Chapter 1.1 properties of-real-numbers

Properties of Real Numbers Properties of Real Numbers

1) real numbers2) rational numbers3) irrational numbers

Classify real numbers.

Use the properties of real numbers to evaluate expressions.

All of the numbers that you use in everyday life are real numbers.

Properties of Real Numbers Properties of Real Numbers

All of the numbers that you use in everyday life are real numbers.

Each real number corresponds to exactly one point on the number line, and

Properties of Real Numbers Properties of Real Numbers

All of the numbers that you use in everyday life are real numbers.

Each real number corresponds to exactly one point on the number line, and

x

Properties of Real Numbers Properties of Real Numbers

All of the numbers that you use in everyday life are real numbers.

Each real number corresponds to exactly one point on the number line, and

x

0 1 2 3 4 5-5 -4 -2 -1-3

Properties of Real Numbers Properties of Real Numbers

All of the numbers that you use in everyday life are real numbers.

Each real number corresponds to exactly one point on the number line, and

x

0 1 2 3 4 5-5 -4 -2 -1-3

every point on the number line represents one real number.

Properties of Real Numbers Properties of Real Numbers

All of the numbers that you use in everyday life are real numbers.

Each real number corresponds to exactly one point on the number line, and

x

0 1 2 3 4 5-5 -4 -2 -1-3

21

2 2

every point on the number line represents one real number.

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

The decimal form of a rational number is either a terminating or repeating decimal.

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

The decimal form of a rational number is either a terminating or repeating decimal.

Examples: ratio form decimal form

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

The decimal form of a rational number is either a terminating or repeating decimal.

Examples: ratio form decimal form

9 0.3

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

The decimal form of a rational number is either a terminating or repeating decimal.

Examples: ratio form decimal form

9 0.3

83

375.0

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

zeroRational numbers can be expressed as a ratio , where a and b areintegers and b is not ____! b

a

The decimal form of a rational number is either a terminating or repeating decimal.

Examples: ratio form decimal form

9 0.3

83

375.0

73

428571.0

or . . . 714285714285714285.0

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Examples:

. . . 141592654.3

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Examples:

. . . 141592654.3 More Digits of PI?

e . . . 718281828.2

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Examples:

. . . 141592654.3 More Digits of PI?

e . . . 718281828.2

2 3 5 7 11 13

Do you notice a pattern within this group of numbers?

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Examples:

. . . 141592654.3 More Digits of PI?

e . . . 718281828.2

2 3 5 7 11 13

Do you notice a pattern within this group of numbers?

Properties of Real Numbers Properties of Real Numbers

Real numbers can be classified a either _______ or ________.rational irrational

A real number that is not rational is irrational.

The decimal form of an irrational number neither __________ nor ________.terminates repeats

Examples:

. . . 141592654.3 More Digits of PI?

e . . . 718281828.2

2 3 5 7 11 13

Do you notice a pattern within this group of numbers?

They’re all PRIME numbers!

Properties of Real Numbers Properties of Real Numbers

- Numbers Real

Relationships among the real numbers - (sets and subsets).

Properties of Real Numbers Properties of Real Numbers

- Numbers Real

Q = rationals

Q I

I = irrationals

Relationships among the real numbers - (sets and subsets).

Properties of Real Numbers Properties of Real Numbers

- Numbers Real

Q = rationals

Q I

I = irrationals

Z

Z = integers

Relationships among the real numbers - (sets and subsets).

Properties of Real Numbers Properties of Real Numbers

- Numbers Real

Q = rationals

Q I

I = irrationals

Z

Z = integers

W

W = wholes

Relationships among the real numbers - (sets and subsets).

Properties of Real Numbers Properties of Real Numbers

The square root of any whole number is either whole or irrational.

Properties of Real Numbers Properties of Real Numbers

The square root of any whole number is either whole or irrational.

For example, is a whole number, but , since it lies between 5 and 6, must be irrational.

36 30

Properties of Real Numbers Properties of Real Numbers

The square root of any whole number is either whole or irrational.

x

0 1 32 4 5 6 7 98 10

For example, is a whole number, but , since it lies between 5 and 6, must be irrational.

36 30

36

. . . 477225575.5

25

30

Properties of Real Numbers Properties of Real Numbers

The square root of any whole number is either whole or irrational.

x

0 1 32 4 5 6 7 98 10

For example, is a whole number, but , since it lies between 5 and 6, must be irrational.

36 30

36

. . . 477225575.5

25

30

Common Misconception:

Do not assume that a number is irrational just because it is expressed using the square root symbol. Find its value first!

Properties of Real Numbers Properties of Real Numbers

The square root of any whole number is either whole or irrational.

x

0 1 32 4 5 6 7 98 10

For example, is a whole number, but , since it lies between 5 and 6, must be irrational.

36 30

36

. . . 477225575.5

25

30

Common Misconception:

Do not assume that a number is irrational just because it is expressed using the square root symbol. Find its value first!

Study Tip:

KNOW and recognize (at least) these numbers,

169644936251694 14412110081

Properties of Real Numbers Properties of Real Numbers

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Properties of Real Numbers Properties of Real Numbers

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Real Number Properties

For any real numbers a, b, and c.

Property Addition Multiplication

Associative

Identity

Inverse

Distributive

Properties of Real Numbers Properties of Real Numbers

Commutative

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Real Number Properties

For any real numbers a, b, and c.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

ba ab ba ab

Properties of Real Numbers Properties of Real Numbers

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Real Number Properties

For any real numbers a, b, and c.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

ba ab ba ab

cba cba cba cba

Properties of Real Numbers Properties of Real Numbers

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Real Number Properties

For any real numbers a, b, and c.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

ba ab ba ab

cba cba cba cba

0a a a0 1a a a1

Properties of Real Numbers Properties of Real Numbers

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Real Number Properties

For any real numbers a, b, and c.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

ba ab ba ab

cba cba cba cba

0a a a0 1a a a1

aa 0 aa then0, a If

a

a1

1 aa1

Properties of Real Numbers Properties of Real Numbers

The real number system is an example of a mathematical structure called a field.

Some of the properties of a field are summarized in the table below:

Real Number Properties

For any real numbers a, b, and c.

Property Addition Multiplication

Commutative

Associative

Identity

Inverse

Distributive

ba ab ba ab

cba cba cba cba

0a a a0 1a a a1

aa 0 aa then0, a If

a

a1

1 aa1

)( cba and acab acb )( caba

Properties of Real Numbers Properties of Real Numbers

Reciprocals

• The Reciprocal of a is providing a does NOT equal 0.

• Definition of Subtraction:– Adding the opposite:

• Definition of Division:– Multiplying by the reciprocal:

a

1

)( baba

ba

b

a 1

Operations with Real Numbers

A. The sum of -5 and -13 is

B.The difference of -17 and -8 is

C. The product of -3 and -6 is

D. The quotient of 28 and -7 is

18

)8(17

9817

1863

47

28

Real Life Wind Farms• One barrel of oil can generate 545 kilowatt-hours of

electricity. In 1990, the 17,000 windmills in California could generate up to 1.5 million kilowatt-hours per hour. At peak capacity, how many barrels of oil could be saved each hour? Operating at 75% of peak efficiency, how many barrels of oil could be saved in a year?

)(545

)(

barrelhourperhoursKilowatt

windhourperhoursKilowattOilofBarrels

545

000,500,1OilofBarrels yearperbarrels752,2

At 75% peak capacity for a year

year

days

day

hours

hour

barrels 36524275275.0

year

barrels61018

Consider these Java Applets to better understand theDistributive Property

Algebra Tiles 1

Algebra Tiles 2