Simplest way to approximate sqrt of complex numbers

[jkg0]

I have a ton of homework with square roots of complex numbers. Like sqrt(2 + 3i)

What is the fastest way to break these down into its approximates like 1.67 + 0.895i without using a TI89/Maple/Matlab/Mathmatica.

 

[rock.freak667]

You could just do a general case for x+iy

\sqrt{x+iy} = a+bi \Rightarrow x+iy = (a+bi)^2and then just equate the real and imaginary parts. Though you will have to solve for 'a' and 'b'. Then you could just either put the numbers into a spreadsheet or calculate by hand when you get 'a' and 'b' in terms of 'x' and 'y'.

 

[HallsofIvy]

Or use the fact that \sqrt{r(cos(\theta)+ i sin(\theta))}= \sqrt{r}(cos(\theta/2)+ i sin(\theta/2)).

For x+ iy, r= \sqrt{x^2+ y^2} and \theta= arctan(y/x)