# Homework Help: Simplest way to approximate sqrt of complex numbers

1. Feb 27, 2010

### 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.

2. Feb 27, 2010

### rock.freak667

You could just do a general case for x+iy

$$\sqrt{x+iy} = a+bi \Rightarrow x+iy = (a+bi)^2$$

and 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'.

3. Feb 28, 2010

### 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 $\theta= arctan(y/x)$