Objectives Students will learn:

1. Objectives Students will learn: Complex Numbers Basic Concepts of Complex Numbers Operations on Complex Numbers

2. Basic Concepts of Complex Numbers To extend the real number system to include such numbers as, the number i is defined to have the following property; There are no real numbers for the solution of the equation

3. Basic Concepts of Complex Numbers The number iis called the imaginary unit. Numbers of the form a + bi, where a and b are real numbers are called complex numbers. In this complex number, a is the real part and b is the imaginary part. So…

4. Nonreal complex numbers a + bi, b ≠ 0 Complex numbers a + bi, a and b real Irrational numbers Real numbers a + bi, b = 0 Integers Rational numbers Non-integers

5. Basic Concepts of Complex Numbers if and only if and Two complex numbers are equal provided that their real parts are equal and their imaginary parts are equal;

6. Basic Concepts of Complex Numbers For complex number a + bi, if b = 0, then a + bi = a So, the set of real numbers is a subset of complex numbers.

7. Basic Concepts of Complex Numbers If a = 0 and b ≠ 0, the complex number is pure imaginary. A pure imaginary number or a number, like 7 + 2i with a ≠ 0 and b ≠ 0, is a nonreal complex number. The forma+ bi (or a + ib) is called standard form.

8. THE EXPRESSION

9. Example 1 WRITING AS Write as the product of a real number and i, using the definition of a. Solution:

10. Example 1 WRITING AS Write as the product of a real number and i, using the definition of b. Solution:

11. Example 1 WRITING AS Write as the product of a real number and i, using the definition of c. Solution: Product rule for radicals

12. Operations on Complex Numbers Then the properties of real numbers are applied, together with the fact that Products or quotients with negative radicands are simplified by first rewriting for a positive number.

13. Operations on Complex Numbers Caution When working with negative radicands, use the definition… before using any of the other rules for radicands.

14. Operations on Complex Numbers Caution In particular, the rule is valid only when c and d are not both negative. while so

15. Example 2 FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Multiply or divide, as indicated. Simplify each answer. a. Solution: First write all square roots in terms ofi. i 2 = −1

16. Example 2 FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Multiply or divide, as indicated. Simplify each answer. b. Solution:

17. Example 2 FINDING PRODUCTS AND QUOTIENTS INVOLVING NEGATIVE RADICALS Multiply or divide, as indicated. Simplify each answer. c. Solution: Quotient rule for radicals

18. Example 3 SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND Write in standard form a + bi. Solution:

19. Example 3 SIMPLIFYING A QUOTIENT INVOLVING A NEGATIVE RADICAND Write in standard form a + bi. Solution: Be sure to factor before simplifying Factor. Lowest terms

20. Addition and Subtraction of Complex Numbers For complex numbers a + bi and c + di, and

21. Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. a. Add imaginary parts. Add real parts. Solution: Commutative, associative, distributive properties

22. Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. b. Solution:

23. Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. c. Solution:

24. Example 4 ADDING AND SUBTRACTING COMPLEX NUMBERS Find each sum or difference. d. Solution:

25. Multiplication of Complex Numbers FOIL Distributive property; i 2 =–1 The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i2= –1, as follows.

26. Multiplication of Complex Numbers For complex numbers a + bi and c + di,

27. Example 5 MULTIPLYING COMPLEX NUMBERS Find each product. a. Solution: FOIL i2 = −1

28. Example 5 MULTIPLYING COMPLEX NUMBERS Find each product. b. Solution: Square of a binomial Remember to add twice the product of the two terms. i 2 = −1

29. Example 5 MULTIPLYING COMPLEX NUMBERS Find each product. c. Solution: Product of the sum and difference of two terms i 2 = −1 Standard form

30. Simplifying Powers of i Powers of i can be simplified using the facts

31. Example 6 SIMPLIFYING POWERS OF i Simplify each power of i. a. Solution: Since i 2 = –1 and i 4 = 1, write the given power as a product involving i 2 or i 4. For example, Alternatively, using i4 and i3 to rewrite i15 gives

32. Example 6 SIMPLIFYING POWERS OF i Simplify each power of i. b. Solution:

33. Powers of i and so on.

34. The numbers differ only in the sign of their imaginary parts and are called complex conjugates. The product of a complex number and its conjugate is always a real number. This product is the sum of squares of real and imaginary parts. Ex 5c. showed that…

35. Property of Complex Conjugates For real numbers a and b,

36. Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: Multiply by the complex conjugate of the denominator in both the numerator and the denominator. Multiply.

37. Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: Multiply. i 2 = −1

38. Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: i 2 = −1

39. Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. a. Solution: Lowest terms; standard form

40. Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. b. Solution: –i is the conjugate of i.

41. Example 7 DIVIDING COMPLEX NUMBERS Write each quotient in standard form a + bi. b. Solution: Standard form i 2 = −1(−1) = 1