1.3 Complex Numbers Definition The imaginary unit i is defined to be √ 1
1
Integer powers of i: If n is an integer, then is exactly one of the following: 1, 1, ,
. 1. To simplify , rewrite in terms of and use the fact Examples: Simplify each power of i. 1. i18
2. i 24
    1
i18  i 2
9
9
3. i 7
 
 1
i 24  i 2
12
 
  1  1
i 7  i 2 i   1 i  1  i  i
12
3
3
4. i 83
 
i83  i 2
41
i   1 i  1  i  i
41
If b > 0, then the principal square root of the negative number –b is √
√ Examples: Simplify the following 5. 9
6. 18
9  i 9  3i
7. 48
18  i 18
48  i 48
 i 9  2
 i 16  3
 3i 2
 4i 3
8. 5 27  3 108
5 27  3 108  5i 27  3i 108
9. 5  15
5  15  i 5  i 15
 5i 9  3  3i 36  3
 i 2 75
 15i 3  18i 3
 1 25  3
 33i 3
 5 3
10.
12  6
8
11.
12  6 i 12  i 6

8
8



5  50
10
5  50 5  i 50

10
10
5  i 25  2 
i 2 72
8

 36  2 
5  5i 2

10
4  2
6 2
2 2

 3

10

5 1 i 2

10
1 i 2

2
2
Definition The set of all numbers of the form with a and b al numbers, is called the set of complex numbers A complex number is said to be simplified if it is written in the standard form Examples: Simplify 12.  7  5i   9  11i 
. 13. 8i  2i  7 
7  5i   9  11i   7  5i  9  11i
8i  2i  7   16i 2  56i
 16  1  56i
 2  16i
 16  56i
14.  8  4i  3  9i 
8  4i  3  9i   24  72i  12i  36i 2
 24  84i  36(1)
 2  3i 
2
  2  3i  2  3i 
 24  84i  36
 4  6i  6i  9i 2  4  12i  9(1)
 12  84i
 5  12i
Definition The conjugate of the complex number is 15.  2  3i 
2
is and the conjugate of the complex number Illustration: Number 7 5 3 2 Conjugate 7 5 3 2 To simplify a ratio of two complex numbers, multiply both the numerator and denominator by the denominator’s conjugate. 3  4i
5  2i
3  4i  3  4i   5  2i 


5  2i  5  2i   5  2i 
16.
15  6i  20i  8i 2
25  10i  10i  4i 2
15  14i  8(1)

25  4(1)
15  14i  8

25  4
23  8i

29
23 8

 i
29 29

17.
2  7i
3  5i
2  7i  2  7i   3  5i 


3  5i  3  5i   3  5i 
6  10i  21i  35i 2
9  15i  15i  25i 2
6  31i  35(1)

9  25(1)
6  31i  35

9  25
29  31i

34
29 31

 i
34 34
