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
Example 6 :
Which of the following is equal to
√-1 - √-4 + √-9 ?
a) i b) 2i c) 3i d) 4i
Solution :
= √-1 - √-4 + √-9
= i - i √4 + i√9
= i - i √(2 x 2) + i√(3 x 3)
= i - 2i + 3i
= 4i - 2i
= 2i
So, the answer is option b.
Example 7 :
If (4 + i)2 = a + ib, what is the value of a + b ?
Solution :
(4 + i)2 = a + ib
Using algebraic identity (a + b)2 = a2 + 2ab + b2 we get
42 + 2(4)(i) + i2 = a + ib
16 + 8i - 1 = a + ib
15 + 8i = a + ib
Comparing the corresponding terms, we get
a = 15 and b = 8
Example 8 :
If the expression (3 - i) / (1 - 2i) is rewritten in the form a + ib, in which a and b are real numbers, what is the value of a + b?
Solution :
= (3 - i) / (1 - 2i)
Since we have to two complex numbers in the numerator and denominator, we have to multiply both numerator and denominator by the conjugate of the denominator.
Conjugate of 1 - 2i is 1 + 2i
= [(3 - i) / (1 - 2i)] [(1 + 2i) / (1 + 2i)]
= (3 - i)(1 + 2i) / (1 - 2i) (1 + 2i)
= (3 + 6i - i - 2i2) / (12 - (2i)2)
= (3 + 5i - 2(-1)) / (12 - 4i2)
= (3 + 5i + 2) / (1 - 4(-1))
= (5 + 5i) / (1 + 4)
= (5 + 5i) / 5
= (5/5) + (5i/5)
= 1 + i
a = 1, b = 1
a + b = 1 + 1
= 2
Example 9 :
Which of the following complex numbers is equivalent to
(1 - i)2/ (1 + i)
a) -i/2 - 1/2 b) -i/2 + 1/2 c) -i - 1 d) -i + 1
Solution :
= (1 - i)2/ (1 + i)
= (12 - 2i + i2) / (1 + i)
= (1 - 2i - 1) / (1 + i)
= - 2i / (1 + i)
= [- 2i/(1 + i)] x [(1 - i) / (1 - i)]
= -2i (1 - i) / (1 + i)(1 - i)
= -2i(1 - i) / (12 - i2)
= -2i(1 - i) / (1 + 1)
= -2i(1 - i) / 2
= -i(1 - i)
= -i + i2
= -i - 1
= -1 - i
So, option c is correct.
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