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
After having gone through the stuff given above, we hope that the students would have understood how to divide complex numbers in rectangular form.
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