Chapter 6Chapter 6Irrational and Irrational and

Complex NumbersComplex Numbers

Section 6-1Section 6-1

Roots of Real Roots of Real NumbersNumbers

Square RootSquare Root A square root of a A square root of a number number bb is a solution is a solution of the equation xof the equation x22 = b. = b. Every positive number Every positive number bb has two square roots, has two square roots, denoted √b and -√b.denoted √b and -√b.

Principal Square RootPrincipal Square Root The The positivepositive square square root of root of b b is the principal is the principal square rootsquare root

The principal square The principal square root of 25 is 5root of 25 is 5

Examples – Square RootExamples – Square Root SimplifySimplify

xx22 = 9 = 9 xx22 + 4 = 0 + 4 = 0 5x5x22 = 15 = 15

Cube RootCube Root A cube root of A cube root of bb is a is a solution of the equationsolution of the equation

xx33 = b. = b.

Examples – Cube RootExamples – Cube Root SimplifySimplify 33√8√8 33√27√27 33√10√1066

33√a√a99

nnth rootth root

1.1. is the solution of xis the solution of xnn = b = b

2.2. If n is even, there If n is even, there could be two, one or could be two, one or no no nnth rootth root

3.3. If n is odd, there is If n is odd, there is exactly one exactly one nnth rootth root

Examples – Examples – nnth rootth root SimplifySimplify 44√81√81 55√32√32 55√-32√-32 66√-1√-1

RadicalRadical The symbol The symbol nn√b is called √b is called a a radicalradical

Each symbol has a nameEach symbol has a name n = indexn = index √ √ = radical= radical b = radicandb = radicand

Section 6-2Section 6-2

Properties of Properties of RadicalsRadicals

Product and Quotient Product and Quotient Properties of RadicalsProperties of Radicals

1.1. n n√ab = √ab = nn√a · √a · nn√b√b

2. 2. nn√a÷b = √a÷b = nn√a ÷ √a ÷ nn√b√b

ExamplesExamples

SimplifySimplify 33√25 · √25 · 33√10√10 33√(81/8)√(81/8) √√2a2a22bb √√36w36w33

Rationalizing the Rationalizing the DenominatorDenominator

Create a perfect Create a perfect square, cube or other square, cube or other power in the power in the denominator in order denominator in order to simplify the answer to simplify the answer without a radical in without a radical in the denominatorthe denominator

Examples Examples SimplifySimplify

√√(5/3)(5/3) 44

33√c√c

TheoremsTheorems1.1. If each radical represents If each radical represents

a real number, then a real number, then nqnq√b = √b = nn√(√(qq√b).√b).

2. If 2. If nn√b represents a real √b represents a real number, then number, then

nn√b√bmm = ( = (nn√b)√b)mm

ExamplesExamples Give the decimal Give the decimal approximation to the approximation to the nearest hundredth.nearest hundredth.

44√100√100 33√170√17022

Section 6-3Section 6-3

Sums of RadicalsSums of Radicals

Like RadicalsLike Radicals Two radicals with the Two radicals with the same index and same same index and same radicandradicand

You add and subtract like You add and subtract like radicals in the same way radicals in the same way you combine like termsyou combine like terms

ExamplesExamples

SimplifySimplify

√√8 + √988 + √98 33√81 - √81 - 33√24√24 √√32/3 + √2/332/3 + √2/3

ExamplesExamples SimplifySimplify √√12x12x55 - x√3x - x√3x33 + 5x + 5x22√3x√3x

AnswerAnswer 6x6x22√√3x3x

Section 6-4Section 6-4

Binomials Binomials Containing Containing

RadicalsRadicals

Multiplying BinomialsMultiplying Binomials You multiply binomials You multiply binomials with radicals just like with radicals just like you would multiply you would multiply any binomials. any binomials.

Use the FOIL method Use the FOIL method to multiply binomialsto multiply binomials

ExamplesExamples SimplifySimplify (4 + √7)(3 + 2√7)(4 + √7)(3 + 2√7)

AnswerAnswer 26 + 11√726 + 11√7

ConjugateConjugate Expressions of the Expressions of the form a√b + c√d and form a√b + c√d and a√b - c√d a√b - c√d

Conjugates can be Conjugates can be used to rationalize used to rationalize denominatorsdenominators

Example - ConjugateExample - Conjugate SimplifySimplify3 + √5 3 + √5 3 - √5 3 - √5

AnswerAnswer7 + 3√57 + 3√5

22

Example - ConjugateExample - Conjugate SimplifySimplify 11 4 - √154 - √15 AnswerAnswer 4 + √15 4 + √15

Section 6-5Section 6-5

Equations Equations Containing Containing

RadicalsRadicals

Radical EquationRadical Equation An equation which An equation which contains a radical contains a radical with a variable in the with a variable in the radicand.radicand.

40 22d

Solving a Radical Solving a Radical EquationEquation

First isolate the First isolate the radical term on one radical term on one side of the equationside of the equation

Solving a Radical Solving a Radical Equation - ContinuedEquation - Continued

If the radical term is a If the radical term is a square root, square square root, square both sidesboth sides

If the radical term is a If the radical term is a cube root, cube both cube root, cube both sidessides

Example 1Example 1 Solve Solve

AnswerAnswer X = 5X = 5

2 1 3x

Example 2Example 2 Solve Solve

AnswerAnswer X = 9X = 9

32 1 4x

Example 3Example 3 Solve Solve

AnswerAnswer X = 2/9X = 2/9

2 5 2 2 1x x

Section 6-6Section 6-6

Rational and Rational and Irrational Irrational NumbersNumbers

Completeness Property Completeness Property of Real Numbersof Real Numbers

Every real number Every real number has a decimal has a decimal representation, and representation, and every decimal every decimal represents a real represents a real numbernumber

Remember…Remember… A A rational rational number is number is any number that can any number that can be expressed as the be expressed as the ratio or quotient of ratio or quotient of two integerstwo integers

Decimal RepresentationDecimal Representation

Every rational number Every rational number can be represented can be represented by a by a terminating terminating decimaldecimal or a or a repeating decimalrepeating decimal

Example 1Example 1 Write each Write each terminating decimal terminating decimal as a fraction in lowest as a fraction in lowest terms.terms.

2.5712.571 0.00360.0036

Example 2Example 2 Write each repeating Write each repeating decimal as a fraction decimal as a fraction in lowest terms.in lowest terms.

0.32727…0.32727… 1.89189189…1.89189189…

Remember…Remember… An An irrational numberirrational number is a real number that is a real number that is not rationalis not rational

Decimal RepresentationDecimal Representation Every irrational number is Every irrational number is represented by an infinite represented by an infinite and nonrepeating decimaland nonrepeating decimal

Every infinite and Every infinite and nonrepeating decimal nonrepeating decimal represents an irrational represents an irrational numbernumber

Example 3Example 3 Classify each number as Classify each number as either rational or irrationaleither rational or irrational

√√22 √4/9 √4/9

2.0303…2.0303… 2.030030003…2.030030003…

Section 6-7Section 6-7

The Imaginary The Imaginary Number Number ii

DefinitionDefinition

i = i = √-1√-1 and and

ii22 = -1 = -1

DefinitionDefinition If If rr is a positive real is a positive real number, then number, then

√√-r-r = = ii√√rr

Example 1Example 1

SimplifySimplify

√√-5-5 √√-25-25 √√-50-50

Combining imaginary Combining imaginary NumbersNumbers

Combine the same Combine the same way you combine like way you combine like termsterms

√√-16 - √-49-16 - √-49 ii√2 + 3√2 + 3ii√2√2

Multiply - ExampleMultiply - Example SimplifySimplify

√√-4 ▪ √-25-4 ▪ √-25 ii√2 ▪ √2 ▪ ii√3√3

Divide - ExampleDivide - Example SimplifySimplify 22

33ii 66

√ √-2-2

ExampleExample SimplifySimplify

√√-9x-9x22 + √-x + √-x22

√√-6y ▪ √-2y -6y ▪ √-2y

Section 6-8Section 6-8

The Complex The Complex NumberNumber

Complex NumbersComplex Numbers Real numbers and Real numbers and imaginary numbers imaginary numbers together form the set of together form the set of complex numberscomplex numbers

The form The form aa + bi + bi, , represents a complex represents a complex numbernumber

Equality of Complex Equality of Complex NumbersNumbers

a + a + bibi = c + = c +didi if and only if if and only if

a = c and a = c and b = db = d

Sum of Complex Sum of Complex NumbersNumbers

((a + a + bibi ) +(c + ) +(c +didi ) = ) = (a + c) + (b + d)(a + c) + (b + d)ii

Product of Complex Product of Complex NumbersNumbers

((a + a + bibi )▪(c + )▪(c +didi )= )= (ac – bd) + (ad + bc)(ac – bd) + (ad + bc)ii

Example 1Example 1 SimplifySimplify

(3 + 6(3 + 6ii) + (4 – 2) + (4 – 2ii)) (3 + 6(3 + 6ii) - (4 – 2) - (4 – 2ii))

Example 2Example 2 SimplifySimplify (3 + 4(3 + 4ii)(5 + 2)(5 + 2ii))

(3 + 4(3 + 4ii))22

(3 + 4(3 + 4ii)(3 - 4)(3 - 4ii))

Using ConjugatesUsing Conjugates Simplify using Simplify using conjugatesconjugates

5 – 5 – ii

2 + 32 + 3ii

ReciprocalsReciprocals Find the reciprocal of Find the reciprocal of

3 – 3 – ii Remember…Remember…

the reciprocal of x = 1/xthe reciprocal of x = 1/x

THE END!THE END!