chapter 2 working with real numbers. 2-1 basic assumptions

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Chapter 2 Chapter 2 Working with Real Working with Real Numbers Numbers

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Page 1: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Chapter 2Chapter 2

Working with Real Working with Real NumbersNumbers

Page 2: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-12-1

Basic AssumptionsBasic Assumptions

Page 3: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

CLOSURE PROPERTIESCLOSURE PROPERTIES

a + b and ab are uniquea + b and ab are unique7 + 5 = 127 + 5 = 127 x 5 = 357 x 5 = 35

Page 4: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

COMMUTATIVE COMMUTATIVE PROPERTIESPROPERTIES

a + b = b + a

ab = ba

2 + 6 = 6 + 22 x 6 = 6 x 2

Page 5: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ASSOCIATIVE ASSOCIATIVE PROPERTIESPROPERTIES

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

+c)(ab)c = a(bc)(5 + 15) + 20 = 5 + (15

+20)(5·15)20 = 5(15·20)

Page 6: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Properties of Properties of EqualityEquality

Page 7: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Reflexive Property - Reflexive Property - a = aa = a Symmetric Property –Symmetric Property –

IfIf a = ba = b,, thenthen b = ab = a Transitive Property –Transitive Property –

If If a = ba = b, and , and b = cb = c, then , then a = a = cc

Page 8: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-22-2

Addition on a Number Addition on a Number LineLine

Page 9: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

IDENTITY PROPERTIESIDENTITY PROPERTIES

There is a unique real number 0 such that:a + 0 = 0 + a = a

-3 + 0 = 0 + -3 = -3

Page 10: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

For each For each aa, there is a unique , there is a unique real number real number – – aa such that: such that:

a + (-a) = 0a + (-a) = 0 and and (-(-aa)+ )+ aa = 0 = 0 (-(-aa)) is called the is called the oppositeopposite or or additive inverseadditive inverse of of aa

PROPERTY OF OPPOSITESPROPERTY OF OPPOSITES

Page 11: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Property of the opposite of Property of the opposite of a Suma Sum

For all real numbers a and b:-(a + b) = (-a) + (-b)

The opposite of a sum of real numbers is equal to the sum

of the opposites of the numbers.

-(8 +2) = (-8) + (-2)

Page 12: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-32-3

Rules for AdditionRules for Addition

Page 13: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Addition Rules

1.1. If a and b are both If a and b are both positive, then positive, then

a + b = a + b = aa + + bb

3 + 7 = 103 + 7 = 10

Page 14: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Addition Rules

2.2. If a and b are both If a and b are both negative, then negative, then

a + b = -(a + b = -(aa + + bb))

(-6) + (-2) = -(6 +2) = -8(-6) + (-2) = -(6 +2) = -8

Page 15: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Addition Rules3.3. If If aa is positive and is positive and bb is is

negative and negative and aa has the has the greater absolute value, greater absolute value, then then

a + b = a + b = aa - - bb

6 + (-2) = (6 - 2) = 46 + (-2) = (6 - 2) = 4

Page 16: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Addition Rules4.4. If If aa is positive and is positive and bb is is

negative and negative and bb has the has the greater absolute value, greater absolute value, then then

a + b = -(a + b = -( b b - - aa))

4 + (-9) = -(9 -4) = -54 + (-9) = -(9 -4) = -5

Page 17: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Addition Rules

5.5. If If aa and and bb are opposites, are opposites, then then a + ba + b = 0 = 0

2 + (-2) = 02 + (-2) = 0

Page 18: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-42-4

Subtracting Real Subtracting Real NumbersNumbers

Page 19: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

DEFINITION of DEFINITION of SUBTRACTIONSUBTRACTION

For all real numbers For all real numbers a a andand bb,,

aa – b = – b = aa + (-b) + (-b)

To subtract any real To subtract any real number, add its oppositenumber, add its opposite

Page 20: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExamplesExamples

1.1. 3 – (-4)3 – (-4)

2.2. -y – (-y + 4)-y – (-y + 4)

3.3. -(f + 8)-(f + 8)

4.4. -(-b + 6 – a)-(-b + 6 – a)

5.5. m – (-n + 3)m – (-n + 3)

Page 21: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-52-5

The Distributive PropertyThe Distributive Property

Page 22: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

DISTRIBUTIVE DISTRIBUTIVE PROPERTYPROPERTY

a(b + c) = ab + ac(b +c)a = ba + ca

5(12 + 3) = 5•12 + 5 •3 = 75

(12 + 3)5 = 12• 5 + 3 • 5 = 75

Page 23: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExamplesExamples

1.1. 2(3x + 4)2(3x + 4)

2.2. 5n + 7(n – 3)5n + 7(n – 3)

3.3. 2(x – 6) + 92(x – 6) + 9

4.4. 8 + 3(4 – y)8 + 3(4 – y)

5.5. 8(k + m) - 15(2k + 5m)8(k + m) - 15(2k + 5m)

Page 24: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-62-6

Rules for MultiplicationRules for Multiplication

Page 25: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

IDENTITY PROPERTY IDENTITY PROPERTY of MULTIPLICATIONof MULTIPLICATION

There is a unique real

number 1 such that for every real number a,

a · 1 = a and 1 · a = a

Page 26: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

MULTIPLICATIVE MULTIPLICATIVE PROPERTY OF 0PROPERTY OF 0

For every real number a,

a · 0 = 0 and 0 · a = 0

Page 27: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

MULTIPLICATIVE MULTIPLICATIVE PROPERTY OF -1PROPERTY OF -1

For every real number a,a(-1) = -a and (-1)a = -a

Page 28: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

PROPERTY of OPPOSITES PROPERTY of OPPOSITES in PRODUCTSin PRODUCTS

For all real number a and

b,-ab = (-a)(b)

and-ab = a(-b)

Page 29: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExamplesExamples

1.1. (-1)(3d – e + 8)(-1)(3d – e + 8)

2.2. -6(7n – 6)-6(7n – 6)

3.3. -[-4(x – y)]-[-4(x – y)]

Page 30: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-72-7 Problem Solving: Problem Solving:

Consecutive IntegersConsecutive Integers

Page 31: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

EVEN INTEGEREVEN INTEGER An integer that is the product of 2 and any

integer.

…-6, -4, -2, 0, 2, 4, 6,…

Page 32: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ODD INTEGERODD INTEGER

An integer that is not even.

…-5, -3, -1, 1, 3, 5,…

Page 33: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

Consecutive IntegersConsecutive Integers

Integers that are listed Integers that are listed in in natural ordernatural order, from , from

least to greatestleast to greatest

……,-2, -1, 0, 1, 2, …,-2, -1, 0, 1, 2, …

Page 34: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExampleExampleThree consecutive integers Three consecutive integers have the sum of 24. Find have the sum of 24. Find all three integers.all three integers.

Page 35: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

CONSECUTIVE EVEN CONSECUTIVE EVEN INTEGERINTEGER

Integers obtained by

counting by twos beginning with any even

integer.

12, 14, 16

Page 36: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExampleExampleFour consecutive even Four consecutive even integers have a sum of 36. integers have a sum of 36. Find all four integers. Find all four integers.

Page 37: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

CONSECUTIVE ODD CONSECUTIVE ODD INTEGERINTEGER

Integers obtained by

counting by twos beginning with any odd

integer.

5,7,9

Page 38: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExampleExampleThere are three There are three consecutive odd consecutive odd integers. The largest integers. The largest integer is 9 less than the integer is 9 less than the sum of the smaller two sum of the smaller two integers. Find all three integers. Find all three integers.integers.

Page 39: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-82-8 The Reciprocal of a Real The Reciprocal of a Real

NumberNumber

Page 40: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

PROPERTY OF PROPERTY OF RECIPROCALSRECIPROCALS

For each For each aa except 0, there is except 0, there is a unique real number 1/a a unique real number 1/a such that:such that:

a a ·· (1/a) = 1 (1/a) = 1 and and (1/a)(1/a)·· a = 1 a = 1 1/a1/a is called the is called the reciprocalreciprocal or or multiplicative inversemultiplicative inverse of of aa

Page 41: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

PROPERTY of the PROPERTY of the RECIPROCAL of the OPPOSITE RECIPROCAL of the OPPOSITE

of a Numberof a Number

For each For each aa except 0, except 0, 1/-1/-aa = -1/ = -1/aa The reciprocal of The reciprocal of ––aa is is -1/-1/aa

Page 42: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

PROPERTY of the PROPERTY of the RECIPROCAL of a PRODUCTRECIPROCAL of a PRODUCT

For all nonzero numbers For all nonzero numbers aa and and bb,,

1/1/abab = 1/ = 1/aa ·1/ ·1/bbThe reciprocal of the product The reciprocal of the product of two nonzero numbers is of two nonzero numbers is the product of their the product of their reciprocals.reciprocals.

Page 43: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2-9 2-9

Dividing Real Dividing Real NumbersNumbers

Page 44: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

DEFINITION OF DIVISIONDEFINITION OF DIVISION

For every real number For every real number aa and and every nonzero real number every nonzero real number b,b, the quotient is defined by:the quotient is defined by:

aa÷b = a·1/b÷b = a·1/b

To divide by a nonzero number, To divide by a nonzero number, multiply by its reciprocalmultiply by its reciprocal

Page 45: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

1.1. The quotient of two The quotient of two positive numbers or positive numbers or two negative two negative numbers is a positive numbers is a positive numbernumber-24/-3 = 8-24/-3 = 8 andand 24/3 = 24/3 = 88

Page 46: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

2.2. The quotient of two The quotient of two numbers when one is numbers when one is positive and the other positive and the other negative is a negative negative is a negative number.number.

24/-3 = -824/-3 = -8 andand -24/3 = -8-24/3 = -8

Page 47: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

PROPERTY OF DIVISIONPROPERTY OF DIVISION

For all real numbers For all real numbers a, b,a, b, andand c c such thatsuch that cc 0, 0,

a + b = a + b anda + b = a + b and c c cc c c

a - b = a - ba - b = a - b c c cc c c

Page 48: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

ExamplesExamples

1.1. 4 ÷ 164 ÷ 16

2.2. 8x8x

1616

3.3. 5x + 255x + 25

55

Page 49: Chapter 2 Working with Real Numbers. 2-1 Basic Assumptions

The EndThe End