1. ## Complex Argument?

Hi,

I am currently doing an online self learn course and seem to have come to a bump in the exercises.

Here is the question.

Indicate if the statement is True or False. If true prove it, if false give a counter example.

arg z1z2 = arg z1 + arg z2 if z1,z2 is not equal to 0.

I have never encountered 'arg' before and have no idea what to make of it. I am currently googling away but some explanation would be helpful. No need to help me solve this, just an explanation of what 'arg z1z2' is would mean a lot!. I can't even find any reference to this in the given book or study guide.

Thanks
Rio

2. ## Re: Complex Argument?

Originally Posted by Crzyrio
Hi,

I am currently doing an online self learn course and seem to have come to a bump in the exercises.

Here is the question.

Indicate if the statement is True or False. If true prove it, if false give a counter example.

arg z1z2 = arg z1 + arg z2 if z1,z2 is not equal to 0.

I have never encountered 'arg' before and have no idea what to make of it. I am currently googling away but some explanation would be helpful. No need to help me solve this, just an explanation of what 'arg z1z2' is would mean a lot!. I can't even find any reference to this in the given book or study guide.

Thanks
Rio
The argument (arg) of a complex number is the angle made with the positive real axis when taken in the anticlockwise direction. Usually we use the principal argument (i.e. within the first revolution) which is defined to be in \displaystyle \begin{align*} (-\pi, \pi] \end{align*}.

3. ## Re: Complex Argument?

Originally Posted by Crzyrio
Hi,

I am currently doing an online self learn course and seem to have come to a bump in the exercises.

Here is the question.

Indicate if the statement is True or False. If true prove it, if false give a counter example.

arg z1z2 = arg z1 + arg z2 if z1,z2 is not equal to 0.

I have never encountered 'arg' before and have no idea what to make of it. I am currently googling away but some explanation would be helpful. No need to help me solve this, just an explanation of what 'arg z1z2' is would mean a lot!. I can't even find any reference to this in the given book or study guide.

Thanks
Rio
According to De Moivre any complex number $z \ne 0$ can be written as $z=e^ {\alpha + i\ \theta}$ where...

$\alpha= \ln |z|\ ,\ \theta= \angle (z)$ (1)

From (1) You derive...

$\alpha = \text{Re} \{\ln z\}\ ,\angle (z)= \text{Im} \{\ln z\}$ (2)

Now for the basic property of the logarithm function if $z_{1} \ne 0$ and $z_{2} \ne 0$ are two complex variables, then...

$\ln (z_{1}\ z_{2})= \ln z_{1}+ \ln z_{2}= \alpha_{1}+\alpha_{2} + i\ (\angle_{1} + \angle_{2})$ (3)

Kind regards

$\chi$ $\sigma$

4. ## Re: Complex Argument?

Thank you for the quick reply, but I am still completely lost.

what does arg z1z2 equal too? What does the specific phrase mean ?

5. ## Re: Complex Argument?

The notation $\arg z_1 \cdot z_2$ means the argument of the product of two complex numbers.

Argument is defined as the angle with the horizontal axis, and so the assertion that $\arg z_1 \cdot z_2 = \arg z_1 + \arg z_2$ is saying that when you multiply two complex numbers, the result is basically a rotation consisting of the sum of the angles.

6. ## Re: Complex Argument?

Originally Posted by MarceloFantini
The notation $\arg z_1 \cdot z_2$ means the argument of the product of two complex numbers.

Argument is defined as the angle with the horizontal axis, and so the assertion that $\arg z_1 \cdot z_2 = \arg z_1 + \arg z_2$ is saying that when you multiply two complex numbers, the result is basically a rotation consisting of the sum of the angles.
Thank You, that makes a lot more sense. I can see how previous posters were trying to say something similar as well. Thanks to all.

Just so I know I understand this fully. arg z1*z2 is like saying 2CIS(40) * 2CIS(22) = 1CIS(30) + 2CIS(33) . Not saying the statement is mathematically correct but that is what the question is implying?

Also how do you add in the terms like the way you guys are doing it?

7. ## Re: Complex Argument?

I'm afraid I didn't follow. What did you mean with 2CIS(40) * 2CIS(22) = 1CIS(30) + 2CIS(33)? Let's take some specific examples: if $z_1 = i$ and $z_2 = \frac{\sqrt{3}}{2} + i \frac{1}{2}$, we have that:

$z_1 \cdot z_2 = i \cdot \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \frac{-1}{2} + i \frac{\sqrt{3}}{2}$.

We see that $\arg z_1 \cdot z_2 = \frac{2 \pi}{3}$ or 120 degrees. Notice that $\arg z_1 = \frac{\pi}{2}$ and $\arg z_2 = \frac{\pi}{6}$, and so $\arg z_1 + \arg z_2 = \frac{\pi}{2} + \frac{\pi}{6} = \frac{2 \pi}{3} = \arg z_1 \cdot z_2$.

The underlying thought is this: a complex number can be represented by a vector starting at the origin, and thus he has an angle within $[0, 2 \pi)$. When you multiply two complex numbers, the result will always sum the angles and their magnitudes will be multiplied.

In our case, if instead of $z_1 = i$ and $z_2 = \frac{\sqrt{3}}{2} + i \frac{1}{2}$ we had $z_1 = 2i$ and $z_2 = \sqrt{3} + i$, the result would be $z_1 \cdot z_2 = -2 + 2i \sqrt{3}$, showing that magnitudes multiplied, however the sum is unchanged.

I hope this clarified it further.

8. ## Re: Complex Argument?

Originally Posted by MarceloFantini
I'm afraid I didn't follow. What did you mean with 2CIS(40) * 2CIS(22) = 1CIS(30) + 2CIS(33)? Let's take some specific examples: if $z_1 = i$ and $z_2 = \frac{\sqrt{3}}{2} + i \frac{1}{2}$, we have that:

$z_1 \cdot z_2 = i \cdot \left( \frac{\sqrt{3}}{2} + i \frac{1}{2} \right) = \frac{-1}{2} + i \frac{\sqrt{3}}{2}$.

We see that $\arg z_1 \cdot z_2 = \frac{2 \pi}{3}$ or 120 degrees. Notice that $\arg z_1 = \frac{\pi}{2}$ and $\arg z_2 = \frac{\pi}{6}$, and so $\arg z_1 + \arg z_2 = \frac{\pi}{2} + \frac{\pi}{6} = \frac{2 \pi}{3} = \arg z_1 \cdot z_2$.

The underlying thought is this: a complex number can be represented by a vector starting at the origin, and thus he has an angle within $[0, 2 \pi)$. When you multiply two complex numbers, the result will always sum the angles and their magnitudes will be multiplied.

In our case, if instead of $z_1 = i$ and $z_2 = \frac{\sqrt{3}}{2} + i \frac{1}{2}$ we had $z_1 = 2i$ and $z_2 = \sqrt{3} + i$, the result would be $z_1 \cdot z_2 = -2 + 2i \sqrt{3}$, showing that magnitudes multiplied, however the sum is unchanged.

I hope this clarified it further.
Yes that makes a lot of sense. Sorry my example given was in polar form.

Thank you very much

EDIT : Removed confusing question I asked, I think i get the concept. Thanks