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Chapter 9

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Chapter 9

Section 4

Complex Numbers

Write complex numbers as multiples of i.

Add and subtract complex numbers.

Multiply complex numbers.

Divide complex numbers.

Solve quadratic equations with complex number solutions.

9.4

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3

4

5

Some quadratic equations have no real number solutions. For example, the numbers

are notreal numbers because – 4 appears in the radicand. To ensure that every quadratic equation has a solution, we need a new set of numbers that includes the real numbers. This new set of numbers is defined with a new number i, call the imaginary unit, such that

and

Slide 9.4-3

Write complex numbers as multiples of i.

Slide 9.4-4

For any positive real number b,

We can write numbers such as and as multiples of i, using the properties of ito define any square root of a negative number as follows.

Slide 9.4-5

Write as a multiple of i.

It is easy to mistake for with the i under the radical. For this reason, it is customary to write the factor i first when it is multiplied by a radical. For example, we usually write rather than

EXAMPLE 1

Simplifying Square Roots of Negative Numbers

Solution:

Slide 9.4-6

Write complex numbers as multiples if i. (cont’d)

Numbers that are nonzero multiples of iare pure imaginary numbers. The complex numbers include all real numbers and all imaginary numbers.

Complex Number

A complex number is a number of the form a + bi, where a and bare real numbers. If a = 0 and b≠ 0, then the number bi is a pure imaginary number.

In the complex number a+ bi,a is called the real part and b is called the imaginary part. A complex number written in the form a + bi (or a + ib) is in standard form. See the figure on the following slide which shows the relationship among the various types of numbers discussed in this course.

Slide 9.4-7

Slide 9.4-8

Add and subtract complex numbers.

Slide 9.4-9

Adding and subtracting complex numbers is similar to adding and subtracting binomials.

To add complex numbers, add their real parts and add their imaginary parts.

To subtract complex numbers, add the additive inverse (or opposite).

Slide 9.4-10

Add or subtract.

EXAMPLE 2

Adding and Subtracting Complex Numbers

Solution:

Slide 9.4-11

Multiply complex numbers.

Slide 9.4-12

We multiply complex numbers as we do polynomials. Since i2 = –1 by definition, whenever i2appears, we replace it with –1.

Slide 9.4-13

Find each product.

EXAMPLE 3

Multiplying Complex Numbers

Solution:

Slide 9.4-14

Divide complex numbers.

Slide 9.4-15

The quotient of two complex numbers is expressed in standard form by changing the denominator into a real number.

The complex numbers 1 + 2i and 1 – 2i are conjugates. That is, the conjugate of the complex number a + bi is a – bi. Multiplying the complex number a + bi by its conjugate a – bi gives the real number a2 + b2.

Product of Conjugates

That is, the product of a complex number and its conjugate is the sum of the squares of the real and imaginary part.

Slide 9.4-16

Write the quotient in standard form.

EXAMPLE 4

Dividing Complex Numbers

Solution:

Slide 9.4-17

Solve quadratic equations with complex number solutions.

Slide 9.4-18

Quadratic equations that have no real solutions do have complex solutions.

Slide 9.4-19

Solve (x– 2)2 = –64.

EXAMPLE 5

Solving a Quadratic Equation with Complex Solutions (Square Root Property)

Solution:

Slide 9.4-20

Solve x2– 2x = –26.

EXAMPLE 6

Solving a Quadratic Equation with Complex Solutions (Quadratic Formula)

Solution:

Slide 9.4-21