Complex Numbers
A complex number is a number in the form a + bi where a and b are real
numbers and i = −1 . This form, a + bi is referred to as standard form.
{a + bi | a and b are real numbers and i = −1 }
1)
If b = 0, then the complex number is a real number.
2)
If b€≠ 0 and a = 0, then the complex number is an imaginary
number (some books call this a pure imaginary number).
€ real part of the complex number. The
3)
The number a represents the
number b represents the imaginary part of the complex number.
4)
The complex number a – bi is the complex conjugate of a + bi
Addition, subtraction. and multiplication of complex numbers works a lot
like simplifying algebraic expression. Pretend that i is a variable and
distribute and combine like terms. The only steps we need to add are to
replace i 2 by – 1, simplify, and to write the answer in standard form.
Simplify:
( 23
13
7
i ) – ( 76 +
a) (– 2.1 + 1.4i ) + (0.07 – 3i )
b)
c) 9.2 – (6 – 3.1i ) – 4.2i
e) 7i (2 – 3i )
g) (– 4 + 3i )2
d) (– 5i )(3i )
f) (4 – 5i )(– 2 + 6i )
h)
€ (– 3 +€7i )(–€3 – 7i )
–
3
i
14
)
a)
€
(– 2.1 + 1.4i ) + (0.07 – 3i ) = – 2.1 + 0.07 + 1.4i – 3i = – 2.03 – 1.6i
b)
( 23 –
13
7
=
i ) – ( 76 +
4
6
–
7
6
–
26
14
3
i
14
)=
i –
3
i
14
2
3
–
=–
7
6
3
6
–
–
13
i
7
29
i
14
–
3
i
14
=–
1
2
–
29
14
i.
€ c) € 9.2 – (6
€ – 3.1
€ i ) – 4.2
€ i =€9.2€– 6 + 3.1
€ i – 4.2i = 3.2 – 1.1i .
2
d) €(– 5€
i )(3i€) = – 15
1) = 15. € €
€i = – 15(–
€ €
e)
7i (2 – 3i ) = 14i – 21i 2 = 14i – 21(– 1) = 21 + 14i .
f)
(4 – 5i )(– 2 + 6i ) = – 8 + 24i + 10i – 30i 2
= – 8 + 34i – 30(– 1) = – 8 + 34i + 30 = 22 + 34i .
g)
(– 4 + 3i )2 = (– 4)2 – 4(3i ) – 4(3i ) + (3i )2 = 16 – 24i + 9i 2
= 16 – 24i + 9(– 1) = 16 – 24i – 9 = 7 – 24i .
h)
(– 3 + 7i )(– 3 – 7i ) = (– 3)2 – (7i )2 = 9 – 49i 2 = 9 + 49 = 58
When dividing by a complex number, we must make the denominator into a
real number. Since i = −1 , we treat the division of complex numbers like
we are rationalize the denominator of a radical expression. If the real part
of the complex number in the denominator is zero, we multiply the top and
the bottom by i to “rationalize the denominator” so to speak. Otherwise, we
€
multiply the top and bottom by the complex conjugate of the denominator.
In either case, we simplify and write the answer in standard form.
Simplify:
5−3i
i)
j)
k)
L)
2i
8+3i
5−6i
3
−4+7i
11−2i
11+2i
€ i)
Since the denominator, 2i , has €
no real part, multiply top and bottom
€
by i :
j)
5−3i
2i
(−4+7i )
=
−12−21i
65
(5−6i )
=
i
i
=
5i −3i 2
2i
2
€=
5i −3(−1)
2(−1)
=
5i +3
−2
= – 1.5 – 2.5i .
(−4−7i )
=–
12
65
–
(−4) −(7i )
21
i.
65
16−49(−1)
16−49i
16+49
(5+6i )
40+63i +18(−1)
25−36(−1)
=
(5) −(6i )
22+63i
40+63i −18
=
61
25+36
=
22
61
25−36i
63
+
i
61
.
conjugate
of 11 + 2i is€11 – 2i . Multiply top and bottom
€The complex
€
€
by 11 – 2i, and simplify:
€
€
€
€ 11−2i
(11−2i )€ (11−2i )
(11)2 −(2i )(11)− (2i )(11)+ (2i )2
•
=
=
2
2
11+2i
(11+2i )
=
€
•
conjugate
of 5 – €
6i is 5 + 6€
i . Multiply top
€The complex
€
€ and bottom by
€
5 + 6i, and simplify:
€ 8+3i
€ ) (5+6i
€ )
8•5+8•6i +3i•5+3i•6i
(8+3i
40+48i +15i +18i 2
•
=
=
=
2
2
2
5−6i
€ L)
(5−3i )
2i
The complex conjugate of – 4 + 7i is – 4 – 7i . Multiply top and
bottom by – 4 – 7i, and simplify:
€
€ −12−21i € −12−21i
3 €
3€
(−4−7i )
−12−21i
−12−21i
•€
=
=
=
=
=
2
2
2
−4+7i
€ k)
=
€
=
121−22i − 22i + 4i 2
2
€117
125
121−4i
44
– €i
125
€
(11) −(2i )
=
121−44i + 4(−1)
121−4(−1)
=
121−44i − 4
121+4
.
€
€
€
(11−2i )
€
€
=
117−44i
125
Problems
Simplify:
1)
(5 – 3i ) – (8 + 7i )
3)
(5 + 4i )(8 – 3i )
5)
( −7 +
€
€
7)
€
€
−5 )( −7 –
€
−5 )
2)
(2 + 5i ) – (– 2 + 4i ) + (6 + i )
4)
(3.2 – 4.7i )2
6)
4+3i
2i
8)
−4+11i
6−5i
€
8−9i
3+2i
€