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Can 1 = 2 ?

  1. Sep 4, 2006 #1
    can 1 = 2 ??

    is this possible
    1 = 2
    ??
     
  2. jcsd
  3. Sep 4, 2006 #2
    Consider the equation,

    x + 1 = x + 2

    And you are asked to solve for x.

    By looking at it, you can say there is no solutions.

    Subtract x from both sides, and you get

    1 = 2

    But what if you could subtitute 1 = 2 in one eqation

    you get x + 1 = x + 1

    or x + 2 = x + 2

    That makes more sense now.
     
  4. Sep 4, 2006 #3

    CRGreathouse

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    Sure, if the underlying assumptions are inconsistent.
     
  5. Sep 4, 2006 #4
    i didnt get it
    for any value of x how can x+1 = X+2
    the assumption itself is wrong
    right??
     
  6. Sep 4, 2006 #5

    radou

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    This really has nothing to do with math.
     
  7. Sep 4, 2006 #6

    arildno

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    Sure 1=2 in some number systems.
    In a number system that denies the axiom [itex]0\neq{1}[/itex], for example, 1=2 will be a true statement.
     
  8. Sep 4, 2006 #7

    StatusX

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    1 and 2 are just symbols. By their standard definition, they correspond to two distinct real numbers, and so the statement '1=2' is false. If you want to use these symbols in a different way, then the statement '1=2' can be true, false, or meaningless, depending on how you define them.
     
  9. Sep 5, 2006 #8
    1 is not equal to 2 in any of the standard formulations of natural numbers (Peano axioms, set theory etc). However you can very easily define two symbols 1 and 2 in a number system such that they are equal.
     
  10. Sep 5, 2006 #9
    cant we use complex numbers to prove it
    like :

    i = i

    root -1 = root -1

    hence ,

    root -1/ root 1 = root -1 / root 1

    hence,
    whole root [-1/1] = whole root [ -1/1]

    hence,
    whole root [-1/1] = whole root [ 1/-1]

    [ -5/4 can also be written as 5/-4...cant it??? i mean -1/1 is the same as 1/-1 right??]

    hence,
    root -1 / root 1 = root 1 / root -1

    hence,

    i / 1 = 1 / i

    i square = 1

    hence,
    -1 = 1

    adding 3/2 on both sides,

    3/2 + ( -1 ) = 3/2 + 1

    3/2 - 1 = 3/2 + 1

    2/ 2 = 4/2

    therefore,

    1 = 2

    can this be possible
     
  11. Sep 5, 2006 #10

    J77

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    Nope.

    You can't take roots on both the top and bottom like that.

    [tex]i=e^{i\pi/2}[/tex]

    [tex]1/i=e^{-i\pi/2}[/tex]

    With these 1=2 things, there's always a mistake/trick.
     
  12. Sep 5, 2006 #11
    um...
    i kinda understood a little
    but i didnt understand
    the

    i = e raised to (-i Pie/ 2)
    what is that
     
  13. Sep 5, 2006 #12

    chroot

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    He's expressing complex numbers as complex exponentials.

    [tex]i=e^{i\pi/2}[/tex]

    is the value on the complex unit circle corresponding to an angle of pi/2. It is equal to i.

    His point is that 1/i and i/1 are quite different numbers, on opposite sides of the unit circle, so your "proof" contains an error.

    - Warren
     
  14. Sep 5, 2006 #13
    ohk
    thank you
     
  15. Sep 5, 2006 #14

    radou

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    A rational number is, by definition, a number of the form [tex]\frac{p}{q}[/tex], where [tex]p \in Z[/tex] and [tex]q \in N[/tex]. So, you can't write -5/4 as 5/-4.
     
  16. Sep 5, 2006 #15

    arildno

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    Eeeh, wherever do you have this limitation from??
    Not saying you might not be right, but I really don't see the necessity of this limitation.
     
  17. Sep 5, 2006 #16
    i remember my father showing me a proof once that 1=2, but cant quite recall it. but, if you start with the equation:
    x^2 -1 = 0, you can factor x^2 - 1 into (x+1)(x-1)=0
    then divide both sides by x-1, and get x+1=0.
    for a value of x=1, you have shown that 2=0.
    :-)
     
  18. Sep 5, 2006 #17

    chroot

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    You should tell your father to take more math classes. You cannot divide both sides by x-1, when x=1, because that is equivalent to division by zero. Division by zero is not a "legal" mathematical operation.

    - Warren
     
  19. Sep 5, 2006 #18
    That is NOT the definition of a rational number

    http://mathworld.wolfram.com/RationalNumber.html

    A rational number is a number p/q where p and q are Integers and [itex]q \neq 0[/itex]

    So, you can write (-5)/4 as 5/(-4)
     
  20. Sep 5, 2006 #19

    CRGreathouse

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    How quaint.
     
  21. Sep 5, 2006 #20

    HallsofIvy

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    A rational number is, by definition, a number which can be written in the form [itex]\frac{p}{q}[/itex] where [itex]p \in Z[/itex] and [itex]q \in N[/itex]. Whether or not the number is written that way is irrelevant.

    Yes, you can write -5/4 as 5/-4 just as you could write it as -1.25.
     
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