Question

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

Answer 1

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

Answer 2

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

Answer 3

where does a and b go?

Answer 4

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

Answer 5

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

Answer 6

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

Answer 7

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

Answer 8

that is correct

Answer 9

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

Answer 10

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)

Answer 11

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) ?

Answer 12

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

Answer 13

oh ok :)

Answer 14

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

Answer 15

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?

Answer 16

i understand how to get r and theta now, but i was wondering if it is a vector and why is there the 'e'?

Answer 17

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

Answer 18

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"

Answer 19

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