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Complex numbers: Why is the modulus of z

  1. May 26, 2012 #1
    Why is the modulus of z, a complex number, |z| = √(a^2+b^2)?

    Why is it not |z| = √(a^2+(ib)^2)?
     
  2. jcsd
  3. May 26, 2012 #2

    HallsofIvy

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    Because in that case, the modulus of a+ ai, for any a, would be 0. And the modulus is supposed to measure the "size" of the number- specifically, its distance from 0.

    Any concept of "modulus", or, more generally, "norm", should satisfy
    1) |0|= 0 and if x is not 0, |x|> 0
    2) If a is a real number, |ax|= |a||x| where "|a|" is the usual absolute value of a real number
    3) [itex]|a+ b|\le |a|+|b|[/itex]

    Your suggestion, |a+ ib|= √(a^2- b^2) would not satisfy those.
     
  4. May 26, 2012 #3
    Thank you, [strike]WallsofIvy[/strike] HallsofIvy. :smile:
     
  5. May 27, 2012 #4

    HallsofIvy

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    Well, the halls have walls!
     
  6. May 27, 2012 #5

    micromass

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    The modulus is supposed to be the distance between (0,0) and (a,b). You are suggestion does not give the distance.
     
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