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

1. joemath314159

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

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

3. manjuthottam

where does a and b go?

4. manjuthottam

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

5. joemath314159

|dw:1359087681895:dw|

6. joemath314159

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

7. joemath314159

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

8. manjuthottam

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

9. joemath314159

that is correct

10. manjuthottam

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

11. joemath314159

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

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

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

14. manjuthottam

oh ok :)

15. joemath314159

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

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

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

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

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

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