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Question about linear mapping

  1. Nov 10, 2012 #1

    loy

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    1. The problem statement, all variables and given/known data
    i recently saw a question about complex number, and its answer about the center of a circle is 1/2*(1+i). what does that mean?


    2. Relevant equations
    f:ℂ → ℝ^2


    3. The attempt at a solution
    since we define z=x+yi is an element of ℂ, so by the mapping above,
    we can say that f:ℂ→ℝ^2 = f(x+yi)=(x,y)?
    and by the inverse function of f,which is f-1(x,y)→(x+yi)?
    From the statement above, f-1(1/2,1/2)=1/2+i/2 =1/2*(1+i) ?
     
  2. jcsd
  3. Nov 10, 2012 #2

    loy

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    help me ~~~
     
  4. Nov 10, 2012 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Yes, that is the "complex plane" where we represent the complex number z= x+iy by the point (x, y). So the number 1/2(1+ i)= 1/2+ (1/2)i is represented by the point (1/2, 1/2).

    A circle with center at (a, b) and radius r has equation [itex](x- a)^2+ (y- b)^2= r^2[/itex] so a circle with center at (1/2, 1/2) has equation [itex](x- 1/2)^2+ (y- 1/2)^2= r^2[/itex]. You may also know that the "modulus" or "absolute value" of z= x+ iy is [itex]|z|= \sqrt{x^2+ y^2}[/itex] so that circle can also be written as [itex]|z- (1/2)(1+ i)|= r[/itex].
     
    Last edited: Nov 10, 2012
  5. Nov 10, 2012 #4

    loy

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    thanks very much
    =)
     
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