# HOW TO WRITE THE GIVEN  COMPLEX NUMBER IN RECTANGULAR FORM

How to Write the Given Complex Number in Rectangular Form :

Here we are going to see some example problems to understand writing the given complex number in rectangular form.

## How to Write the Given Complex Number in Rectangular Form - Questions

Question 1 :

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. After having gone through the stuff given above, we hope that the students would have understood, "How to Write the Given Complex Number in Rectangular Form".

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