# 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
625
122
$\frac{2-4i}{3+5i} = -\frac{7}{17} - \frac{11}{17}i \neq 5 + 3i$

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
625
122
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
20,228
5,755
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}
$

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
20,228
5,755
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} = $$-\frac{7}{17}$$ \; + \; $$-j \frac{11}{17}$$
$

7. ### MrAl Distinguished Member

Jun 17, 2014
3,741
791
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