Complex conjugates

Thread Starter

Biggsy100

Joined Apr 7, 2014
88
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?
 

tjohnson

Joined Dec 23, 2014
611
\(\frac{2-4i}{3+5i} = -\frac{7}{17} - \frac{11}{17}i \neq 5 + 3i\)

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: \(m = \sqrt{x^2+y^2}\)
 

tjohnson

Joined Dec 23, 2014
611
Please clarify one thing so that I can be sure I am not misleading you.
2-j4/3+j5
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:

WBahn

Joined Mar 31, 2012
29,979
so I have a question, find the suitable complex conjugates to determine the magnitude of;

2-j4/3+j5
Once again, you appear to be being very sloppy with your notation. What you have written above is

\(
2 \; - \; \frac{j4}{3} \; + \; j5
\)

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

\(
\frac{2-j4}{3+j5}
\)

As I understand I can show both variables as:
What variables? You don't have any variables at all.

2-j4/3+j5 = /5-3i =5+3i
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
 

WBahn

Joined Mar 31, 2012
29,979
Let's walk through putting this into rectangular form

\(
\frac{2-j4}{3+j5}
\)

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).

\(
\(\frac{2-j4}{3+j5}\) \(\frac{3-j5}{3-j5}\) \; = \; \frac{\(2-j4\)\(3-j5\)}{\(3+j5\)\(3-j5\)}
\)

Multiply out the polynomials in the numerator and denominator

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

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

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

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

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

MrAl

Joined Jun 17, 2014
11,389
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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