HOW TO WRITE THE GIVEN  COMPLEX NUMBER IN RECTANGULAR FORM

Question :

Write the following in the rectangular form:

(i)  [(5 + 9i) + (2 − 4i)] whole bar

Solution : 

[(5 + 9i) + (2 − 4i)] whole bar  =  (5 + 9i) bar + (2 − 4i) bar

  =  (5 - 9i) + (2 + 4i)

  =  (5 + 2) + (-9i + 4i)

  =  7 - 5i

(ii)  (10 - 5i)/(6 + 2i)

Solution :

(10 - 5i)/(6 + 2i)

Multiplying both numerator and denominator by the conjugate of of denominator, we get 

  =   [(10 - 5i)/(6 + 2i)] [(6 - 2i)/(6 - 2i)]

  =  (10 - 5i) (6 - 2i) / (36 - 4(-1))

  =  (60 - 20i - 30i + 10 (-1)) / (36 + 4)

  =  (60 - 20i - 30i - 10) / 40

  =  (50 - 50i) / 40

  =  (5 - 5i)/4

  =  (5/4) (1 - i)

(iii)  3i bar  + 1/(2 - i)

Solution :

3i bar  + 1/(2 - i)

  =  - 3i + { (1/(2 - i)) ((2 + i)/(2 + i)) }

  =  - 3i + { (2 + i)/(4 - (-1)) }

  =  - 3i + { (2 + i)/5 }

  =  (2/5) + {(1/5) - 3}

  =  (2/5) - (14/5)

Question 2 :

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

(i)  Re (1/z)

Solution :

z = x + iy

1/z  =  1/(x + iy)  

  =  [1/(x + iy)]  [(x - iy)/(x - iy)]

  =  (x - iy)/(x² + y²)

 =  (x/(x² + y²)) - i (y/(x² + y²))

Hence the Re (1/z) is (x/(x² + y²)) - i (y/(x² + y²)).

(ii)  Re (i z bar)

Solution :

  =  i (x + iy) bar

  =  i(x - iy)

  =  ix - i²y

  =  ix + y

Hence the Re (i z bar) is y.

(iii)  Im(3z + 4zbar − 4i)

Solution :

(3z + 4zbar − 4i)  =  [3(x + iy) + 4(x + iy) bar - 4i]

  =  3x + i3y + 4(x - iy) - 4i]

  =  3x + i3y + 4(x - iy) - 4i

  =  3x + i3y + 4x - i4y - 4i

  =  (3x + 4x) + i(-4y + 3y - 4)

  =  (3x + 4x) + i(-y - 4)

Hence the value of Im(3z + 4zbar − 4i) is - y - 4.

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