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manjuthottam

  • 3 years ago

COMPLEX VARIABLES: Can someone explain how to find the Arg z and what does it mean?

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  1. joemath314159
    • 3 years ago
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    If I give you a complex number in \[a+bi\]form, do you know how to convert it to its "polar" form\[re^{i\theta}\]?

  2. joemath314159
    • 3 years ago
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    Because when written in its polar form:\[\arg(re^{i\theta})=\theta\]

  3. manjuthottam
    • 3 years ago
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    where does a and b go?

  4. manjuthottam
    • 3 years ago
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    I don't know how to convert it to its polar form, could you help me understand that formula?

  5. joemath314159
    • 3 years ago
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    |dw:1359087681895:dw|

  6. joemath314159
    • 3 years ago
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    r is the length of the complex number, given by using pythagorean theorem:\[r=\sqrt{a^2+b^2}\]

  7. joemath314159
    • 3 years ago
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    the angle is the argument. You use inverse tangent to get it.\[\theta=\tan ^{-1}(\frac{a}{b})\]

  8. manjuthottam
    • 3 years ago
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    oh so r = modulus of the point (a,b)?

  9. joemath314159
    • 3 years ago
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    that is correct

  10. manjuthottam
    • 3 years ago
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    so when asked arg z, it is the theta that they are asking for?

  11. joemath314159
    • 3 years ago
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    yes, they want the angle. Be careful when using that formula though. You need to make sure the answer you get reflects the right quadrant. Ex. arg(1+i) vs. arg(-1-i)

  12. manjuthottam
    • 3 years ago
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    so when asked the arg of 1st quadrant, theta is by the formula you gave me; when asked the arg of quadrant 3, theta is tan inverse of (b/a) ?

  13. joemath314159
    • 3 years ago
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    ack! thanks for pointing out my mistake, it should be b/a in the formula i posted.

  14. manjuthottam
    • 3 years ago
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    oh ok :)

  15. joemath314159
    • 3 years ago
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    the problem arises because inverse tangent always gives an angle between -pi/2 and pi/2, but you want an angle between 0 and 2pi. So if you were asked for the argument of -1-i, the formula would give pi/4 as an answer, but you know you want something in the 3rd quadrant, so you add pi to get the correct answer of 5\pi/4

  16. manjuthottam
    • 3 years ago
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    oh ok that makes sense! Thanks!! may i ask another question related? How do i put it into the form of Re^(i theta)? actually what is that formula mean? is it just the form of the vector?

  17. manjuthottam
    • 3 years ago
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    i understand how to get r and theta now, but i was wondering if it is a vector and why is there the 'e'?

  18. joemath314159
    • 3 years ago
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    Once you know the modulus and argument of a complex number, you know what r and theta are. Euler's formula says:\[e^{i\theta}=\cos\theta+i\sin\theta\]

  19. joemath314159
    • 3 years ago
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    Note that the modulus of e^(itheta)=1 since:\[|\cos\theta+i\sin\theta|=\cos ^2\theta+\sin^2\theta=1\]So the polar form of a complex number\[re^{i\theta}\]it really just saying, "i want a complex number whose length is r, in the direction of theta"

  20. manjuthottam
    • 3 years ago
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    oh i get it now!! I really appreciate you helping me!! Thank you so much!!

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