## anonymous 3 years ago COMPLEX VARIABLES: Can someone explain how to find the Arg z and what does it mean?

1. anonymous

If I give you a complex number in $a+bi$form, do you know how to convert it to its "polar" form$re^{i\theta}$?

2. anonymous

Because when written in its polar form:$\arg(re^{i\theta})=\theta$

3. anonymous

where does a and b go?

4. anonymous

I don't know how to convert it to its polar form, could you help me understand that formula?

5. anonymous

|dw:1359087681895:dw|

6. anonymous

r is the length of the complex number, given by using pythagorean theorem:$r=\sqrt{a^2+b^2}$

7. anonymous

the angle is the argument. You use inverse tangent to get it.$\theta=\tan ^{-1}(\frac{a}{b})$

8. anonymous

oh so r = modulus of the point (a,b)?

9. anonymous

that is correct

10. anonymous

so when asked arg z, it is the theta that they are asking for?

11. 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)

12. 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) ?

13. anonymous

ack! thanks for pointing out my mistake, it should be b/a in the formula i posted.

14. anonymous

oh ok :)

15. 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

16. 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?

17. 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'?

18. anonymous

Once you know the modulus and argument of a complex number, you know what r and theta are. Euler's formula says:$e^{i\theta}=\cos\theta+i\sin\theta$

19. 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"

20. anonymous

oh i get it now!! I really appreciate you helping me!! Thank you so much!!