# complex numbers I

#### Punch

##### New member
The complex number w has modulus $$\sqrt{2}$$ and argument $$-\frac{3\pi}{4}$$, and the complex number $$z$$ has modulus $$2$$ and argument $$-\frac{\pi}{3}$$. Find the modulus and argument of $$wz$$, giving each answer exactly.
By first expressing w and $$z$$ is the form $$x+iy$$, find the exact real and imaginary parts of $$wz$$.
I have a problem with finding the argument of $$wz$$ and expressing $$w$$ and $$z$$ in the form $$x+iy$$

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#### Mr Fantastic

##### Member
Re: complex numbers

The complex number w has modulus \sqrt{2} and argument -\frac{3\pi}{4}, and the complex number z has modulus 2 and argument -\frac{\pi}{3}. Find the modulus and argument of wz, giving each answer exactly.
By first expressing w and z is the form x+iy, find the exact real and imaginary parts of wz.
I have a problem with finding the argument of wz and expressing w and z in the form x+iy
Review how to multiply two complex numbers when they are written in polar form.

z = r cis(theta). You need to review polar form.

#### Prove It

##### Well-known member
MHB Math Helper
The complex number w has modulus $$\sqrt{2}$$ and argument $$-\frac{3\pi}{4}$$, and the complex number $$z$$ has modulus $$2$$ and argument $$-\frac{\pi}{3}$$. Find the modulus and argument of $$wz$$, giving each answer exactly.
By first expressing w and $$z$$ is the form $$x+iy$$, find the exact real and imaginary parts of $$wz$$.
I have a problem with finding the argument of $$wz$$ and expressing $$w$$ and $$z$$ in the form $$x+iy$$
If you need to solve this problem by converting to Cartesians, then

\displaystyle \begin{align*} w &= \sqrt{2}\left[\cos{\left(-\frac{3\pi}{4}\right)} + i\sin{\left(-\frac{3\pi}{4}\right)}\right] \\ &= \sqrt{2}\left(-\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right) \\ &= -1 - i \end{align*}

and

\displaystyle \begin{align*} z &= 2\left[\cos{\left(-\frac{\pi}{3}\right)} + i\sin{\left(-\frac{\pi}{3}\right)}\right] \\ &= 2\left(\frac{1}{2} - \frac{i\sqrt{3}}{2} \right) \\ &= 1 - i\sqrt{3} \end{align*}

So multiplying them together gives...

\displaystyle \begin{align*} wz &= \left(-1-i\right)\left(1-i\sqrt{3}\right) \\ &= -1 + i\sqrt{3} - i + i^2\sqrt{3} \\ &= \left(-1 - \sqrt{3}\right) + i\left(-1 + \sqrt{3}\right) \end{align*}

Can you evaluate the modulus and argument of this complex number?