anonymous
  • anonymous
COMPLEX VARIABLES: Can someone explain how to find the Arg z and what does it mean?
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
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anonymous
  • anonymous
If I give you a complex number in \[a+bi\]form, do you know how to convert it to its "polar" form\[re^{i\theta}\]?
anonymous
  • anonymous
Because when written in its polar form:\[\arg(re^{i\theta})=\theta\]
anonymous
  • anonymous
where does a and b go?

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anonymous
  • anonymous
I don't know how to convert it to its polar form, could you help me understand that formula?
anonymous
  • anonymous
|dw:1359087681895:dw|
anonymous
  • anonymous
r is the length of the complex number, given by using pythagorean theorem:\[r=\sqrt{a^2+b^2}\]
anonymous
  • anonymous
the angle is the argument. You use inverse tangent to get it.\[\theta=\tan ^{-1}(\frac{a}{b})\]
anonymous
  • anonymous
oh so r = modulus of the point (a,b)?
anonymous
  • anonymous
that is correct
anonymous
  • anonymous
so when asked arg z, it is the theta that they are asking for?
anonymous
  • anonymous
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)
anonymous
  • anonymous
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) ?
anonymous
  • anonymous
ack! thanks for pointing out my mistake, it should be b/a in the formula i posted.
anonymous
  • anonymous
oh ok :)
anonymous
  • anonymous
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
anonymous
  • anonymous
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?
anonymous
  • anonymous
i understand how to get r and theta now, but i was wondering if it is a vector and why is there the 'e'?
anonymous
  • anonymous
Once you know the modulus and argument of a complex number, you know what r and theta are. Euler's formula says:\[e^{i\theta}=\cos\theta+i\sin\theta\]
anonymous
  • anonymous
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"
anonymous
  • anonymous
oh i get it now!! I really appreciate you helping me!! Thank you so much!!

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