# Thread: modulus and argument

1. ## modulus and argument

find the modulus and arguments of the complex number $(1+\sqrt 3i)^3$.

my solution;
$(1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)$
$= 1+2\sqrt3i+3i^2(1+\sqrt 3i)$
$= -2+2\sqrt3i(1+\sqrt 3i)$
$= -2-2\sqrt3i+2\sqrt3i+2(3)i^2$
$= -2+6i^2$
$= -8$

is there anything wrong with my working above?
to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
then |-8+0i| = 8
and arg |-8+0i| = 0

2. Originally Posted by z1llch
find the modulus and arguments of the complex number $(1+\sqrt 3i)^3$.

my solution;
$(1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)$
$= {\color{red}(}1+2\sqrt3i+3i^2{\color{red})}(1+\sqr t 3i)$
$= {\color{red}(}-2+2\sqrt3i{\color{red})}(1+\sqrt 3i)$
$= -2-2\sqrt3i+2\sqrt3i+2(3)i^2$
$= -2+6i^2$
$= -8$

is there anything wrong with my working above?
Well technically, you need the parentheses I added above.

to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
then |-8+0i| = 8
and arg |-8+0i| = 0
If you look at the trig form of the complex number: $r(\cos \theta + i\sin \theta)$, r is the modulus and theta is the argument. Anyway, your modulus is right, but the argument is not. The argument should be pi.

01

3. Originally Posted by z1llch
find the modulus and arguments of the complex number $(1+\sqrt 3i)^3$.

my solution;
$(1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)$
$= 1+2\sqrt3i+3i^2(1+\sqrt 3i)$
$= -2+2\sqrt3i(1+\sqrt 3i)$
$= -2-2\sqrt3i+2\sqrt3i+2(3)i^2$
$= -2+6i^2$
$= -8$

is there anything wrong with my working above?
to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
then |-8+0i| = 8
and arg |-8+0i| = 0

yeongill has answered ..

4. i do not understand the argument part.. can please explain on it a little bit?

5. Originally Posted by z1llch
find the modulus and arguments of the complex number $(1+\sqrt 3i)^3$.

my solution;
$(1+\sqrt 3i)(1+\sqrt 3i)(1+\sqrt 3i)$
$= 1+2\sqrt3i+3i^2(1+\sqrt 3i)$
$= -2+2\sqrt3i(1+\sqrt 3i)$
$= -2-2\sqrt3i+2\sqrt3i+2(3)i^2$
$= -2+6i^2$
$= -8$

is there anything wrong with my working above?
to find modulus and arguments of the complex number i need to express it in a+bi form first right? or can i assume b=0? being -8+0i
then |-8+0i| = 8
and arg |-8+0i| = 0
ANATHOR RESOLUTION :

THANKS

6. Here are some simple facts the would have made this an easy problem.
$\begin{gathered}
\left| {z^n } \right| = \left| z \right|^n \; \Rightarrow \;\left| {\left( {1 + \sqrt 3 i} \right)^3 } \right| = \left| {1 + \sqrt 3 i} \right|^3 \hfill \\
\arg \left( {z^n } \right) = n\arg (z) \hfill \\
\end{gathered}$

7. Originally Posted by z1llch
i do not understand the argument part.. can please explain on it a little bit?
Have you learned about the trig form of the complex number? Plotting the number -8 + 0i on the complex plane is like plotting the point (-8, 0) on the Cartesian plane. You got an angle in standard position (where the initial side is the positive x-axis and the terminal side is a ray where the point is on), and this angle is pi radians. So the argument is pi. We've already figured that the modulus is 8, so
$-8 + 0i = 8(\cos \pi + i\sin \pi)$.

01