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Math Help - Complex Numbers help

  1. #1
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    Complex Numbers help

    Consider the complex numbers:
    z1 = 3 + j and z2 = −5 + 5j
    (i) (2 pts) Plot both numbers on the complex plane.
    Answer:I have done.

    (ii) (2 pts) Evaluate |zi| and angle zi for i = 1, 2. *i is a subscript

    My attempt: I know that |zi|is the magnitude and is bascially the square root of z x z*. But here we have z1 and z2. So what is zi. Are we supposed to combine z1 and z2 and then take the magnitude. Please give me a headstart on this problem.

    (iii) (2 pts) Without using a calculator, show that angle:z1 and anglez1 + z2) differ by an integer multiple of pi/2.

    My attempt: So I know the angle is arctan of y/x. I dont know how to prove this without a calculator. Please help on this also.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Complex Numbers help

    Quote Originally Posted by kmalik001 View Post
    Consider the complex numbers:
    z1 = 3 + j and z2 = −5 + 5j
    (i) (2 pts) Plot both numbers on the complex plane.
    Answer:I have done.

    (ii) (2 pts) Evaluate |zi| and angle zi for i = 1, 2. *i is a subscript

    My attempt: I know that |zi|is the magnitude and is bascially the square root of z x z*. But here we have z1 and z2. So what is zi. Are we supposed to combine z1 and z2 and then take the magnitude. Please give me a headstart on this problem.

    (iii) (2 pts) Without using a calculator, show that angle:z1 and anglez1 + z2) differ by an integer multiple of pi/2.

    My attempt: So I know the angle is arctan of y/x. I dont know how to prove this without a calculator. Please help on this also.
    Hi kmalik001!

    For (ii) you are supposed to calculate both |z1| and |z2| and the same for the angle.
    With zi they mean that you should find |zi| for each value of i. The possible values of i are 1 and 2. So that means both |z1| and |z2|.

    For (iii), what did you find for the angle of z1 and for the angle of (z1 + z2)?
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  3. #3
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    Re: Complex Numbers help

    Lol you're so right I don't know why I didn't see that. Thanks.
    So for angle z1 I get arctan(1/3) and for angle(z1+z2) I get arctan(-3)
    SO does that imply a pi/2 difference?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Complex Numbers help

    Quote Originally Posted by kmalik001 View Post
    Lol you're so right I don't know why I didn't see that. Thanks.
    Good!

    So for angle z1 I get arctan(1/3) and for angle(z1+z2) I get arctan(-3)
    SO does that imply a pi/2 difference?
    The question is how to figure that out.
    Can you draw a rectangular triangle with an angle such that the tangent will be (1/3)?
    How long should its sides be?
    And how about a rectangular triangle with a tangent of -3?
    How do they relate?
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  5. #5
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    Re: Complex Numbers help

    Quote Originally Posted by kmalik001 View Post
    Consider the complex numbers:
    z1 = 3 + j and z2 = −5 + 5j
    (i) (2 pts) Plot both numbers on the complex plane.
    Answer:I have done.

    (ii) (2 pts) Evaluate |zi| and angle zi for i = 1, 2. *i is a subscript
    (iii) (2 pts) Without using a calculator, show that angle:z1 and anglez1 + z2) differ by an integer multiple of pi/2.
    Here are some matters of notation.
    This is a mathematics forum, not a physics forum, based in North America.
    As such we use i as the complex base not j.
    We want all who use this site to be able to benefit from the replies, therefore consistent notation is necessary.


    It is the case that \overline{z}=z^* in your notation.

    |z|^2=z\cdot\overline{z}, |z|=\sqrt{\text{Re}^2(z)+\text{Im}^2(z)}.

    What you write as "angle:z1" is \text{arg}(z_1).

    Given any complex number z=a+bi then
    Arg(z) = \left\{ {\begin{array}{rl}   {\arctan \left( {\frac{b}{a}} \right),} & {a > 0}  \\   {\arctan \left( {\frac{b}{a}} \right) + \pi ,} & {a < 0\;\& \,b > 0}  \\    {\arctan \left( {\frac{b}{a}} \right) - \pi ,} & {a < 0\;\& \,b < \pi }  \\ \end{array} } \right.
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: Complex Numbers help

    Quote Originally Posted by Plato View Post
    Here are some matters of notation.
    This is a mathematics forum, not a physics forum, based in North America.
    As such we use i as the complex base not j.
    We want all who use this site to be able to benefit from the replies, therefore consistent notation is necessary.

    Gee, now you're telling me.
    I thought it was an international forum.
    And I thought it was open for anyone, whether in engineering, physics, or anything else, who needed help with the math part of his/her education.

    I'd like to think of math as something that transcends nationality or specialty.

    Given any complex number z=a+bi then
    Arg(z) = \left\{ {\begin{array}{rl}   {\arctan \left( {\frac{b}{a}} \right),} & {a > 0}  \\   {\arctan \left( {\frac{b}{a}} \right) + \pi ,} & {a < 0\;\& \,b > 0}  \\    {\arctan \left( {\frac{b}{a}} \right) - \pi ,} & {a < 0\;\& \,b < \pi }  \\ \end{array} } \right.
    That is not quite correct.
    The case a=0 is missing which consists of 2 or 3 distinct sub cases depending on how you count them.
    The bound on b in the last case is also incorrect.
    See for instance wiki for a proper definition.
    Last edited by ILikeSerena; February 9th 2013 at 07:13 AM.
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  7. #7
    Super Member ILikeSerena's Avatar
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    Re: Complex Numbers help

    @Plato: Please try not to be so pedantic, especially if you're responding in a thread that I replied to.
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  8. #8
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    Re: Complex Numbers help

    Complex Numbers help-image.jpg
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  9. #9
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    Re: Complex Numbers help

    is this what you meant?
    Please don't mind my slowness, I suck at Math.
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    Re: Complex Numbers help

    I still don't see how they relate I mean they are scaled by a factor of 3 but still...
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  11. #11
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    Re: Complex Numbers help

    Quote Originally Posted by ILikeSerena View Post
    Gee, now you're telling me.
    I'd like to think of math as something that transcends nationality or specialty.
    That totally misses the point. It is not about mathematics. It is about the notation as well as the language used. You may not like it, but this is a mathematics forum written is English using notation standard for most users. Again, you may not think so, but that means students in the Americas and the British Isles.

    Quote Originally Posted by ILikeSerena View Post
    That is not quite correct.
    The case a=0 is missing which consists of 2 or 3 distinct sub cases depending on how you count them.
    Look, anyone doing this level of work should know that was not meant to be inclusive. Of a=0\text{ or }b=0 are special and easy cases.
    You are just being too picky. But I guess you got pi** off at my remarks about the intent of this forum.
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  12. #12
    Super Member ILikeSerena's Avatar
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    Re: Complex Numbers help

    Yes. That is what I mean. Good!

    The scale is a little bit off though, making it a bit harder to see what's important.
    The 2 triangles you have drawn are the same!
    The sides of both triangles have the same lengths, meaning the sharp angle is the same.
    The one is just rotated by 90 degrees.

    Since we measure angles starting from the positive x-axis, it means that the angle up to the second triangle is pi/2 more than the angle of the first triangle.
    Since you might also have drawn the triangles at other positions relative to the axes, more generally, the difference is a multiple of pi/2.
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  13. #13
    Super Member ILikeSerena's Avatar
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    Re: Complex Numbers help

    Quote Originally Posted by Plato View Post
    That totally misses the point. It is not about mathematics. It is about the notation as well as the language used. You may not like it, but this is a mathematics forum written is English using notation standard for most users. Again, you may not think so, but that means students in the Americas and the British Isles.
    Not true.

    The choice to write "i" or "j" varies internationally, by school, by topic, by class, and by specialty.
    Are you so sure the same notation is used by all schools in the United States (not the Americas) and the British Isles?
    Often the choice is switched if otherwise there would be a conflict in notation (in a particular specialty).
    In particular it is totally irrelevant which choice you make. The math is the same.
    Both are valid!

    And yes, the language is English, but this does not just include the United States and Great Britain.
    It also includes Australia, New Zealand, India, and all other countries that can read and write English, such as all countries in Europe.
    Do consider that math notation comes from Europe, most of which is natively non-English.


    Look, anyone doing this level of work should know that was not meant to be inclusive. Of a=0\text{ or }b=0 are special and easy cases.
    You are just being too picky. But I guess you got pi** off at my remarks about the intent of this forum.
    Math is about being precise and accurate.
    It ticks me off when people are not.
    Normally I would have let it go, but yes, I got pi** of by your remarks about what you think the intent of the forum is.
    Please do not claim that they represent the intent of this forum.
    As far as I know they are yours alone.
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  14. #14
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    Re: Complex Numbers help

    Thank you so much!
    I understand it.

    I am also kind of stumped on this question.

    Determine the only real values a, b, c and d such that the equation:
    z^4 + az^3 + bz^2 + cz + d = 0
    has both z1 and z2 as roots.

    I don't really get what this is asking.
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  15. #15
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    Re: Complex Numbers help

    Quote Originally Posted by ILikeSerena View Post
    @Plato: Please try not to be so pedantic, especially if you're responding in a thread that I replied to.
    @ ILikeSerena
    I don't care that you responded to this thread.

    Quote Originally Posted by kmalik001 View Post
    I am also kind of stumped on this question.
    Determine the only real values a, b, c and d such that the equation:
    z^4 + az^3 + bz^2 + cz + d = 0
    has both z1 and z2 as roots.
    As long as the coefficients of a polynomial are real, if w is a root then so is \overline{w}.

    Thus you know four factors:
    (z-z_1)(z-\overline{z_1})(z-z_2)(z-\overline{z_2})
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