MathCalculate z+1/z if z=(-1+i*3^1/2)/2.  

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have z = (-1+i*3^1/2)/2

z + 1/z has to be determined.

z + 1/z = (-1+i*3^1/2)/2 + 1 / (-1+i*3^1/2)/2

=> (-1+i*3^1/2)/2 + 2/(-1+i*3^1/2)

=> (-1 + i*sqrt 3)/2 + 2/(-1 + i*sqrt 3)

=> [(-1 + i*sqrt 3)(-1 + i*sqrt 3) + 4]/(2*(-1 + i*sqrt 3))

=> (-2 - 2*sqrt 3*i + 4)/(2*(-1 + i*sqrt 3))

=> (1 - i*sqrt 3*)/(-1 + i*sqrt 3)

=> -1

z + 1/z = -1

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Since 1/z = z' or the conjugate of the complex number z, we'll put

z' = (-1-i*sqrt3)/2 (cubic roots of the unit)

Now, we'll add z + z' = 2Re(z)

We'll identify Re(z) = x:

z = (-1+i*sqrt3)/2 for z = x + i*y

We'll compare and we'll get:

Re(z) = -1/2

z + 1/z = 2Re(z)

z + 1/z = -2/2

z + 1/z = -1

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