Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Complex values and size

  1. Oct 30, 2011 #1

    Pythagorean

    User Avatar
    Gold Member

    complex values and "size"

    The only measure of size I can think of would be to imagine a complex number as a vector and calculate its length via pythagoras. In that case, I would find i+1 and i-1 to be of equal length.. however using matlab's min function and < operand, I get two completely different answers (and different from each other too):

    according to min, the first entry in A (i+1) is the "smallest"
    Code (Text):
    A =

      1.000000000000000 + 1.000000000000000i
     -1.000000000000000 + 1.000000000000000i

    >> min(A)

    ans =

      1.000000000000000 + 1.000000000000000i
    but according to the lessthan sign, the second entry (i-1) is the "smallest"

    Code (Text):
     >> if A(1) < A(2)
    dips('YES')
    end
    >>if A(2) < A(1)
    disp('YES')
    end
    YES
    are there alternative ways of measuring/comparing the "size" of complex values?
     
  2. jcsd
  3. Oct 31, 2011 #2
    Last edited by a moderator: Apr 26, 2017
  4. Oct 31, 2011 #3
    Re: complex values and "size"

    there is no order among the complex numbers, but usually what I have seen is that complex functions have min and max values that refer to the min and max modulus the functions attain (this is what you were talking about for example |i+1| = |i-1| but I don't think it makes since to say if we have two complex numbers z and v, ask which one is "smaller" or the minimum of the two.
     
  5. Nov 1, 2011 #4

    Pythagorean

    User Avatar
    Gold Member

    Re: complex values and "size"

    but with modulus you use the conjugate don't you? Hrm... guess it's the same result, just different definition. I can't think of complex numbers without thinking of it as a vector. i is just an orthogonal transform operator as far as I can tell.
     
  6. Nov 1, 2011 #5
    Re: complex values and "size"

    by modulus i was just referring to the distance from the complex number to the origin. so in that case, yeah a complex number z would have the same modulus as its conjugate. all you really have to work with in terms of telling two complex numbers apart are its modulus and Argument, but i still can't think of that showing one numbers being greater than the other


    just out of curiosity because i am just beginning to learn complex analysis but, why think of a complex number as a vector? i guess what i mean to say is, why not just think of a complex number as a complex number? i guess im just curious because i am new to the subject and i had never thought of them as vectors before. It seems interesting
     
  7. Nov 1, 2011 #6

    Pythagorean

    User Avatar
    Gold Member

    Re: complex values and "size"

    My background is more physics, I guess the way I intuitively understand things is through spatial metaphor. "Complex number" is just a title. I know it's a real part and an imaginary part, but "imaginary part" is a terribly unhelpful name.

    What is a complex number then?

    Here's an explanation I heard that sat with me, I guess. It starts with the solution of:

    x^2 + 1 = 0

    which is

    x = +/- sqrt(-1)
    http://galileo.phys.virginia.edu/classes/152.mf1i.spring02/ComplexNumbers.htm
     
  8. Nov 1, 2011 #7

    Pythagorean

    User Avatar
    Gold Member

    Re: complex values and "size"

    ah, ok. I thought the modulus was defined as:

    sqrt(zz*)

    where z* is the conjugate of z.

    where I would just imagine i as an operator, (say transforming an i-hat to a j-hat) and you'd just take the magnitude of the vector (the distance to the origin). I mean, if you're talking about "distance" you're seeing it as a vector through two spatial dimensions aren't you?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook