# Thread: complex number

1. ## complex number

Given that z=-2+i , express$\displaystyle z+\frac{1}{\sqrt{z}}$ in the form of a+bi

2. Write -2 + i as x + i y ; r = (x^2 + y^2)^ (1/2)

Z = x + i y = r e^ i theta = r ( cos theta + i sin theta) from De Moivre;s
Theorem

where sin theta = y / (x^2 + y^2)^ (1/2) and

cos theta = x / (x^2 + y^2)^ (1/2)

Just substitute for Z + 1/Z and remember sin(-theta) = - sin theta.

3. Is there a simpler way of doing this ?

This is what i tried :

$\displaystyle \sqrt{-2+i}=a+bi$

$\displaystyle -2+i=a^2-b^2+2abi$

$\displaystyle -2=a^2-b^2$ and $\displaystyle 2ab=1$

i am not sure whether i am correct up to this step but after solving , i get weird values for a and b ..

4. Actually the problem is because there are roots of a complex no involved I think and we have to involve the unit circle etc.

I tried it too . I got 4a^4 + 8a^2 - 1 = 0 . Put a^2 = x and then solve

4x^2 + 8 x - 1 = 0 to get x = -1 +or- 1/2 sqrt 5

Choose the +ve value and x = 0.118 or a = 0.3435 . Similarly try for b.
We have to ignore the value which is -ve else we end up with a being a complex no starting from the assumption it is not!

5. Originally Posted by thereddevils
Given that z=-2+i , express $\displaystyle z+\frac{1}{\sqrt{z}}$ in the form of a+bi
Just do the simplification:

$\displaystyle -2\, +\, i\, +\, \frac{1}{-2\, +\, i}$

$\displaystyle -2\, +\, i\, +\, \left(\frac{1}{-2\, +\, i}\right)\left(\frac{-2\, -\, i}{-2\, -\, i}\right)$

$\displaystyle -2\, +\, i\, +\, \frac{-2\, -\, i}{(-2)^2\, -\, i^2}$

$\displaystyle -2\, +\, i\, +\, \frac{-2\, -\, i}{4\, +\, 1}$

$\displaystyle -2\, +\, i\, +\, \frac{-2\, -\, i}{5}$

$\displaystyle -2\, -\, \frac{2}{5}\, +\, i\, -\, \frac{1}{5}i$

$\displaystyle -\frac{12}{5}\, +\, \frac{4}{5}i$

6. You seem to have solved for Z + 1/Z and not Z + 1/sqrt Z

7. Originally Posted by thereddevils
Given that z=-2+i , express$\displaystyle z+\frac{1}{\sqrt{z}}$ in the form of a+bi
$\displaystyle \sqrt z = \sqrt { - 2 + i} = a + bi,$

$\displaystyle \therefore - 2 + i = (a + bi)^2 = a^2 + 2abi - b^2 ........\langle 1\rangle$
$\displaystyle a^2 - b^2 = - 2,2ab = 1 \Rightarrow b^2 = \frac{1} {{4a^2 }}$

$\displaystyle \text{Substitute into }\langle 1\rangle \text{: }4a^4 + 8a^2 - 1 = 4(a^2 )^2 + 8a^2 - 1\text{ }.........\langle \text{2}\rangle$

$\displaystyle \text{Put }a^2 = x \Rightarrow \langle \text{2}\rangle = 4x^2 + 8x - 1$
$\displaystyle \text{Set }\langle \text{2}\rangle = 0\text{ and solve: }a^2 = - 1 + \frac{1} {2}\sqrt 5 \Rightarrow a = \sqrt { - 1 + \frac{1} {2}\sqrt 5 } \text{ (only take + ve part)}$
$\displaystyle \text{From }2ab = 1\text{, }b = \frac{1} {{2a}} = \frac{1} {2}\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{ - \tfrac{1} {2}}$

$\displaystyle \therefore z = - 2 + i = (a + bi)^2 =$

$\displaystyle = \left[ {\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{\tfrac{1} {2}} + \frac{1} {2}\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{ - \tfrac{1} {2}} i} \right]^2$

$\displaystyle \text{and }z + \frac{1} {{\sqrt z }} =$
$\displaystyle = \left[ {\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{\tfrac{1} {2}} + \frac{1} {2}\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{ - \tfrac{1} {2}} i} \right]^2 +$
$\displaystyle + \left[ {\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{\tfrac{1} {2}} + \frac{1} {2}\left( { - 1 + \frac{1} {2}\sqrt 5 } \right)^{ - \tfrac{1} {2}} i} \right]^{ - 1}$

$\displaystyle \text{which we need to reduce in the form of }a + bi.\text{ Any hints are highly appreciated}\text{.}$