Evaluate Complex Numbers: z_1

In summary: What?No: (3 - 4\,j)^{*} \neq -3 + 4\, j; taking the complex conjugate changes the sign of the imaginary part only, and does not affect the real part.
  • #1
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Homework Statement



Evaluate (find the real and complex components) of the following complex numbers, in either rectangular or polar form:

[itex]z_{1}[/itex] = [itex]\frac{j(3-j4)^{*}}{(-1+6j)(2+j)^{2}}[/itex]

Homework Equations



[itex]e^{jθ}[/itex] = cosθ + j sinθ

The Attempt at a Solution



I sadly don't even know where to begin here. I understand that my textbook uses j instead of i for imaginary number. I understand that a star superscript means the complex conjurgate so...

[itex]z_{1}[/itex] = [itex]\frac{j(3-j4)^{*}}{(-1+6j)(2+j)^{2}}[/itex] = [itex]\frac{j(j4-3)}{(-1+6j)(2+j)^{2}}[/itex]

Is this true? Where do I go from here thanks.

Just a note.

The prerequisite for this course was an introductory differential equations course, multivariable calculus and a second semester of physics, all of which I have taken. I still have no idea what half of this stuff is. I've never studied complex numbers before but dove into euler's formula and stuff of the like in high school out of curiosity. I however am not sure what a phasor is exactly but have some understand of the concept. Thanks for any help or suggestions on how to proceed to solve this problem thanks.
 
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  • #2
The first thing you can do is work out all the brackets. What do you get once you do that?
 
  • #3
Your first step is a little off. The complex conjugate of 3-j4 is 3+j4. Then you can just start multiplying the numerator and denominator out using j*j=(-1). Once you've got something in the form (a+bj)/(c+dj) you multiply numerator and denominator by the complex conjugate of (c+dj) to make the denominator real. There's nothing really subtle involved. It's just a lot of arithmetic.
 
  • #4
[itex]\frac{4-3j}{106j-41}[/itex]

oh ok let me see here
 
  • #5
GreenPrint said:

Homework Statement



Evaluate (find the real and complex components) of the following complex numbers, in either rectangular or polar form:

[itex]z_{1}[/itex] = [itex]\frac{j(3-j4)^{*}}{(-1+6j)(2+j)^{2}}[/itex]

Homework Equations





The Attempt at a Solution



I sadly don't even know where to begin here. I understand that my textbook uses j instead of i for imaginary number. I understand that a star superscript means the complex conjurgate so...

[itex]z_{1}[/itex] = [itex]\frac{j(3-j4)^{*}}{(-1+6j)(2+j)^{2}}[/itex] = [itex]\frac{j(j4-3)}{(-1+6j)(2+j)^{2}}[/itex]

Is this true? Where do I go from here thanks.

Just a note.

The prerequisite for this course was an introductory differential equations course, multivariable calculus and a second semester of physics, all of which I have taken. I still have no idea what half of this stuff is. I've never studied complex numbers before but dove into euler's formula and stuff of the like in high school out of curiosity. I however am not sure what a phasor is exactly but have some understand of the concept. Thanks for any help or suggestions on how to proceed to solve this problem thanks.

NO: [itex] (3 - 4\,j)^{*} \neq -3 + 4\, j;[/itex] taking the complex conjugate changes the sign of the imaginary part only, and does not affect the real part.

Anyway: express the whole numerator in the form [itex] a + j \, b[/itex] for real 'a' and 'b', and express the whole denominator in the form [itex] c + j \, d[/itex] for real 'c' and 'd'. Then use the standard quotient rule to evaluate
[tex] \frac{a + j\,b}{c + j\, d}[/tex]
in whatever final form you choose (either as [itex] A + j\, B[/itex] or as [itex] r\, e^{j \theta}[/itex]). Basically, that is how such questions are always done.

RGV
 
  • #6
[itex]\frac{266+371j}{9555}[/itex]

where do i go from here?
 
  • #7
GreenPrint said:
[itex]\frac{266+371j}{9555}[/itex]

where do i go from here?

I have no idea what you are doing, but those numbers don't look anything like what I'm getting. Can you spell out your calculation in detail?
 
  • #8
Ya I think I screwed it up I get

2/315 + 47/315 j

I hope that is correct. There's nothing left to do at this point?
 
  • #9
GreenPrint said:
Ya I think I screwed it up I get

2/315 + 47/315 j

I hope that is correct. There's nothing left to do at this point?

Nothing left to do except try and get the numbers right. I still don't agree with you. What did you get for the numerator and the denominator?
 
  • #10
What :cry:

[itex]\frac{{(3-4j)}^{*}j}{(-1+6j)(2+j)^{2}}[/itex]

[itex]{(3-4j)}^{*}=(3+4j)[/itex]
[itex]{(2+j)}^{2}=2^{2}+j^{2}+2(2)j=4+1+4j=5+4j[/itex]


[itex]\frac{(3+4j)j}{(-1+6j)(5+4j)}[/itex]

[itex](3+4j)j=3j+4j^{2}=3j+4[/itex]
[itex](-1+6j)(5+4j)=-5-4j+30j+24j^{2}=-5+26j+24=19+26j[/itex]

[itex]\frac{3j+4}{19+26j}[/itex]

[itex]\frac{3j+4}{19+26j}*\frac{19-26j}{19-26j}[/itex]

(19+26j)(19-26j)=[itex]19^{2}-26(19)j+26(19)j-26^{2}j^{2} = 19^{2} - 26^{2} = 361 - 676 = -315[/itex]

[itex](3j+4)(19-26j)=3(19)j-3(26)j^{2}+4(19)-26(4)j=57j-78+76-104j = -2-47j[/itex]

[itex]\frac{-2-47j}{-351} = \frac{2}{315}+\frac{47j}{315}[/itex]

don't see what i did wrong
 
  • #11
GreenPrint said:
What :cry:

[itex]\frac{{(3-4j)}^{*}j}{(-1+6j)(2+j)^{2}}[/itex]

[itex]{(3-4j)}^{*}=(3+4j)[/itex]
[itex]{(2+j)}^{2}=2^{2}+j^{2}+2(2)j=4+1+4j=5+4j[/itex]

You did [itex]j^2=1[/itex]. It should be [itex]j^2=-1[/itex].
 
  • #12
Um, j^2=(-1). Not +1. Check (2+j)^2 again. It's not 5+4j, is it? Etc.
 
  • #13
6/37 - 1/37 j ?
 
  • #14
GreenPrint said:
6/37 - 1/37 j ?

That's what I get.
 
  • #15
Thank you much
 

1. What is a complex number?

A complex number is a number that contains both a real and an imaginary part. It is written in the form a + bi, where a is the real part and bi is the imaginary part (with i representing the square root of -1).

2. How do you evaluate a complex number?

To evaluate a complex number, simply add or subtract the real parts and the imaginary parts separately. For example, to evaluate z1 = 3 + 2i, the real part is 3 and the imaginary part is 2i. Therefore, z1 = 3 + 2i.

3. What is the purpose of evaluating complex numbers?

Evaluating complex numbers allows us to perform mathematical operations on them, such as addition, subtraction, multiplication, and division. It also helps us to understand the properties of complex numbers and their relationships with real numbers.

4. Can complex numbers be graphed on a number line?

No, complex numbers cannot be graphed on a number line because they have two components (real and imaginary) and therefore require a two-dimensional graph, such as the complex plane.

5. How can complex numbers be used in real life?

Complex numbers have various applications in real life, such as in electrical engineering, physics, and signal processing. They can also be used to represent and solve problems in geometry and are often used in computer graphics and game development.

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