HOW TO DIVIDE COMPLEX NUMBERS IN RECTANGULAR FORM

To divide the complex number which is in the form

(a + ib)/(c + id)

we have to multiply both numerator and denominator by  the conjugate of the denominator.

That is,

[ (a + ib)/(c + id) ] ⋅ [ (c - id) / (c - id) ]

 =  [ (a + ib) (c - id) / (c + id) (c - id) ]

Example 1 :

Dividing the complex number (3 + 2i) by (2 + 4i)

Solution :

(3 + 2i) by (2 + 4i)  =  (3 + 2i) /(2 + 4i)

Conjugate of 2 + 4i is 2 - 4i

  =  [(3 + 2i) /(2 + 4i)] ⋅ [(2 - 4i)/(2 - 4i)]

  =  [(3 + 2i)(2 - 4i) /(2 + 4i) (2 - 4i)]

Simplifying the numerator, we get

(3 + 2i)(2 - 4i)  =  6 - 12i + 4i - 8i²

  =  6 - 8i - 8(-1)

  =  6 - 8i + 8

  =  14 - 8i

Simplifying the denominator , we get

(2 + 4i) (2 - 4i)  =  2² - (4i)²

  =  4 - 16i²

=  4 - 16(-1)

=  4 + 16

=  20

(3 + 2i)(2 - 4i) /(2 + 4i) (2 - 4i)  =  (14 - 8i)/20

Example 2 :

Divide the complex number (2 + 3i) by (3 - 2i)

Solution :

(2 + 3i) by (3 - 2i)  =  (2 + 3i) / (3 - 2i)

Conjugate of 3 - 2i is 3 + 2i

  =  [(2 + 3i) / (3 - 2i)] ⋅ [(3 + 2i) / (3 + 2i)]

  =  [(2 + 3i)(3 + 2i) / (3 - 2i) (3 + 2i)]

Simplifying the numerator, we get

(2 + 3i)(3 + 2i)  =  6 + 4i + 9i + 6i²

=  6 + 13i + 6(-1)

=  6 + 13i - 6

=  13i

Simplifying the denominator , we get

(3 - 2i) (3 + 2i) =  3² - (2i)²

  =  9 - 4i²

=  9 - 4(-1)

=  9 + 4

=  13

(2 + 3i)(3 + 2i) / (3 - 2i) (3 + 2i)  =  13i/13

  =  i

Example 3 :

Divide the complex number (7 - 5i) by (4 + i)

Solution :

(7 - 5i) by (4 + i)  =  (7 - 5i) / (4 + i)

Conjugate of (4 + i) is (4 - i)

  =  [(7 - 5i) / (4 + i)] ⋅ [(4 - i) / (4 - i)

  =  [(7 - 5i) (4 - i) / (4 + i) (4 - i)]

Simplifying the numerator, we get

(7 - 5i) (4 - i)  =  28 - 7i - 20i + 5i²

=  28 - 27i + 5(-1)

=  28 - 5 - 27i

=  23 - 27i

Simplifying the denominator , we get

 (4 + i) (4 - i)  =  4² - i²

  =  16  + 1

=  17

(7 - 5i) (4 - i) / (4 + i) (4 - i)  =  (23 - 27i)/17

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