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Tricky complex numbers proof

  1. Nov 8, 2005 #1
    I recently was confronted by this monstrosity of a question in one of my mock exams.

    |Z1 + Z2| ≤ |Z1| + |Z2|

    I made a few attempts at it before becoming demoralized with the lack of progress..
    |Z^2| was equal to Z1(conjugate)Z1
    Hence equaling X^2 + Y^2

    However even when expanding into x+iy form etc no avail, help is appreciated.
  2. jcsd
  3. Nov 8, 2005 #2
    square both side and see if you notice anything and remember
    if A=A then, surely A is less than or equal to A
    Last edited: Nov 8, 2005
  4. Nov 9, 2005 #3


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    Geometrically this is just the triangle inequality. It just says that the sum of two sides of a triangle is always greater or equal than the third.

    Since you know |z|^2=z*z. Why not write the left side out in this form?
    ie: |z+w|^2=(z*+w*)(z+w)
  5. Nov 17, 2005 #4
    I'm sorry but I had to laugh when I read "I recently was confronted by this monstrosity of a question..." only to find the triangle inequality beneathe. lol.
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