Properties of Real Numbers

2

Sets

• In mathematics, a set is a collection of things

• Sets can be studies as a topic all on its own (known as set theory), but we only need to know a few basic ideas

• We often use braces { } to indicate a set

• Sets can have a finite number of objects, an infinite number of objects, or no objects

• This last set is called the empty set

3

Examples

• The set of coins in my piggy bank

• The set of students at DRHS

• The set of first names in the world

4

Examples

• Sometimes we are concerned, not with an entire set, but some small part of the set

• This small part is also a set

• Because it is contained in a bigger set, we call it a subset of the larger set

• Every element in a subset is also in the larger set, but not every element in the larger set is in the subset

5

Examples

• The set of coins in my piggy bank

• The set of 5 dimes in my piggy bank is a subset of the coins in my piggy bank

6

Examples

• The set of students at DRHS

• The set of football players at DRHS is a subset of the set of students at DRHS

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Examples

• The set of first names in the world

• The set of first names beginning with the letter Q is a subset of the set of first names in the world

8

Examples

• If we name a set that has no elements, we say that it is an empty set

• We signify that a set is empty by using or

9

Examples

• The set of students at DRHS

• The set of students at DRHS who are age 30 or over

• There are no students at DRHS who are age 30 or over, so this is an empty set

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The Real Numbers

• In algebra we will be dealing with the elements of a set called the set of real numbers

• This set has a number of subset

• Today you will learn the names of these subsets and how to know if a given number belongs in the subset or not

The Real Numbers

• In algebra you will be working with real numbers

• Unless otherwise indicated, any variable that you see or work with will represent some real number

• The real numbers form a set (a collection) and there are several subsets (a collection within a collection) of the real numbers that will be important for you to know about and to identify

The Real Numbers

• The real numbers contain both the rational numbers and the irrational numbers

• Every real number is either rational or irrational; it cannot be both!

• You must be able to tell whether a number is rational or irrational

• The next slide shows one way to think about how these sets are related

The Real Numbers

The Rational Numbers

• The rational numbers include three other subsets

• These are• The integers• The whole numbers• The natural or counting numbers

• The next slide shows how these sets are related

The Rational Numbers

Natural Numbers

• The natural numbers are the set of numbers

• Note that• Natural numbers don’t have decimal or fraction parts• Natural numbers are all positive (no negative numbers and no zero)• Natural numbers are sometimes called counting numbers

Whole Numbers

• The whole numbers are the set

• Note that• The whole numbers include all the natural numbers and also zero• The whole numbers have no decimal or fraction part• The whole numbers are non-negative

The Integers

• These integers extend the whole numbers to the left of zero on the number line

The Integers

• We can represent the set of integers as

• Note that• Integers do not have a fraction or decimal part• Unlike the whole numbers, the integers do not have a “first” number

The Rational Numbers

• Missing on our number line is values between the integers

• We create these numbers by dividing up each space between two integers into equal parts

• In that way we create the rational numbers

• The rational numbers are all those numbers that can be written as , where and are integers

The Rational Numbers

• Rational numbers can also be expressed as decimals• that terminate (for example, 1.5)• that have a repeating pattern (for example, 1.33333…)

• Whole numbers and integers are also rational numbers because each can be written as

The Irrational Numbers

• It was discovered by a follower of Pythagoras that, no matter into how many parts we divide the interval between two integers, there will always be “holes”

• These holes represent numbers that cannot be written as the ratio of two integers

• We therefore call them irrational numbers

• Some numbers that are irrational include

Example 1: Classify Real Numbers

Write each number in its most inclusive set using the figure represented below.

a)

b)

c)

d)

e)

f)

Example 1: Classify Real NumbersReal Numbers

Rational Numbers

Integers

Irrational Numbers

Whole Numbers

Exercise 1.1

• Handout

The Field of Real Numbers

• In the previous lesson you learned how the real numbers can be classified

• Now we must learn how to work with the real numbers

• In mathematics, what I will be describing is called a field

• A field consists of a set, two operations (i.e., two different ways to “act on” the things in the set), and the set of properties that you are about to study

The Field of Real Numbers

• In algebra II our field will consist of the set of real numbers, the operations of addition and multiplication, and these rules (one of each for addition and multiplication)• Closure• Commutativity• Associativity• Identity Element• Inverse Element• Distributive Property

The Field of Real Numbers

• In algebra II our field will consist of the set of real numbers, the operations of addition and multiplication, and these rules (one of each for addition and multiplication)• Closure—how to think of or of • Commutativity and associativity—what we can do with or without changing

the value• Identity Element—a unique number with a special property (one each for

addition and multiplication)• Inverse Element—a number that is uniquely paired with another number and

that together have a special property• Distributive Property—how to deal with expressions that include addition and

multiplication

Closure Property

If a and b are real numbers, then If a and b are real numbers, then

Closure Property

If a and b are real numbers, then

We should think of any combination of addition as a single, real number

If a and b are real numbers, then

We should think of any combination of multiplication as a single, real number

Commutative Property

If a and b are real numbers, then If a and b are real numbers, then

Commutative Property

If a and b are real numbers, then

We can change the order of addition without changing the result

If a and b are real numbers, then

We can change the order of multiplication without changing the result

Associative Property

If a, b, and c are real numbers, then

If a, b, and c are real numbers, then

Associative Property

If a, b, and c are real numbers, then

Under addition, we can place parentheses wherever we please, or choose not to use parentheses

If a, b, and c are real numbers, then

Under multiplication, we can place parentheses wherever we please or choose not to use parentheses

Identity Property

There exists a unique number called zero (0) such that, for any number a

There exists a unique number called one (1) such that, for any number a

Identity Property

There exists a unique number called zero (0) such that, for any number a

If we ever end up with zero plus a number, we can drop the zero

There exists a unique number called one (1) such that, for any number a

If we ever end up with one times a number, we can drop the 1

Inverse Property

For every non-zero real number a, there exists the number such that

For every non-zero real number a, there exists the number such that

Inverse Property

For every non-zero real number a, there exists the number such that

The “canceling” property for addition

For every non-zero real number a, there exists the number such that

The “canceling” property for multiplication

Distributive Property

If a, b, and c are real numbers, then

and

Distributive Property

If a, b, and c are real numbers, then

and

The top equation is multiplication. The bottom is factoring.

Examples

Use the:

a) Commutative Property for Addition

b) Commutative Property for Multiplication

c) Associative Property for Addition

d) Associative Property for Multiplication

Examples

Use the:

a) Commutative Property for Addition

Examples

Use the:

b) Commutative Property for Multiplication

Examples

Use the:

c) Associative Property for Addition

Examples

Use the:

d) Associative Property for Multiplication

Examples

What is the multiplicative inverse for

What is the additive inverse for

Examples

What is the multiplicative inverse for

Multiplicative inverse (reciprocal) is

What is the additive inverse for

Examples

What is the multiplicative inverse for

What is the additive inverse for

Additive inverse is

Definitions of Subtraction and Division

DEFINITION:

For real numbers a and b, we define subtraction to be

For real numbers a and b, with , we define division to be

Examples

Show that each equation is a true statement. Justify each step using the number properties.

a)

b)

c)

d)

e)

Examples

• Justification

• Definition of division

• Distributive Property

• Commutative Property for Multiplication

• Associative Property for Multiplication

• Multiplication

• Commutative Property for Addition

Examples

• Justification

• Definition of subtraction

• Commutative Property for Addition

• Associative Property for Addition

• Inverse Property of Addition

• Identity Property of Addition

Examples

• Justification

• Definition of Division

• Commutative Property for Multiplication

• Associative Property for Multiplication

• Inverse Property for Multiplication

• Identity Property for Multiplication

Examples

• Justification

• Definition of Division

• Distributive Property

• Commutative Property for Multiplication

• Associative Property for Multiplication

• Inverse Property for Multiplication

• Identity Property for Multiplication

• Addition

Examples

• Justification

• Commutative Property of Addition

• Associative Property of Addition

• Distributive Property (Factoring)

• Addition

• Commutative Property for Addition

Exercise 1.2: Number Properties

• Handout