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Complex variable question

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I have been given a question, one that should be fairly simple but I cant seem to figure this one out and not to sure where to start. The question is to solve the equation z^4 = -1 where z is a complex variable. They also state that modulus argument form is needed to solve this. I know this is |z| which is equal to 1 obviously but i don't know where to go from there. I know the answer is +- (1/sqrt(2)) +- (1/sqrt(2))i but that's cause they gave us the answer. i understand that if you apply the modulus form to either of the answers you get 1. What i don't understand is how to get to the answer from z^4 = -1. Not to sure if we're supposed to just see that or if there is a more practical way of solving this that will help me to make sense of it
 
in general fourth order polynomial equations are not trivial to solve and there may not be a general method for the case you have. But, this type of problem is fairly easy once you have seen them a few times. You know there are 4 solutions because it is a fourth order equation. Anything that looks lil z^n=C will have n solutions which are evenly spaced around a circle in the complex plane, and that circle has a radius of the n'th root of |C|. So, once you find one solution, the remaining n-1 solutions are easily found by evenly spacing the solutions around the circle. You can always find one solution considering that z equals the n'th root of C. When C=1, this is really easy because one answer is obviously z=1, and the rest are evenly spaced on the circle of radius equal to one. It's also easy if C is real and positive.

But, generally you know that z equals the n'th root of the magnitude of C times exp(jt/n), where j=sqrt(-1) and t is the angle of C.
 
I have been given a question, one that should be fairly simple but I cant seem to figure this one out and not to sure where to start. The question is to solve the equation z^4 = -1 where z is a complex variable. They also state that modulus argument form is needed to solve this. I know this is |z| which is equal to 1 obviously but i don't know where to go from there. I know the answer is +- (1/sqrt(2)) +- (1/sqrt(2))i but that's cause they gave us the answer. i understand that if you apply the modulus form to either of the answers you get 1. What i don't understand is how to get to the answer from z^4 = -1. Not to sure if we're supposed to just see that or if there is a more practical way of solving this that will help me to make sense of it

Noodles141,

Yes, put the equation into polar form and use De Moivre's theorem.

Noodles141.JPG


Ratch
 
When I saw this problem, the Z^4 screamed out to be factorised into Z^2 * Z^2.
Z^4 = Z^2 *Z^2 = j * j
Therefore Z^2 = +j, and -j
Z= sqrt j and sqrt -j
Since j = 1 @ pi/2 and -j= 1 @-pi/2
Then sqrt j = 1 @pi/4 and 1 @ pi*3/4 (45 and 135 degree)
and sqrt -j = 1 @pi *5/4 and 1 @ -pi/4 (-45 and -135 degree)
 
When I saw this problem, the Z^4 screamed out to be factorised into Z^2 * Z^2.
Z^4 = Z^2 *Z^2 = j * j
Therefore Z^2 = +j, and -j
Z= sqrt j and sqrt -j
Since j = 1 @ pi/2 and -j= 1 @-pi/2
Then sqrt j = 1 @pi/4 and 1 @ pi*3/4 (45 and 135 degree)
and sqrt -j = 1 @pi *5/4 and 1 @ -pi/4 (-45 and -135 degree)

"Then sqrt j = 1 @pi/4 and 1 @ pi*3/4 (45 and 135 degree)"

Not true, sqrt(j) = 1/_45°, not 1/_135°. 135° gives a negative real value, which sqrt(j) does not have.

Ratch
 
apologies to ratch and thanks for the comment.
The root of +j is at 1, pi/4 and 1, 5pi/4.
The root of -j is 1, 3pi/4 and 1, -pi/4.
rumpfy.
 
I have been given a question, one that should be fairly simple but I cant seem to figure this one out and not to sure where to start. The question is to solve the equation z^4 = -1 where z is a complex variable. They also state that modulus argument form is needed to solve this. I know this is |z| which is equal to 1 obviously but i don't know where to go from there. I know the answer is +- (1/sqrt(2)) +- (1/sqrt(2))i but that's cause they gave us the answer. i understand that if you apply the modulus form to either of the answers you get 1. What i don't understand is how to get to the answer from z^4 = -1. Not to sure if we're supposed to just see that or if there is a more practical way of solving this that will help me to make sense of it

"They also state that modulus argument form is needed to solve this". No idea what this means, so this may not be apropos. Sometimes, teachers will call it a "wrong" answer, even if it's right, but you solved it in a manner they didn't like. SUX, but what can you do?

The key to solving: z^4= -1 is to remember that: -1= 1.0/_180deg (Steinmetz representation where it's understood that this stands for: 1.0 X e^(j * pi) and Euler's Constant and j are understood, and that degrees are preferred since they're whole numbers, but that the only arguments to trig functions are in radians.)

Since it's a fourth order polynomial, we need four answers. To get them, modify accordingly:

z^4= 1.0/_(180 + 360n) Going 360deg (2pi rad) always brings you right back where you started. Then extract the fourth root of the right side of the equation:

z= [1.0/_(180 + 360n)]^0.25
1.0^0.25= 1.0
180*0.25= 45
360*0.25= 90

z= 1.0/_(45 + 90n)
z_0= 1.0/_45= (1/sqrt(2)) + j(1/sqrt(2))
z_1= 1.0/_(45 + 90)= -(1/sqer(2)) + j(1/sqrt(2))
z_2= 1.0/_(45 + 180)= -(1/sqrt(2)) - j(1/sqrt(2))
z_3= 1.0/_(45 + 270)= (1/sqrt(2)) - j(1/sqrt(2))

Four answers, and done. If you need this in Cartesian form, you can extract the answer via Euler's Identity, or get the answer from a scientific calculator. You just need to figure once, since these answers appear as complex conjugates.
 
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