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Math Help - modulus and argument

  1. #1
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    modulus and argument

    find the modulus and arguments of the complex number (1+\sqrt 3i)^3.

    my solution;
    (1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)
    = 1+2\sqrt3i+3i^2(1+\sqrt 3i)
    = -2+2\sqrt3i(1+\sqrt 3i)
    = -2-2\sqrt3i+2\sqrt3i+2(3)i^2
    = -2+6i^2
    = -8

    is there anything wrong with my working above?
    to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
    then |-8+0i| = 8
    and arg |-8+0i| = 0
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  2. #2
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    Quote Originally Posted by z1llch View Post
    find the modulus and arguments of the complex number (1+\sqrt 3i)^3.

    my solution;
    (1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)
    = {\color{red}(}1+2\sqrt3i+3i^2{\color{red})}(1+\sqr  t 3i)
    = {\color{red}(}-2+2\sqrt3i{\color{red})}(1+\sqrt 3i)
    = -2-2\sqrt3i+2\sqrt3i+2(3)i^2
    = -2+6i^2
    = -8

    is there anything wrong with my working above?
    Well technically, you need the parentheses I added above.

    to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
    then |-8+0i| = 8
    and arg |-8+0i| = 0
    If you look at the trig form of the complex number: r(\cos \theta + i\sin \theta), r is the modulus and theta is the argument. Anyway, your modulus is right, but the argument is not. The argument should be pi.


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  3. #3
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    Quote Originally Posted by z1llch View Post
    find the modulus and arguments of the complex number (1+\sqrt 3i)^3.

    my solution;
    (1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)
    = 1+2\sqrt3i+3i^2(1+\sqrt 3i)
    = -2+2\sqrt3i(1+\sqrt 3i)
    = -2-2\sqrt3i+2\sqrt3i+2(3)i^2
    = -2+6i^2
    = -8

    is there anything wrong with my working above?
    to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
    then |-8+0i| = 8
    and arg |-8+0i| = 0

    yeongill has answered ..
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  4. #4
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    i do not understand the argument part.. can please explain on it a little bit?
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  5. #5
    Super Member dhiab's Avatar
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    Quote Originally Posted by z1llch View Post
    find the modulus and arguments of the complex number (1+\sqrt 3i)^3.

    my solution;
    (1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)
    = 1+2\sqrt3i+3i^2(1+\sqrt 3i)
    = -2+2\sqrt3i(1+\sqrt 3i)
    = -2-2\sqrt3i+2\sqrt3i+2(3)i^2
    = -2+6i^2
    = -8

    is there anything wrong with my working above?
    to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
    then |-8+0i| = 8
    and arg |-8+0i| = 0
    ANATHOR RESOLUTION :

    THANKS
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  6. #6
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    Here are some simple facts the would have made this an easy problem.
    \begin{gathered}<br />
  \left| {z^n } \right| = \left| z \right|^n \; \Rightarrow \;\left| {\left( {1 + \sqrt 3 i} \right)^3 } \right| = \left| {1 + \sqrt 3 i} \right|^3  \hfill \\<br />
  \arg \left( {z^n } \right) = n\arg (z) \hfill \\ <br />
\end{gathered}
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  7. #7
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    Quote Originally Posted by z1llch View Post
    i do not understand the argument part.. can please explain on it a little bit?
    Have you learned about the trig form of the complex number? Plotting the number -8 + 0i on the complex plane is like plotting the point (-8, 0) on the Cartesian plane. You got an angle in standard position (where the initial side is the positive x-axis and the terminal side is a ray where the point is on), and this angle is pi radians. So the argument is pi. We've already figured that the modulus is 8, so
    -8 + 0i = 8(\cos \pi + i\sin \pi).


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