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Math Help - Real and Complex number systems

  1. #1
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    Unhappy Real and Complex number systems

    Hi there... I'm back in school after ten years, and I must say my math is a little rusty. I could use some help-with some explanation....so I can truly follow and understand.

    I am given this:

    x=r(cos u + i sin u) and y=t(cos v + i sin v)

    and I'm being asked to (1) prove that the modulus of (xy) is the product of their moduli, and (2) prove that the amplitude of (xy) is the sum of their amplitudes, showing each step of both proofs.

    I understand modulus with respect to the time of day using a 12 hour clock, but I'm finding it hard to apply, nevermind prove this situation. Just need some direction!!!! Thanks so much.
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  2. #2
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    Hello, Conorsmom!

    Given: . \begin{array}{ccc}<br />
x &=&r(\cos u + i\sin u) \\y &=&t(\cos v + i\sin v) \end{array}

    (a) Prove that the modulus of xy is the product of their moduli

    (b) Prove that the amplitude of xy is the sum of their amplitudes.
    x\!\cdot\!y \:=\:\bigg[r(cos u + i\sin u)\bigg]\,\bigg[t(\cos v + i\sin v)\bigg]

    . . . = \;rt\bigg[\cos u\cos v + i\sin u\cos v + i\sin v\cos u + i^1\sin u\sin v\bigg]

    . . . = \;rt\bigg[\underbrace{(\cos u\cos v - \sin u\sin v)}_{\text{This is }\cos(u+v)} + i\underbrace{(\sin u\cos v + \sin v\cos u)}_{\text{This is }\sin(u+v)}\bigg]

    . . . = \;rt\bigg[\cos(u+v) + i\sin(u+v)\bigg]


    \text{Then: }\;|x\!\cdot\!y| \:=\:\sqrt{r^2t^2\underbrace{\left(\cos^2(u+v) + \sin^2(u+v)\right)}_{\text{This is 1}}} \quad\Rightarrow\quad |x\!\cdot\!y| \:=\:rt<br />
 .[1]



    |x| \:=\:\sqrt{(r\cos u)^2 + (r\sin u)^2} \:=\:\sqrt{r^2\underbrace{(\cos^2\!u + \sin^2\!u)}_{\text{This is 1}}} \:=\: \sqrt{r^2} \:=\:r

    |y| \:=\:\sqrt{(t\cos v)^2 + (t\sin v)^2} \:=\:\sqrt{t^2(\cos^2\!v+\sin^2\!v)} \:=\:\sqrt{t^2} \:=\:t

    \text{Then: }\;|x|\!\cdot|y| \;=\;rt .[2]


    (a) Therefore: . |x\!\cdot\!y| \;=\;|x|\!\cdot\!|y|


    ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~


    We have: . x \:=\:r(\cos u + i\sin u) \qquad y \:=\:t(\cos v + i\sin v)
    . . . . . . . . . . . . . . \uparrow\qquad\quad\; \uparrow \qquad\qquad\qquad\quad \uparrow\qquad\quad\,\uparrow<br />
    . . . . . . . . . . . . .
    amplitude of x . . . . . . . . amplitude of y


    And:. . x\!\cdot\!y \;=\;rt\bigg[\cos(u+v) + i\sin(u+v)\bigg]
    . . . . . . . . . . . . . . . . \uparrow \qquad\qquad\quad\; \uparrow
    . . . . . . . . . . . . . . . .
    amplitude of xy


    (b) Therefore, the amplitude of xy is the sum of the amplitudes of x\text{ and }y.

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  3. #3
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    Quote Originally Posted by Conorsmom View Post
    Hi there... I'm back in school after ten years, and I must say my math is a little rusty. I could use some help-with some explanation....so I can truly follow and understand.

    I am given this:

    x=r(cos u + i sin u) and y=t(cos v + i sin v)

    and I'm being asked to (1) prove that the modulus of (xy) is the product of their moduli, and (2) prove that the amplitude of (xy) is the sum of their amplitudes, showing each step of both proofs.

    I understand modulus with respect to the time of day using a 12 hour clock, but I'm finding it hard to apply, nevermind prove this situation. Just need some direction!!!! Thanks so much.
    By modulus, I don't think it means modular arithmetic. I think it means modulus in the sense that if  z = x+iy , then the modulus of z is given by  |z| = \sqrt{x^2+y^2} . I.e., by modulus they mean absolute value!
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  4. #4
    MHF Contributor

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    Also "amplitude" is not the correct word here. What is meant is the "argument" of the complex number.
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  5. #5
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    as it turns out amplitude and argument are the same thing.
    one is older text, and one is newer text.
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