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

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

Basic Concepts of Complex Numbers The number i is 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.

THE EXPRESSION

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

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

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

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

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

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

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

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

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

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

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

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

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

Powers of i and so on.

Ex 5c. showed that… 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.

Property of Complex Conjugates For real numbers a and b,

Write each quotient in standard form a + bi. DIVIDING COMPLEX NUMBERS Example 7 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.

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

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