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.

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

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

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.

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 form a + bi (or a + ib) is called standard form.

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

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. b. Solution:

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

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

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

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

Multiplication of Complex Numbers 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. FOIL Distributive property; i 2 = – 1

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

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

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

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

Simplify each power of i. SIMPLIFYING POWERS OF i Example 6 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

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

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

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

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

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