Download 7-9: MORE About Complex Numbers

7-9: MORE About Complex Numbers
Goal: Be able to solve equations with complex numbers, multiply
complex numbers, find conjugates of complex numbers, and divide
and find reciprocals of complex numbers.
Equality for Complex Numbers
If a + bi = c + di
then a = c
and
b=d
Ex 1: Solve for x and y.
a.)
5 x  6i  10  2 yi
5 x  10
x2
6i 2 yi

2i
2i
3 y
b.)
4 x  3 yi  8  6i
4x  8
x2
3 yi 6i

3i
3i
y2
Ex 2: Multiply.
Remember: i2 = -1
a.) 5i  2i  10i 2  10(1)   10
2
3
i

6
i

18
i
 18(1)   18
b.)
c.) (4i)2  16i 2  16(1)   16
d.) (7i)
2
 49i
2
 49(1)   49
2
(2

5
i
)(3

4
i
)

6

8
i

15
i

20
i
e.)
 6  23i  20(1)
  14  23i
2
(

2

3
i
)(6

5
i
)


12

10
i

18
i

15
i
f.)
  12  28i  15(1)
 3  28i
Complex Conjugates
Conjugates: a + bi ; a - bi
Ex 3: Find the conjugate of each number.
a.) 2  5i ; 2  5i
b.) 3  10i ;  3  10i
c.) 8i ;  8i
d.) 4 ; 4
(since 0 + 8i, 0 – 8i are conjugates)
(since 4 + 0i, 4 – 0i are conjugates)
Consider(a  bi)(a  bi)
 a 2  abi  abi  b2i 2  a 2  b2 (1)
 a 2  b2
(a  bi)(a  bi)  a 2  b2
Ex 4: Multiply.
a.) (4  2i)(4  2i)  (4) 2  (2) 2
 16  4
 20
b.) (3  7i)(3  7i)  (3) 2  (7) 2
 9  49
 58
c.) (5  i 3)(5  i 3)  (5)2  ( 3) 2
 25  (3)
 28
Division and Reciprocals
Multiply by “1;”
Ex 5: Divide.
use the conjugate
2

3
i
1

i
a.)

1i 1 i

2  2i  3i  3i
2
(1)  (1)
2
2
2  5i  3(1)

2
1 5i
 
2 2
Division and Reciprocals
Ex 5: Divide.
1

2
i
3

4
i
b.)

3  4i 3  4i

3  4i  6i  8i 2
(3)2  (4)2
Multiply by “1;”
use the conjugate
3  2i  8(1)

9  16
11 2i


25 25
Ex 6: a.) Find the reciprocal of 2  7i.
1
2  7i

2  7i 2  7i
2  7i
Multiply by “1;”
use the conjugate
2  7i


2
2
4  49
(2)  (7)
2
7i


53 53
Ex 6: b.) Find the reciprocal of 5  4i.
1
5  4i

5  4i 5  4i
5  4i
Multiply by “1;”
use the conjugate
5  4i


2
2
25  16
(5)  (4)
5
4i


41 41