How to Multiply and Divide Complex Numbers ?
In this section, we will see how to multiply and divide complex numbers.
Multiplying complex numbers :
Suppose a, b, c, and d are real numbers. Then,
Division of complex numbers :
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 :
Multiply the following complex numbers
(2 + 3i) (4 - 7i)
Solution :
(2 + 3i) (4 - 7i) = 2(4) + 2(-7i) + 4(3i) + 3i(-7i)
= 8 - 14i + 12i - 21i2
= 8 - 2i - 21(-1)
= 8 - 2i + 21
= 29 - 2i
Example 2 :
Multiply the following complex numbers
(4 - 2i) (3 - 5i)
Solution :
(4 - 2i) (3 - 5i) = 4(3) + 4(-5i) + 3(-2i) - 2i(-5i)
= 12 - 20i - 6i + 10i2
= 12 - 26i + 10(-1)
= 12 - 10 - 26i
= 2 - 26i
Example 3 :
Multiply the following complex numbers
(-5 + 3i)(-2 + i)
Solution :
(-5 + 3i)(-2 + i) = -5(-2) - 5(i) + 3i(-2) + 3i(i)
= 10 - 5i - 6i + 3i2
= 10 - 11i + 3(-1)
= 10 - 3 - 11i
= 7 - 11i
Example 4 :
Multiply the following complex numbers
(3 - i) (8 + 7i)
Solution :
(3 - i) (8 + 7i) = 3(8) + 3(7i) - i(8) - i(7i)
= 24 + 21i - 8i - 7i2
= 24 + 13i - 7(-1)
= 24 + 13i + 7
= 31 + 13i
Example 5 :
Divide the complex number (3 + 2i) by (2 + 4i)
Solution :
(3 + 2i) by (2 + 4i) = (3 + 2i)/(2 + 4i)
Whenever we have complex numbers in the denominator, we have to multiply the numerator and denominator by the conjugate of the denominator of the given complex number.
= [(3 + 2i)/(2 + 4i)] ⋅[(2 - 4i)/(2 - 4i)]
= [(3 + 2i)(2 - 4i)/(2 + 4i) (2 - 4i)]
Multiplying the numerator, we get
(3 + 2i)(2 - 4i) = 3(2) + 3(-i) + 2i(2) + 2i(-4i)
= 6 - 3i + 4i - 8i2
= 6 - 8(-1) + i
= 6 + 8 + i
= 14 + i
Multiplying the denominator, we get
(2 + 4i) (2 - 4i) = 2(2) + 2(-4i) + 4i(2) + 4i(-4i)
= 4 - 8i + 8i - 16i2
= 4 - 16(-1)
= 4 + 16
= 20
(3 + 2i)/(2 + 4i) = (14 + i)/20
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