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Math Help - Set

  1. #1
    Junior Member
    Joined
    Aug 2009
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    62

    Set

    Hi everybody,

    How to show that exists a set \mathbb{C}={a+ib, a,b \in \mathbb{R}} such as i^2=-1

    Can you help me please??
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  2. #2
    MHF Contributor

    Joined
    Apr 2005
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    1466
    Starting from what basis? If you are given a "symbol", i, such that i^2= -1, then the rest is easy. If not, then the usual construction of the complex numbers is:

    Given the set of ordered pairs of real numbers, (x, y), define (x, y)+ (u, v)= (x+ u, y+ v) and (x, y)(u, v)= (xu- yv, xv+ yu) where x, y, u, v can be any real numbers.

    Show that this set of pairs satisfies all of the properties of the complex numbers with (x, y)= (x, 0)+ (0, y)= x(1, 0)+ y(0, 1) and you identify (1, 0) with the real number 1 and (0, 1) with "i".
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