Conjugate Property of Complex Numbers Practice Worksheet :
Here we are going to see some example problems to understand conjugate properties of complex numbers .
(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
(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
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