Abstractionist is correct!
    I made a mistake in my original post, which was spotted by Chris L T521. I do this quite often, and when someone spots my mistakes, I normally thank the person who spotted them and just go back and edit my post. (To Chris L T521: Thanks! )

    By the way, did you spot the fallacy in the “proof”?
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  10. #10
    Super Member Deadstar's Avatar
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    Is it that we can also write \sqrt{0+\frac{1}{2}} = \frac{-1}{\sqrt{2}} and hence -\frac{1}{2} = -\frac{1}{2} i.e. -1 = -1...
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  11. #11
    Super Member Deadstar's Avatar
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    Here's what could be the simplest -1 = 1 proof.
    1 = \sqrt{1} = -1

    So 1 = -1.
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  12. #12
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by Deadstar View Post
    Is it that we can also write \sqrt{0+\frac{1}{2}} = \frac{-1}{\sqrt{2}} and hence -\frac{1}{2} = -\frac{1}{2} i.e. -1 = -1...
    It’s along those lines, but I wouldn’t write \sqrt{0+\frac{1}{2}} = \frac{-1}{\sqrt{2}}. The LHS is generally taken to mean the positive square root, so the equation is technically incorrect.

    Hint:
    Spoiler:
    If a^2=b^2 (where a and b are real) then a=\ldots?
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  13. #13
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    This is logical 'proof' rather than mathematical....

    Consider the following statement;
    " If this statement is true, then 1= 2. " (let's call this A)

    First, let's assume A to be true. on this supposition;
    As statement A is true, "If statement A is true, then 1=2" is true.
    and again, as statement A is true and
    "if statement A is true, then 1=2" is true, we obtain 1=2.

    Now we have proved that "If this statement is true, then 1=2 " is true.

    Then, as 'this statement' is true, 1=2 (q.e.d.)

    By using the same trick, anything can be proved.
    This is called Curry's paradox..
    see also: http://en.wikipedia.org/wiki/Curry%27s_paradox
    Last edited by joll; January 17th 2010 at 02:17 AM. Reason: to add the link to wikipedia
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  14. #14
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    A is racing against B; A wears sweater#1, B wears sweater#2.
    Thet hit "finish line" at exactly same time.....WELL?
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  15. #15
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    One empty bottle and two empty bottles contain the same amount of liquid.
    If x empty bottles contain zero liquid in total, x is not immediately determineable.

    Quote Originally Posted by blueirony View Post

    Let a = 1
    Let b = 1

    therefore:
    a = b

    multiply both sides by b:
    ab = b^2

    subtract a^2:
    ab - a^2 = b^2 - a^2
    this is now 0=0.

    factorise:
    a(b - a) = (b - a)(b + a)
    this is now 1(0)=(0)2, only non-zero values can be factorised.


    What is wrong with the above proof? (It's not difficult, but it's a good one to pull out to confuse people)
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