# Help with calculating complex modulus

Polar form $z=re^{i\varphi}$, in which $\varphi \in (-\pi, \pi)$. Now $z$ is defined as:

$$z=\frac{4i-2}{5i+4}$$

I would like to know how to calculate $|z|$. You can plug this into wolframalpha and get correct result but it doesn't help understanding what's going on here. Correct answer would be: $$|z|=2\sqrt{\frac{5}{41}}$$

There is formula i found $|z|=\sqrt{x^2+y^2}$ but i dont know how to apply it here.

Another thing is that what would be $\tan \varphi$. This is somehow related to polar form i suppose since $\varphi \in (-\pi,\pi)$ but again i dont have good understanding of what is polar form nor how does it work.

• $\left| \dfrac uv \right| = \dfrac{|u|}{|v|}$ Commented Oct 5, 2017 at 15:22
• We know that $|z|=r$. $\tan\varphi$ would be: $$\tan \varphi = \frac{r \sin \varphi}{r \cos \varphi}$$. We know value for r which is $\frac{\sqrt{20}}{\sqrt{41}}$ so how do you calculate value for $\tan \varphi$ now ?
– Tuki
Commented Oct 5, 2017 at 16:04

$$|z| = \left| \frac{4i-2}{5i+4} \right| = \frac{ |4i-2| }{ |5i+4| } = \frac{\sqrt{4^2 + 2^2}}{\sqrt{5^2 + 4^2}} = \frac{\sqrt{20}}{\sqrt{41}}.$$

• $$=\frac{2\sqrt{205}}{41}$$ if anyone is curious Commented Oct 5, 2017 at 16:18
• I just ran the numbers $$\frac{\sqrt{20}}{\sqrt{41}}\approx 0.698$$ $$\frac{2\sqrt{205}}{41}\approx0.698$$ $$\frac{2\sqrt5}{\sqrt4}\approx2.24$$ $$2\sqrt{\frac{5}{41}}\approx0.698$$ so I don’t know what you’re implying… Commented Oct 5, 2017 at 21:04
• More about simplest radical form Commented Oct 5, 2017 at 21:08

$$z=\frac{4i-2}{5i+4}$$

$$|z|= \sqrt{\dfrac{16+4}{25+16}}=\sqrt{\dfrac{20}{41}}$$

\begin{align} z &= \frac{4i-2}{5i+4} \\ &= \frac{4i-2}{5i+4} \cdot \frac{-5i+4}{-5i+4} \\ &= \dfrac{12}{41} + \dfrac{26}{41}i \\ &= |z| \ \dfrac{6+13i}{\sqrt{205}} \end{align}

So $\tan \varphi = \dfrac{13}{6}$