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Properties of complex numbers

  1. Sep 18, 2009 #1
    Why is |z|^2 = z*z?

    z = a + ib
    z*z = (a - ib)(a + ib) = a^2 + b^2
    z^2 = (a + ib)^2 = a^2 + 2iab - b^2

    So it must have something to do with the absolute value, but I don't understand what or why.
     
  2. jcsd
  3. Sep 18, 2009 #2
    |z| does not define the "Absolute Value" of a complex number. The notation |z| refers to the modulus of z, which is by definition

    |z| = sqrt(a^2 + b^2)

    Geometrically it gives the distance of the complex number from the origin on the Argand Plane.

    And quite obviously |z|^2 is NOT EQUAL TO z^2
     
  4. Sep 18, 2009 #3

    Dick

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    |z|^2 is conjugate(z)*z=a^2+b^2. It's not equal to z^2.
     
  5. Sep 19, 2009 #4
    although the modulus of z when z is of the form a + 0i (i.e. it is only in the reals), then wouldn't that be essentially like an absolute value?
     
  6. Sep 19, 2009 #5

    Dick

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    Sure. Modulus of z is |z| is sqrt(a^2+b^2). It's still not the same as z^2. What's the question again?
     
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