Square root of i


By Brad Rodgers (P1930) on Friday, April 28, 2000 - 03:01 am :

What is the square root of i?
I know it's not i, as then i=-1 and 1=-1.

Brad


By Dan Goodman (Dfmg2) on Friday, April 28, 2000 - 04:33 am :

Try this approach: you know the answer is a complex number of the form a+ib where a and b are real numbers, so you want to solve (a+ib)2 =i. So you need to solve a2 -b2 =0 and 2ab=1. This gives the correct solution very quickly of sqrt(i)=±(1+i)/sqrt(2). This method (which I discussed in the thread "Imaginary Numbers" ) usually works for questions like this.

Another method is to use De Moivre's formula (cos(q)+isin(q))n=cos(nq)+isin(nq). Put n=2 and q = 45 and you get (cos45+isin45)2=cos90+isin90 = i, so Öi=cos45+isin45. You also know that cos2 x+sin2 x=1, and cos45 = sin45 (draw a picture), so cos45 = sin45 = 1/Ö2.