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Write a few sentences comparing Write a few sentences comparing rational and irrational numersrational and irrational numers– Must give an example of eachMust give an example of each– State a topic involving each typeState a topic involving each type

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Algebra 2Algebra 2

Chapter 1Chapter 1Lessons 1.1 Real Number and Lessons 1.1 Real Number and

Number OperationsNumber Operations

1.1 Real Numbers 1.1 Real Numbers Rational NumbersRational Numbers

• Can be written as a quotient of integers.Can be written as a quotient of integers.• Can be written as decimals that terminate Can be written as decimals that terminate

or repeat.or repeat.

Irrational NumbersIrrational Numbers• Cannot be written as quotients of integers.Cannot be written as quotients of integers.• Cannot be written as decimals that Cannot be written as decimals that

terminate or repeat.terminate or repeat.

Real Numbers

Rational Numbers: Any number that can be written as a fraction where the numerator and denominator are both integers and the denominator doesn’t equal zero

Natural (Counting) numbers: N = {1, 2, 3, …}

Whole numbers: W = {0, 1, 2, 3, …}

Integers: Z = {0, 1, 2, 3, …}

Irrational Numbers: Any number that isn’t a rational number

Irrational Numbers

2- e

7

Rational Numbers

Integers

Whole Numbers

Natural Numbers

3.7 21 34.

-5 -2 -1

0

1 2 3

Real Numbers

Example 1

Graph the real numbers – and 3 on a number line.54

SOLUTION

Note that – = –1.25. Use a calculator to approximate

3 to the nearest tenth:

5

4

3 1.7. (The symbol means is approximately equal to.)

So, graph – between –2 and –1, and graph 3 between

1 and 2, as shown on the number line below.

54

EXAMPLE 2 Standardized Test Practice

SOLUTION

From lowest to highest, the elevations are – 408, –156, –86, – 40, –28, and –16.

ANSWER The correct answer is D.

GUIDED PRACTICE for Examples 1 and 2

Graph the numbers – 0.2, , –1, 2 , and – 4 on a number line.

710

1.

0 1 2 3 4 – 4 – 3 – 2 – 1

27

10– 0.2–1–4

ANSWER

GUIDED PRACTICE for Examples 1 and 2

Which list shows the numbers in increasing order?2.

– 0.5, 1.5, – 2, – 0.75, 7

– 0.5, – 2, – 0.75, 1.5, 7

– 2, – 0.75, – 0.5, 1.5, 7

7 , 1.5, – 0.5 , – 0.75, – 2

ANSWER The correct answer is C.

1.1 Properties of Addition and 1.1 Properties of Addition and MultiplicationMultiplication

Let a, b, and c be real numbers.Let a, b, and c be real numbers.

PropertyProperty AdditionAddition ExampleExample MultiplicationMultiplication ExampleExample

ClosureClosure a + b is a real a + b is a real numbernumber

5 + -6 = -15 + -6 = -1 ab is a real ab is a real numbernumber

½ (4) = 2½ (4) = 2

CommutativeCommutative a+b=b+aa+b=b+a -3+7=7+-3-3+7=7+-3 ab=baab=ba -4(3)=3(-4)-4(3)=3(-4)

AssociativeAssociative (a+b)+c=a+(b+c)(a+b)+c=a+(b+c) (2+6)+1=2+(6+1)(2+6)+1=2+(6+1) (ab)c=a(bc)(ab)c=a(bc) (2*7)1=2(7*1)(2*7)1=2(7*1)

IdentityIdentity a+0=a, a+0=a, 0+a=a0+a=a

-2+0=-2, -2+0=-2, 0+-2=-20+-2=-2

a*1=a, 1*a=aa*1=a, 1*a=a ¾ *1= ¾ ¾ *1= ¾

1* ¾ = ¾ 1* ¾ = ¾

InverseInverse a+(-a)=0a+(-a)=0 .5+-.5=0.5+-.5=0 a* 1/a = 1,a* 1/a = 1,

a≠0a≠02* ½ = 12* ½ = 1

DistributiveDistributive a(b+c)=ab+ac a(b+c)=ab+ac (combines adding & multiplying)(combines adding & multiplying) 2(1+4)=2*1+2*42(1+4)=2*1+2*4

EXAMPLE 3 Identify properties of real numbers

Identify the property that the statement illustrates.

a. 7 + 4 = 4 + 7

b. 13 = 11

13

SOLUTION

Inverse property of multiplication

Commutative property of addition

SOLUTION

Identify the property that the statement illustrates.

4. 15 + 0 = 15

SOLUTION

Identity property of addition.

Associative property of multiplication.

SOLUTION

3. (2 3) 9 = 2 (3 9)

GUIDED PRACTICE for Examples 3 and 4

Identify the property that the statement illustrates.

5. 4(5 + 25) = 4(5) + 4(25)

SOLUTION

Identity property of multiplication.

Distributive property.

SOLUTION

6. 1 500 = 500

GUIDED PRACTICE for Examples 3 and 4

Classwork

• Guided practice

• Page 6. #1-14

Homework

• Page 6. #15, 19, 22, 24, 27, 31