# Thread: complex numbers

1. ## complex numbers

Express the comlex number (√3+i) in trigonometric form. Hence find the smallest positive integer n that (√3+i)^n is a real number.

This is how I have started it however im really not sure if this is right hoping someone could help.

ArgZ = tan^-1 (1/√3) = 30

|Z| = √3+1 = 2

Z = 2(cos30+isin30)

Is this correct??

Also how do i complete the question??

Thanks

2. To finish this problem it is best to use number notation.
$\left( {\sqrt 3 + i} \right) = 2\left( {\cos \left( {\frac{\pi }
{6}} \right) + i\sin \left( {\frac{\pi }{6}} \right)} \right)$
.
Now real numbers have arguments that are integral multiples of pi
So the second answer is n=6.

3. Using DeMoivre,

So you're finding n so that $2^n(cos (nx) + i sin (nx))$ is real.

The only way to do that is if your imaginary part is zero, as in $sin (nx) = 0$.

Since x is 30 degrees, your first option is n=6, as $sin (6*30) = sin (180) = 0$.

4. Originally Posted by Plato
Now real numbers have arguments that are integral multiples of pi
Could you please explain this sentence lol, unsure where the 6 came from :/ thanks

5. still on the topic of complex numbers, could anybody tell me if this is correct?

complex number Z and conjugate Zbar satisfy equation

3Z + Zbar = (11+7i)/(1+i)

find Z in form x + iy

This is how i attempt to answer it but unsure if its correct

let Z = x + iy
Zbar = x - iy

3(x+iy) + (x-iy) = (11+7i)/(1+i)

3 (x+iy) + (x-iy) = 10 + 6i

2(x + iy) = 10 + 6i

(x+iy) = (10 + 6i))/2

x + iy = 5 + 3i

is this correct anybody please???

6. Where are you getting your calculations?
$\frac{{11 + 7i}}{{1 + i}} = \frac{{\left( {11 + 7i} \right)\left( {1 - i} \right)}}
{2} = 9 - 4i$
.

$\begin{gathered} 3z + \overline z = 9 - 2i \hfill \\ 3(x + yi) + (x - yi) = 9 - 2i \hfill \\ 4x + 2yi = 9 - 2i \Rightarrow \quad x = \frac{9}{4}\,\& \,y = - 1 \hfill \\
\end{gathered}$