CONJUGATE PROPERTY OF COMPLEX NUMBERS PRACTICE WORKSHEET

(1)  Write the following in the rectangular form:

(i)  [(5 + 9i) + (2 − 4i)] whole bar

(ii)  (10 - 5i)/(6 + 2i)

(iii)  3i bar  + 1/(2 - i)            Solution 

(2)  If z = x + iy , find the following in rectangular form.

(i)  Re (1/z)

(ii)  Re (i z bar)

(iii)  Im(3z + 4zbar − 4i)            Solution

(3)  If z₁ = 2 − i and z₂ = −4 + 3i , find the inverse of z₁ z₂ and z₁/z₂               Solution

(4)  The complex numbers u,v , and w are related by (1/u)  =  (1/v) + (1/w) if v  =  3 - 4i and w  =  4 + 3i, find u in rectangular form.                Solution

(5)  Prove the following properties:

(i) z is real if and only if z = z bar

(ii) Re(z) = (z + z bar)/2

Solution

(6)  Find the least value of the positive integer n for which (√3 + i)ⁿ (i) real (ii) purely imaginary.       Solution

(7)  (i)  Show that (i) (2 + i√3)¹⁰ - (2 - i√3)¹⁰ is purely imaginary

(ii)  [(19 - 7i)/(9 + i)]¹² + [(20 - 5i)/(7 - 6i)]¹² is real

Solution

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