Help! All Help Appreciated

z^2 = 2i

z = +- sqrt 2i =   1+i      or  1 -i    or   -1 -i     or   -1 + i

EP is off-line; I'll try to explain it in a way that makes sense to me.

 

To find the four fourth roots of -4:  z  =  (-4)^1/4

 

I'm assuming that you are familiar with  r·cis( theta ) form; if not, this may not make much sense.

 

If you graph z = -4 on the complex plane, you'll notic that its distance from the origin is 4;

this means that  r  =  4

 

You'll also notice that it's directly west; therefore, its angle is  180^o  (if you wish, you can change this into radians).

 

So, the complex number  -4 =  -4 + 0i  =  4cis(180^o)

 

To find the primary fourth root, find the fourth root of r and divide the angle by 4.

So, the primary fourth root of -4  is  4^1/4cis( 180^o/4 )  =  sqrt(2)·cis( 45^o )

To find the other fourth roots, divide 360^o by 4 to get 90^o and add this amount onto the angle

of the primary fourt root three times:

So, your four fourth roots are:

sqrt(2)·cis( 45^o )

sqrt(2)·cis( 45^o + 90^o )  =  sqrt(2)·cis( 135^o )

sqrt(2)·cis( 135^o  + 90^o )  =  sqrt(2)·cis( 225^o )

sqrt(2)·cis( 225^o + 90^o )  =  sqrt(2)·cis( 315^o )

 

sqrt(2)·cis( 45^o )  =  sqrt(2)·cos( 45^o ) + i·sqrt(2)·sin( 45^o )  =  1 + 1i

sqrt(2)·cis( 135^o )  =  -1 + i

sqrt(2)·cis( 225^o )  =  -1 - i

sqrt(2)·cis( 315^o )  =  1 - i