This document discusses complex numbers. It provides examples of solving quadratic equations using complex numbers, multiplying and dividing complex numbers, and working with powers of i. It also includes exercises involving combining, expanding, dividing, and simplifying complex number expressions. Key points covered include using i=√-1, FOIL method for multiplication, conjugate multiplication, and the cyclic pattern of powers of i.

1. Complex Numbers
Using i = –1, the “solutions” of the equations
Example A. Solve x2 + 49 = 0 using imaginary numbers.
Using the square-root method:
x2 + 49 = 0 → x2 = –49 so
x = ±–49
x = ±49–1 = ±7i
x2 = –r
are x = ± ir
(Multiplication of complex numbers)
To multiply complex numbers, use FOIL, then set i2 to be (-1)
and simplify the result.
Example D. (4 – 3i)(2 + 7i) FOIL
= 8 – 6i + 28i – 21i2 set i2 = (-1)
= 8 – 6i + 28i = 29 + 22i

2. Complex Numbers
Example E.
(4 – 3i)(4 + 3i) = 42 + 32 = 25
(5 – 7i)(5 + 7i) = (5)2 + 72 = 54
(Conjugate Multiplication) The nonzero conjugate product is
(a + bi)(a – bi) = a2 + b2 which is always positive.
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–

3. Complex Numbers
(Division of Complex Numbers)
To divide complex numbers, we write the division as a fraction,
then multiply the top and the bottom of the fraction by the
conjugate of the denominator.
3 – 2i
4 + 3i
Example F. Simplify
Multiply the conjugate of the denominator (4 – 3i) to the top
and the bottom.
(3 – 2i)
(4 + 3i)
=
(4 – 3i)
(4 – 3i)* 42 + 32 =
25
12 – 8i – 9i + 6i2
–6
6 – 17i
=
25
6
25
17i
–
Example G. Solve 2x2 – 2x + 3 = 0 and simplify the answers.
To find b2 – 4ac first: a = 2, b = –2, c = 3, so b2 – 4ac = –20.
x =
2 ± –20
4 =
2 ± 2–5
4 =
2(1 ± i5)
4 =
1 ± i5
2
Using the quadratic formula, we can solve all 2nd degree
equations and obtain their complex number solutions.
Hence

4. The powers of i go in a cycle as shown below:
i = i5 = i9 ..
-1 = i2 = i6 ..
-i = i3 = i7 ..
1 = i4 = i8 ..
Example H. Simplify i59
59 = 4*14 + 3,
hence i59 = i4*14+3 = i4*14+3 = (i4)14 i3 = 114 i3 = i3 = -i
Powers of i

5. Complex Numbers
Exercise A. Write the complex numbers in i’s. Combine the
following expressions.
1. 2 – 2i + 3 + √–4 2. 4 – 5i – (4 – √–9) 3. 3 + 2i + (4 – i√5)
4. 4 – 2i + (–6 + i√3) 5. 4 – √–25 – (9 – √–16)
6. 11 – 9i + (–7 + i√12) 7. ½ – (√–49)/3 – (3/4 – √–16)
Exercise B. Do by inspection.
8. (1 – 2i)(1 + 2i) 9. (1 + 3i)(1 – 3i) 10. (2 + 3i)(2 – 3i)
11. (3 – 4i)(3 + 4i) 12. (9 + i√3)(9 – √3i) 13. (7 – i√5)(7 + i√5)
14. (9 + i√3) (7 – i√5)(9 – i√3) (7 + i√5)
15. (√3 + i√3) (√7 – i√5)(√3 – i√3)(√7 + i√5)
Exercise C. Expand and simplify.
16. (1 – 3i)(1 + 2i) 17. (2 + 3i)(1 – 3i) 18. (2 + 3i)(3 – 2i)
19. (4 – 3i)(3 – 4i) 20. (5 + 3i)(5 + 3i) 21. (1 – i)2
22. (2 + 3i)2 23. (5 + 2i)2

6. Complex Numbers
Exercise D. Divide by rationalizing the denominators.
2 + 3i
i24.
3 – 4i
i25.
3 + 4i
i26.
1 + i
1 – i27. 2 – i
3 – i28. 3 – 2i
2 + i29.
2 + 3i
2 – 3i
30.
3 – 4i
3 – 2i
31.
3 – 4i
2 + 5i
32.
Simplify
33. i92
38. Find a and b if (a + bi) 2 = i.
34. i –25 36. i 205
37. i –102
39. Is there a difference between √4i and 2i?

7. Complex Numbers
(Answers to odd problems) Exercise A.
1. 5 3. 7 + (2 – √5) i
Exercise B.
9. 10 11. 5
Exercise C.
17. 11 – 3i 19. – 25i
5. –5 – i 7. –1/4 + 5/3 i
13. 54 15. 72
21. – 2i 23. 21 + 20i
Exercise D.
– 4 – 3i25. 27.
33. 1
i 29.
1
5
(4 – 7i) 31.
1
13
(17 – 6i)
37. – 1 39. There is no difference