Fundamental Concepts of Algebra11.1 Real Numbers

Objective: Students will be introduced to the real number system that is used throughout mathematics and will be acquainted with the symbols that represent them.

The Real Numbers

The real numbers can be ordered and represented in order on a number line

-3 -2 -1 0 1 2 3 4

-1.87

0

4.552

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

REAL NUMBERS (R)

Definition:

REAL NUMBERS (R)

- Set of all rational and

irrational numbers.

Definition:

REAL NUMBERS (R)

- Set of all rational and

irrational numbers.

SUBSETS of R

Definition:

RATIONAL NUMBERS (Q)

- numbers that can be expressed as a quotient a/b, where a and b are integers.

- terminating or repeating decimals

- Ex: {1/2, .25, 1.3, 5}

Definition:

RATIONAL NUMBERS (Q)

- numbers that can be expressed as a quotient a/b, where a and b are integers.

- terminating or repeating decimals

- Ex: {1/2, .25, 1.3, 5}

SUBSETS of R

Definition:IRRATIONAL NUMBERS (Q´)- infinite and non-repeating decimals- Ex: { ∏, √2, -1.436512…..}

Definition:IRRATIONAL NUMBERS (Q´)- infinite and non-repeating decimals- Ex: { ∏, √2, -1.436512…..}

SUBSETS of R

Definition:

INTEGERS (Z)

- numbers that consist of positive integers, negative integers, and zero,

- {…, -2, -1, 0, 1, 2 ,…}

Definition:

INTEGERS (Z)

- numbers that consist of positive integers, negative integers, and zero,

- {…, -2, -1, 0, 1, 2 ,…}

SUBSETS of R

Definition:

NATURAL NUMBERS (N)

- counting numbers

- positive integers

- {1, 2, 3, 4, ….}

Definition:

NATURAL NUMBERS (N)

- counting numbers

- positive integers

- {1, 2, 3, 4, ….}

SUBSETS of R

Definition:

WHOLE NUMBERS (W)

- nonnegative integers

- {0, 1, 2, 3, 4, …}

Definition:

WHOLE NUMBERS (W)

- nonnegative integers

- {0, 1, 2, 3, 4, …}

The Set of Real NumbersThe Set of Real Numbers

Q

Q'Q'QQ

ZZWW

NN

PROPERTIES of R

Definition:

CLOSURE PROPERTY

Given real numbers a and b,

Then, a + b is a real number (+),

or a x b is a real number (x).

Definition:

CLOSURE PROPERTY

Given real numbers a and b,

Then, a + b is a real number (+),

or a x b is a real number (x).

PROPERTIES of R

Example 1:

12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.

Example 1:

12 + 3 is a real number. Therefore, the set of reals is CLOSED with respect to addition.

PROPERTIES of R

Example 2:

12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.

Example 2:

12 x 4.2 is a real number. Therefore, the set of reals is CLOSED with respect to multiplication.

PROPERTIES of R

Definition:

COMMUTATIVE PROPERTY

Given real numbers a and b,

Addition: a + b = b + a

Multiplication: ab = ba

Definition:

COMMUTATIVE PROPERTY

Given real numbers a and b,

Addition: a + b = b + a

Multiplication: ab = ba

PROPERTIES of R

Example 3:

Addition:

2.3 + 1.2 = 1.2 + 2.3Multiplication:

(2)(3.5) = (3.5)(2)

Example 3:

Addition:

2.3 + 1.2 = 1.2 + 2.3Multiplication:

(2)(3.5) = (3.5)(2)

PROPERTIES of R

Definition:

ASSOCIATIVE PROPERTY

Given real numbers a, b and c,

Addition:

(a + b) + c = a + (b + c)

Multiplication: (ab)c = a(bc)

Definition:

ASSOCIATIVE PROPERTY

Given real numbers a, b and c,

Addition:

(a + b) + c = a + (b + c)

Multiplication: (ab)c = a(bc)

PROPERTIES of R

Example 4:

Addition:

(6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication:

(9 x 3) x 4 = 9 x (3 x 4)

Example 4:

Addition:

(6 + 0.5) + ¼ = 6 + (0.5 + ¼) Multiplication:

(9 x 3) x 4 = 9 x (3 x 4)

PROPERTIES of R

Definition:

DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITION

Given real numbers a, b and c,

a (b + c) = ab + ac

Definition:

DISTRIBUTIVE PROPERTY of MULTIPLICATION OVER ADDITION

Given real numbers a, b and c,

a (b + c) = ab + ac

PROPERTIES of R

Example 5:4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)

Example 6:

2x (3x – b) = (2x)(3x) + (2x)(-b)

Example 5:4.3 (0.11 + 3.02) = (4.3)(0.11) + (4.3)(3.02)

Example 6:

2x (3x – b) = (2x)(3x) + (2x)(-b)

PROPERTIES of R

Definition:

IDENTITY PROPERTY

Given a real number a,

Addition: 0 + a = a

Multiplication: 1 x a = a

Definition:

IDENTITY PROPERTY

Given a real number a,

Addition: 0 + a = a

Multiplication: 1 x a = a

PROPERTIES of R

Example 7:

Addition:

0 + (-1.342) = -1.342 Multiplication:

(1)(0.1234) = 0.1234

Example 7:

Addition:

0 + (-1.342) = -1.342 Multiplication:

(1)(0.1234) = 0.1234

PROPERTIES of R

Definition:

INVERSE PROPERTY

Given a real number a,

Addition: a + (-a) = 0

Multiplication: a x (1/a) = 1

Definition:

INVERSE PROPERTY

Given a real number a,

Addition: a + (-a) = 0

Multiplication: a x (1/a) = 1

PROPERTIES of R

Example 8:

Addition:

1.342 + (-1.342) = 0 Multiplication:

(0.1234)(1/0.1234) = 1

Example 8:

Addition:

1.342 + (-1.342) = 0 Multiplication:

(0.1234)(1/0.1234) = 1

Inequality Graph Interval

3 7x

5x

1

3x

3,7

5,

1,

3

]

( ]

(5

3 7

1

3

) or ( means not included in the solution

] or [ means included in the solution

Inequalities, graphs, and notation

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

IntervalsInterval Graph

( )

[ ]

( ]

[ )

(

)

[

]

a b

Example

(a, b)

[a, b]

(a, b]

[a, b)

(a, )

(- , b)

[a, )

(- , b]

(3, 5)

[4, 7]

(-1, 3]

[-2, 0)

(1, )

(- , 2)

[0, )

(- , -3]

( )

[ ]

( ]

[ )

(

)

[

]

a b

a b

a b

a

a

b

b

3 5

-2 0

4 7

-1 3

-3

2

1

0

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Absolute Value

if 0

if 0

a aa

a a

To evaluate:

3 8 ( 5) 5 5Notice the opposite sign

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Real Number Venn Diagram

1-C

Scientific Notation

A short-hand way of writinglarge numbers without writing all of the zeros.

When using Scientific Notation, there are two kinds of exponents:

positive and negative

Positive Exponent:

2.35 x 108

Negative Exponent:3.97 x 10-7

An easy way to remember this is:

• If an exponent is positive, the number gets larger, so move the decimal to the right.

• If an exponent is negative, the number gets smaller, so move the decimal to the left.

The exponent also tells how many spaces to move the decimal:

4.08 x 103 = 4 0 8

In this problem, the exponent is +3, so the decimal moves 3 spaces to the right.

The exponent also tells how many spaces to move the decimal:

4.08 x 10-3 = 4 0 8

In this problem, the exponent is -3, so the decimal moves 3 spaces to the left.

Try changing these numbers from Scientific Notation to Standard

Notation:1) 9.678 x 104

2) 7.4521 x 10-3

3) 8.513904567 x 107

4) 4.09748 x 10-5

96780

.0074521

85139045.67

.0000409748

When changing from Standard Notation to Scientific Notation:

1) First, move the decimal after the first whole number:

3 2 5 8

123

3

2) Second, add your multiplication sign and your base (10).

3 . 2 5 8 x 10

3) Count how many spaces the decimal moved and this is the exponent. 3 . 2 5 8 x 10

When changing from Standard Notation to Scientific Notation:

4) See if the original number is greater than or less than one.– If the number is greater than one, the exponent

will be positive.

348943 = 3.489 x 105

– If the number is less than one, the exponent will be negative.

.0000000672 = 6.72 x 10-8

Try changing these numbers from Standard Notation to Scientific

Notation:1) 9872432

2) .0000345

3) .08376

4) 5673

9.872432 x 106

3.45 x 10-5

8.376 x 102

5.673 x 103

1-1 Answers (2-40e, 50,52)• 2. -,-,+,+• 4. >,<,=• 6. <,>,>• 8. b > 0, s < 0, w > -4, 1/5< c < 1/3, p < -2, -m > -2, r/s ≥ 1/5, 1/f ≤ 14, |x| < 4• 10. 10, 3, 17• 12. 4, 5/2, 10• 14. √3 -1.7, √3 – 1.7, 2/15• 16. 4,6, 6, 10• 18. 12, 3, 3, ,9• 20. | -√2-x|> 1 • 22. |4-x | < 2• 24. |x + 2| > 2 • 26. x – 5 • 28. 7 + x • 30. a – b • 32. x2 + 1• 34. =• 36 ≠• 38 ≠• 40 ≠• 50. 8.52 x 104 5.5 x 10-6 2.49 x 107

• 52. 23,000,000 .00000000701 12,300,000,000

1.2 Laws of Exponents

m n m na a a

Law Example

nm mna am

m nn

aa

a

n n nab a bn n

n

a a

b b

3 12 3 12 15x x x x

65 5(6) 303 3 3 14

14 12 212

yy y

y

4 4 4 43 3 81r r r 3 3

3 3

4 4 64

x x x

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Exponents

na 35 5 5 5 125 ...na a a a a

Definition

n factors

Examplen,m positive integers

0a

na

0 1 0a a

10n

na a

a

032 1

44

1 12

162

/m na

/m na

/ nm n ma a

/ 1m n

n ma

a

32 / 3 2125 125 25

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

8

27

4

9

9

432/3

http://www.youtube.com/watch?v=QIZTruxt2rQ&feature=related

1.2 Answers: p. 29 (12-30 x3)

12. -12x2

18.

24. -4x12y7

30. -288r8s11

5

12

y

Definitions

The entire expression, including the radical sign and radicand, is called the radical expression.

." ofroot square" theread is xx

x

thecalled issign radical theinside expression The

radicand.

thecalled is The radical sign.

Definitions

The positive or principal square root of a positive number a is written as . The negative square root is written as - .

aa

abba 2 if .00written0, is 0 ofroot square theAlso,

Note that the principal square root of a positive number, a, is the positive number whose square equals a. Whenever the term ‘square root’ is used in this book, the positive or principal square root is meant to be used.

Definitions

The index tells the “root” of the expression. Since square roots have an index of 2, the index is generally not written in a square root.

2 means xxExample:

25) 555 (since 525 2

)16

9

4

3

4

3

4

3 (since

4

3

16

92

Definitions

Square roots of negative numbers are not real numbers. Square roots of negative numbers are called imaginary numbers.

?25 There is no number multiplied by itself that will give you –25.

(Imaginary numbers will be discussed in a later section)

Cube and Fourth Roots

is read “the cube root of a.”3 a4 a is read “the fourth root of a.”

abba 33 if abba 44 if

8222 since 283

8)2)(2)(2( since 283

8133333 since 318 44

Even and Odd Indices

Even Indices

The nth root of a, , where n is an even index and a is a nonnegative real number, is the nonnegative real number b such that bn = a.

n a

813 since 381 44

10010 since 10100 2

Even and Odd Indices

Odd Indices

The nth root of a, , where n is an odd index and a is a any real number, is the real number b such that bn = a.

n a

644 since 464 33

32(-2) since 232 55

Cube and Fourth Roots

Note that the cube root of a positive number is a positive number and the cube root of a negative number is a negative number.

The radicand of a fourth root (or any even root) must be a nonnegative number for the expression to be a real number.

Evaluate by Using Absolute Value

For any real number a,

aa 2

7772

99)9( 2

119)119( 2 baba

)6()6()3612( 22 xxxx

Changing a Radical Expression

When a is nonnegative, n can be any index.When a is negative, n must be odd.

nn aa1

77 21

A radical expression can be written using exponents by using the following procedure:

3143 4 yxyx

9149 4 7373 zxzx

Changing a Radical Expression

When a is nonnegative, n can be any index.When a is negative, n must be odd.

nn aa1

15 15 21

Exponential expressions can be converted to radical expressions by reversing the procedure.

331 bb

Simplifying Radical Expressions

73 2372 9898 yxyx

This rule can be expanded so that radicals of the form can be written as exponential expressions. n ma

For any nonnegative number a, and integers m and n,

nmmnn m aaa

Power

Index

3 232 bb

Definitions

A perfect square is the square of a natural number. 1, 4, 9, 16, 25, and 36 are the first six perfect squares.Variables with exponents may also be perfect squares. Examples include x2, (x2)2 and (x3)2.

A perfect cube is the cube of a natural number. 1, 8, 27, 64, 125, and 216 are the first six perfect cubes.Variables with exponents may also be perfect cubes. Examples include x3, (x2)3 and (x3)3.

Perfect Powers

A quick way to determine if a radicand xn is a perfect power for an index is to determine if the exponent n is divisible by the index of the radical.

Example: 5 20x Since the exponent, 20, is divisible by the index, 5, x20 is a perfect fifth power.

This idea can be expanded to perfect powers of a variable for any radicand.The radicand xn is a perfect power when n is a multiple of the index of the radicand.

Product Rule for Radicals

Examples:

3333 424832

4444 3231648

3333 252125250

and numbers real enonnegativFor nnn abba

ba

,

Product Rule for Radicals

1. If the radicand contains a coefficient other than 1, write it as a product of the two numbers, one of which is the largest perfect power for the index.

2. Write each variable factor as a product of two factors, one of which is the largest perfect power of the variable for the index.

3. Use the product rule to write the radical expression as a product of radicals. Place all the perfect powers under the same radical.

4. Simplify the radical containing the perfect powers.

To Simplify Radicals Using the Product Rule

Product Rule for Radicals

Examples:

2623623672

4 354 34 204 3204 23 || bbbbbbb

3233 633 63 222816 xyyxyx

4 324 28164 3118 21632 yxyxyx 4 3274 2||2 yxyx *When the radical is simplified, the

radicand does not have a variable with an exponent greater than or equal to the index.

Quotient Rule for Radicals

Examples:

10

9

100

81

100

81

0 ,

and numbers real enonnegativFor

bb

a

b

a

ba

nn

n

,

3

75 5251

25Simplify radicand, if possible.

Quotient Rule for Radicals

More Examples:

4

2

3 12

3 6

312

6 46464

y

x

y

x

y

x

2

42

4 8

44 8

4 8

4 8

48

8

4132

56

2

3

16

3

16

3

16

3

16

3

b

a

b

a

b

a

b

a

ba

ba

241

216

1

32

2

64

2

64 22

3

5

3

5

xxx

x

x

x

x

CAUTION!

The product rule does not apply to addition or subtraction!

baba

baba

Rationalizing Denominators

Examples:

To Rationalize a Denominator

3

6

3

3

3

2

3

2

233

32

3

3

3

2

3

2 ||

y

yx

y

yxy

y

yx

y

y

y

x

y

x

Multiply both the numerator and the denominator of the fraction by a radical that will result in the radicand in the denominator becoming a perfect power.

r

prq

r

rpq

r

r

r

pq

r

pq

2

10

2

10

2

2

2

5

2

5 2444

Cannot be simplified further.

Conjugates

When the denominator of a rational expression is a binomial that contains a radical, the denominator is rationalized. This is done by using the conjugate of the denominator. The conjugate of a binomial is a binomial having the same two terms with the sign of the second term changed.

The conjugate of 65 is 65

The conjugate of 44 23 is 23 yxyx

12

125

12

12

12

5

)(

Simplifying Radicals

Simplify by rationalizing the denominator:

12

5

dc

dc

2

dc

dcdcdc

dcdc

dcdc

22

))((

))(2(

dc

dc

dc

dc 2

Simplifying Radicals

A Radical Expression is Simplified When the Following Are All True

1. No perfect powers are factors of the radicand and all exponents in the radicand are less than the index.

2. No radicand contains a fraction.

3. No denominator contains a radical.

Assignment:

• Day 2: Continued…pp. 29-31 (3-9, 33-81 x3, 92, 101/102)

• Day 3: Continued…pp. 29-31 (3-9, 33-81 x3, 92, 101/102)

Even Answers: Day 2: Continued… pp. 29-31 (3-9, 33-81 x3, 92, 101/102)

4. ½ 92. (a) -1.0813

6.5/1 (b) -44.3624

8.243/1

36.4r5/6

42.-y11/2

48.x5/3

54.a) 4+ x √x b) (4+x) √(4+x)

60.4

66.3 r s2 4√r

72.xy3 /5 • 4√5x2

78. 5x2 y5 √2

1.3 Algebraic Expressions

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Polynomials• Addition

3 2 33 2 7 15 5 13 12x x x x x 3 2 3

3 2

3 2 7 15 5 13 12

8 2 6 27

x x x x x

x x x

Combine like terms

• Subtraction

3 2 3 26 1 3 2x x x x x x 3 2 3 2

3

6 1 3 2

2 4 1

x x x x x x

x x

Combine like terms

Distribute

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Polynomials• Multiplication

2 5 3 2x x

Combine like terms

Distribute2 (3 2) 5(3 2)x x x

Distribute26 4 15 10x x x 26 11 10x x

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Polynomials

• Division

http://www.youtube.com/watch?v=uERRlY-WmmU

1.3 (4-44 x 4) Answers Day 1

4.

8.

12.

16.

20.

24.

28.

32.

36.

40.

44.

xxx 2106 23 22 15234 yxyx

24424117 234 xxxx

5103102 23456 xxxxxx

yxz 26629 yx

22 164025 yxyx 4224 2 yyxx

yx 3223 6414410827 yxyyxx

yzxzxyzyx 126494 222

1.3 Factoring Polynomials

3 26 36t t

• Greatest Common Factor

• Grouping

26 6t t

2 2 2mx mx x

1 2 1mx x x

The terms have 6t2 in common

2 1mx x

Factor mx Factor –2

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Factoring Polynomials

• Sum/Difference of Two Cubes:

• Difference of Two Squares:

2 9m

38 1x 22 1 4 2 1x x x

3 3m m

2 2x y x y x y

3 3 2 2x y x y x xy y

Ex.

Ex.

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Factoring Polynomials• Trinomials

2 5 6x x

3 26 27 12x x x

3 2x x

Ex.

Ex.

Trial and Error

23 2 9 4x x x

Trial and Error 3 2 1 4x x x

Greatest Common Factor

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

http://www.youtube.com/watch?v=OFSrINhfNsQ

POLYNOMIAL FUNCTIONS

The DEGREE of a polynomial in one variable is the greatest exponent of its variable.

A LEADING COEFFICIENT is the coefficient of the term with the highest degree.

What is the degree and leading coefficient of 3x5 – 3x + 2 ?

POLYNOMIAL FUNCTIONS

A polynomial equation used to represent a function is called a POLYNOMIAL FUNCTION.

Polynomial functions with a degree of 1 are called LINEAR POLYNOMIAL FUNCTIONS

Polynomial functions with a degree of 2 are called QUADRATIC POLYNOMIAL FUNCTIONS

Polynomial functions with a degree of 3 are called CUBIC POLYNOMIAL FUNCTIONS

1.3 Answers (48-100 x 4) Day 2

48.52.56.60.64.68.72.76.80.84.88.92.96.100.

)32(5 yxy )5711(11 232 rsrssr

Irreducible

)27)(53( xx2)74( z

)49)(49( trtr )5)(5( xxx

)34)(34(4 yxyx )253036)(56( 2363 yyxxyx

)2)(3( xyxay )42)(3)(2( 2 xxxx

)2)(2)(4( 224 xxx

)23)(23( xyxy 2)12( xx

1.4 Rational ExpressionsP, Q, R, and S are polynomials

Addition

Operation

Multiplication

Subtraction

Division

P Q P Q

R R R

P Q P Q

R R R

P Q PQ

R S RS

P Q P S PS

R S R Q RQ

Notice the common denominator

Reciprocal and Multiply

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Rational Expressions• Simplifying

2

2

25

7 10

x

x x

5 5

2 5

x x

x x

Cancel common factorsFactor

• Multiplying

2 2

3 2

2 1 6 6

1

x x x x

x x

3

1 1 6 1

1 1

x x x x

x xx

FactorCancel common factors

2

Multiply Across

5

2

x

x

2

6 1x

x

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Rational Expressions• Adding/Subtracting

3 2

4x x

Combine like terms

3 4 2

( 4) 4

x x

x x x x

Must have LCD: x(x + 4)

3 12 2 5 12

( 4) 4

x x x

x x x x

Distribute and combine fractions

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Other Algebraic Fractions• Complex Fractions

32

94

x

xx

Simplify to get to here

Distribute and reduce to get here

32

94

xx

x xx

2

3 2

9 4

x

x

Multiply by the LCD: x

3 2 1

3 2 3 2 3 2

x

x x x

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

Other Algebraic Fractions

• Rationalizing a Denominator

7

3 y

Simplify

7 3

3 3

y

y y

21 7

9

y

y

Multiply by the conjugate

Notice: a b a b a b

Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc.

1.4 Answers (3-30 x 3)

6.

12.

18.

24.

30.

23

12

x

x

)2(

12 xx

2

2

)25(

425

s

ss

x

x )13(2

)5(

10222

uu

uu

1.4 Answers Day 2 (33-51 x 3)

36.

42.

48.

22 sr

rs

ax

2

)32)(322(

10

xhx

Answers to Ch. Review1. Positive 15. 2. 84 16. 3. 6-x 17. 4. 3.865 x 102

5. 0.000093 6. 1.76 x 1013

7. 4x2y4

8. 9. 10. 11. 17x3 - 6x + 312. 12x3 + 73x2 + 79x – 5213. x4 + 13x2 – 1414. 64x3 + 336x2y + 588xy2 + 343y3

Simplifying Radicals Video

http://www.youtube.com/watch?v=pZSuMBXzEic

Complex Fractions Video

• http://www.wonderhowto.com/how-to-simplify-complex-fractions-algebra-365934/

Negative Exponents Video

http://www.youtube.com/watch?v=c4aiYf3fzVQ

Rational Expressions Video

http://www.youtube.com/watch?v=L1KD-C0lWsY