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Math Help - Complex Numbers

  1. #1
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    Complex Numbers

    Put the following expressions into the shape a+ib with a,b \in \mathbb{R}.

    (a) i^3

    (b) (1-i)i

    The answers should be:

    (a) -i

    (b) 1+i

    I know that the real and imaginary parts of a+ib are a and b respectively. But I don't know how to arrive at this answers, can anyone please explain? Thanks.
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  2. #2
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    Quote Originally Posted by demode View Post
    Put the following expressions into the shape a+ib with a,b \in \mathbb{R}.

    (a) i^3

    (b) (1-i)i

    The answers should be:

    (a) -i

    (b) 1+i

    I know that the real and imaginary parts of a+ib are a and b respectively. But I don't know how to arrive at this answers, can anyone please explain? Thanks.
    Remember the defintion of i is i^2=-1

    For the first i^3=i^2\cdot i

    For the 2nd just distribute and use the property above.
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    Remember the defintion of i is i^2=-1

    For the first i^3=i^2\cdot i

    For the 2nd just distribute and use the property above.

    Thank you, that helps a lot. But I don't understand why i^2=-1! When I look at the imaginary-real axis, i on the imaginary axis corresponds to 1 on the real axis. That's why I thought i^3 = 1^3.
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  4. #4
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    Quote Originally Posted by demode View Post
    Thank you, that helps a lot. But I don't understand why i^2=-1! When I look at the imaginary-real axis, i on the imaginary axis corresponds to 1 on the real axis. That's why I thought i^3 = 1^3.
    i^2 = -1 by DEFINITION of imaginary and complex numbers.


    This is because they needed to create a number (i.e. out of their imaginations) that  = \sqrt{-1}. They called this number i.


    So if i = \sqrt{-1} then that must mean i^2 = -1.
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