complex number plane

**kalemale** · September 9th 2012, 07:36 AM

(a+bi)^2=i What is the squareroot of i solve geometrically in a complex number plane

**Plato** · September 9th 2012, 07:47 AM

Originally Posted by kalemale

(a+bi)^2=i What is the squareroot of i solve geometrically in a complex number plane

There are two square roots of $i$ : $\exp \left( {\frac{{i\pi }}{4}} \right)\& \;\exp \left( {\frac{{ - 3i\pi }}{4}} \right)$

**kalemale** · September 9th 2012, 08:01 AM

I do not really understand how do you get to that?

**Prove It** · September 9th 2012, 08:14 AM

Originally Posted by kalemale

(a+bi)^2=i What is the squareroot of i solve geometrically in a complex number plane

One method:

$\displaystyle \begin{align*} (a + b\,i)^2 &= i \\ a^2 - b^2 + 2ab\,i &= 0 + 1i \\ a^2 - b^2 = 0 \textrm{ and } 2ab &= 1 \end{align*}$

From the second of these equations we have $\displaystyle \begin{align*} b = \frac{1}{2a} \end{align*}$ . Substituting into the first equation we have

$\displaystyle \begin{align*} a^2 - \left(\frac{1}{2a}\right)^2 &= 0 \\ a^2 - \frac{1}{4a^2} &= 0 \\ a^2 &= \frac{1}{4a^2} \\ 4a^4 &= 1 \\ a^4 &= \frac{1}{4} \\ a &= \pm \frac{\sqrt{2}}{2} \end{align*}$

Back-substituting gives

$\displaystyle \begin{align*} b &= \frac{1}{2\left( \pm \frac{\sqrt{2}}{2}\right) } \\ &= \frac{1}{ \pm \sqrt{2}} \\ &= \pm \frac{\sqrt{2}}{2} \end{align*}$

Therefore the solutions to your equation are $\displaystyle \begin{align*} \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i \end{align*}$ and $\displaystyle \begin{align*} -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} i \end{align*}$ .

These are consistent with the solutions given by Plato, who probably used the Exponential form of each number and Index Laws as an alternative method.

**kalemale** · September 9th 2012, 08:16 AM

Yeah but I need to solve in a complex number plane. Sorry if I am slow but your solutions is equational?

**Plato** · September 9th 2012, 08:17 AM

Originally Posted by kalemale

I do not really understand how do you get to that?

Do you understand that $\exp(i\theta)=\cos(\theta)+i\sin(\theta)~?$

Do you know how to do $\left(\cos(\theta)+i\sin(\theta)\right)^2~?$

**Prove It** · September 9th 2012, 08:24 AM

Originally Posted by kalemale

Yeah but I need to solve in a complex number plane. Sorry if I am slow but your solutions is equational?

I don't understand what the problem is. You will always need to use some algebra to at least find one solution. If you wish to use more geometry after that, you can use the fact that there are as many "roots" as its power (i.e. square roots have 2 solutions, cube roots have 3 solutions, fourth roots have 4 solutions, etc) and they are all evenly spaced around a circle centred at the origin.

**kalemale** · September 9th 2012, 08:25 AM

Thanks for all the help!

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