Chapter 2Chapter 2

Working with Real Working with Real NumbersNumbers

2-12-1

Basic AssumptionsBasic 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

COMMUTATIVE COMMUTATIVE PROPERTIESPROPERTIES

a + b = b + a

ab = ba

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

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)

Properties of Properties of EqualityEquality

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

2-22-2

Addition on a Number Addition on a Number LineLine

IDENTITY PROPERTIESIDENTITY PROPERTIES

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

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

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

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)

2-32-3

Rules for AdditionRules for Addition

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

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

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

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

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

2-42-4

Subtracting Real Subtracting Real NumbersNumbers

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

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)

2-52-5

The Distributive PropertyThe Distributive Property

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

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)

2-62-6

Rules for MultiplicationRules for Multiplication

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

MULTIPLICATIVE MULTIPLICATIVE PROPERTY OF 0PROPERTY OF 0

For every real number a,

a · 0 = 0 and 0 · a = 0

MULTIPLICATIVE MULTIPLICATIVE PROPERTY OF -1PROPERTY OF -1

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

PROPERTY of OPPOSITES PROPERTY of OPPOSITES in PRODUCTSin PRODUCTS

For all real number a and

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

and-ab = a(-b)

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)]

2-72-7 Problem Solving: Problem Solving:

Consecutive IntegersConsecutive Integers

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

integer.

…-6, -4, -2, 0, 2, 4, 6,…

ODD INTEGERODD INTEGER

An integer that is not even.

…-5, -3, -1, 1, 3, 5,…

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, …

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

CONSECUTIVE EVEN CONSECUTIVE EVEN INTEGERINTEGER

Integers obtained by

counting by twos beginning with any even

integer.

12, 14, 16

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.

CONSECUTIVE ODD CONSECUTIVE ODD INTEGERINTEGER

Integers obtained by

counting by twos beginning with any odd

integer.

5,7,9

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.

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

NumberNumber

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

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

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.

2-9 2-9

Dividing Real Dividing Real NumbersNumbers

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

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

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

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

ExamplesExamples

1.1. 4 ÷ 164 ÷ 16

2.2. 8x8x

1616

3.3. 5x + 255x + 25

55

The EndThe End