Complex conjugates

Discussion in 'Math' started by Biggsy100, Jun 9, 2015.

  1. Biggsy100

    Thread Starter Member

    Apr 7, 2014
    88
    1
    so I have a question, find the suitable complex conjugates to determine the magnitude of;

    2-j4/3+j5

    As I understand I can show both variables as:

    2-j4/3+j5 = /5-3i =5+3i

    2-j4/3+j5 = /5-3i

    How can I determine the magnitude? Is there further multiplication or addition to be made to these two variables?
     
  2. tjohnson

    Active Member

    Dec 23, 2014
    614
    121
    \frac{2-4i}{3+5i} = -\frac{7}{17} - \frac{11}{17}i \neq 5 + 3i

    The magnitude can be determined the same way as it would be for any other rectangular coordinate by finding the length of the hypotenuse of a triangle with sides of lengths x and y: m = \sqrt{x^2+y^2}
     
  3. Biggsy100

    Thread Starter Member

    Apr 7, 2014
    88
    1
    Ok, I have completley misunderstood this.

    I can't understand how this is broken down,. I thought I cross multiply?
     
  4. tjohnson

    Active Member

    Dec 23, 2014
    614
    121
    Please clarify one thing so that I can be sure I am not misleading you.
    Does that mean "(2-j4) divided by (3+j5)"? And are the numerator and the denominator complex numbers, with both a real and an imaginary part?

    I would suggest that you take a look at Khan Academy's videos on complex numbers: https://www.khanacademy.org/math/precalculus/imaginary_complex_precalc/multiplying-dividing-complex. They cover multiplying and dividing complex numbers, as well as calculating their conjugates.
     
    Last edited: Jun 9, 2015
  5. WBahn

    Moderator

    Mar 31, 2012
    17,747
    4,796
    Once again, you appear to be being very sloppy with your notation. What you have written above is

    <br />
2 \; - \; \frac{j4}{3} \; + \; j5<br />

    You setting yourself up for problem after problem until you start being precise and correct with your notation.

    What you (almost certainly) meant to state was

    (2-j4)/(3+j5)

    which is

    <br />
\frac{2-j4}{3+j5}<br />

    What variables? You don't have any variables at all.

    This makes absolutely no sense. What is "/5"?

    I suggest you read up on working with complex numbers. There are lot's of webpages devoted to this. The one that I link in your other thread (repeated below for convenience) is one place you might start.

    http://www.dragonwins.com/domains/getteched/MathReview.htm
     
  6. WBahn

    Moderator

    Mar 31, 2012
    17,747
    4,796
    Let's walk through putting this into rectangular form

    <br />
\frac{2-j4}{3+j5}<br />

    Multiply numerator and denominator by the complex conjugate of the denominator (see the link in my prior post if you don't know what a complex conjugate is).

    <br />
\(\frac{2-j4}{3+j5}\) \(\frac{3-j5}{3-j5}\) \; = \; \frac{\(2-j4\)\(3-j5\)}{\(3+j5\)\(3-j5\)}<br />

    Multiply out the polynomials in the numerator and denominator

    <br />
\frac{\(2-j4\)\(3-j5\)}{\(3+j5\)\(3-j5\)} \; = \; \frac{6 - j10 - j12 + j^2 20}{9 -j15 + j15 - j^2 25}<br />

    Now leverage the fact that j²=-1, by definition

    <br />
\frac{6 - j10 - j12 + j^2 20}{9 -j15 + j15 - j^2 25} \; = \; \frac{6 - j10 - j12 + (-1)20}{9 -j15 + j15 - (-1)25}<br />

    Now collect real and imaginary components in both the numerator and denominator

    <br />
\frac{6 - j10 - j12 + (-1)20}{9 -j15 + j15 - (-1)25} \; = \; \frac{-14 - j22}{34} \; = \; \frac{-7 - j11}{17} = \(-\frac{7}{17}\) \; + \; \(-j \frac{11}{17}\)<br />
     
  7. MrAl

    Well-Known Member

    Jun 17, 2014
    2,433
    490
    Hi,

    A simple way to explain what the complex conjugate is, is just to say that the sign of the imaginary part changes.

    A couple examples of finding the conjugate:
    1+2j => 1-2j
    1-2j => 1+2j
    3.45+7.3j => 3.45-7.3j
    -29-86j => -29+86j
    a+bj => a-bj
    a-bj => a+bj

    In each case all we did was change the sign of the imaginary part.
    Note also that the conjugate of the conjugate is the original back again:
    conjugate(conjugate(X))=X
     
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