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10.7 Complex Numbers. Objective 1 . Simplify numbers of the form where b > 0 . Slide 10.7- 2. Simplify numbers of the form where b > 0. . Slide 10.7- 3. Simplify numbers of the form where b > 0. . - PowerPoint PPT Presentation

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10.7 Complex Numbers

10.7 Complex NumbersObjective 1 Simplify numbers of the form where b > 0.

Slide 10.7- 2Imaginary Unit i The imaginary unit i is defined as

That is, i is the principal square root of 1.

Slide 10.7- 3Simplify numbers of the form where b > 0.

For any positive real number b,

Slide 10.7- 4Simplify numbers of the form where b > 0.

It is easy to mistake for with the i under the radical. For this reason, we usually write as as in the definition of

Write each number as a product of a real number and i.

Slide 10.7- 5CLASSROOM EXAMPLE 1Simplifying Square Roots of Negative NumbersSolution:Multiply.

Slide 10.7- 6CLASSROOM EXAMPLE 2Multiplying Square Roots of Negative NumbersSolution:

Divide.

Slide 10.7- 7CLASSROOM EXAMPLE 3Dividing Square Roots of Negative NumbersSolution:Objective 2 Recognize subsets of the complex numbers.Slide 10.7- 8Complex NumberIf a and b are real numbers, then any number of the form a + bi is called a complex number. In the complex number a + bi, the number a is called the real part and b is called the imaginary part.Slide 10.7- 9Recognize subsets of the complex numbers.For a complex number a + bi, if b = 0, then a + bi = a, which is a real number.

Thus, the set of real numbers is a subset of the set of complex numbers.

If a = 0 and b 0, the complex number is said to be a pure imaginary number.

For example, 3i is a pure imaginary number. A number such as 7 + 2i is a nonreal complex number.

A complex number written in the form a + bi is in standard form. Slide 10.7- 10Recognize subsets of the complex numbers.The relationships among the various sets of numbers.

Slide 10.7- 11Recognize subsets of the complex numbers.Objective 3 Add and subtract complex numbers.Slide 10.7- 12Add.

Slide 10.7- 14CLASSROOM EXAMPLE 5Subtracting Complex NumbersSolution:

Objective 4 Multiply complex numbers.Slide 10.7- 15Multiply.

Slide 10.7- 16CLASSROOM EXAMPLE 6Multiplying Complex NumbersSolution:

Slide 10.7- 17CLASSROOM EXAMPLE 6Multiplying Complex Numbers (contd)Multiply. Solution:

Slide 10.7- 18CLASSROOM EXAMPLE 6Multiplying Complex Numbers (contd)Multiply. Solution:The product of a complex number and its conjugate is always a real number.

(a + bi)(a bi) = a2 b2( 1) = a2 + b2 Slide 10.7- 19Multiply complex numbers.Objective 5 Divide complex numbers.Slide 10.7- 20Find the quotient.

Slide 10.7- 21CLASSROOM EXAMPLE 7Dividing Complex NumbersSolution:

Slide 10.7- 22CLASSROOM EXAMPLE 7Dividing Complex Numbers (contd)Find the quotient. Solution:Objective 6 Find powers of i.Slide 10.7- 23Because i2 = 1, we can find greater powers of i, as shown below.

i3 = i i2 = i ( 1) = i i4 = i2 i2 = ( 1) ( 1) = 1

i5 = i i4 = i 1 = i

i6 = i2 i4 = ( 1) (1) = 1

i7 = i3 i4 = (i) (1) = I

i8 = i4 i4 = 1 1 = 1Slide 10.7- 24Find powers of i.Find each power of i.

Slide 10.7- 25CLASSROOM EXAMPLE 8Simplifying Powers of iSolution: