The Real Number The Real Number SystemSystem

Real NumbersReal NumbersReal numbers consist of all the rational Real numbers consist of all the rational and irrational numbers.and irrational numbers.

The real number system has many The real number system has many subsets:subsets:

Natural Numbers Natural Numbers

Whole Numbers Whole Numbers

Integers Integers

Natural NumbersNatural NumbersNatural numbersNatural numbers are the set of are the set of counting numbers.counting numbers.

{1, 2, 3,…}{1, 2, 3,…}

Whole NumbersWhole NumbersWhole numbersWhole numbers are the set of are the set of numbers that include 0 plus the numbers that include 0 plus the set of natural numbers.set of natural numbers.

{0, 1, 2, 3, 4, 5,…}{0, 1, 2, 3, 4, 5,…}

IntegersIntegersIntegersIntegers are the set of whole are the set of whole numbers and their opposites.numbers and their opposites.

{…,-3, -2, -1, 0, 1, 2, 3,…}{…,-3, -2, -1, 0, 1, 2, 3,…}

Rational NumbersRational NumbersRational numbers Rational numbers are any numbers are any numbers that can be expressed in the form of that can be expressed in the form of , where , where aa and and bb are integers, and b are integers, and b ≠ 0≠ 0. .

They can always be expressed by They can always be expressed by using terminating decimals or using terminating decimals or repeating decimals. repeating decimals.

b

a

Terminating Terminating DecimalsDecimals

Terminating decimals are decimals Terminating decimals are decimals that contain a finite number of that contain a finite number of digits.digits.

Examples:Examples: 36.836.8 0.1250.125 4.54.5

Repeating DecimalsRepeating DecimalsRepeating decimals are decimals that contain Repeating decimals are decimals that contain a infinite number of digits.a infinite number of digits.

Examples:Examples: 0.333…0.333… 7.689689…7.689689…

FYI…The line above the decimals indicate that numberFYI…The line above the decimals indicate that number

repeats.repeats.

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Irrational NumbersIrrational NumbersIrrational numbersIrrational numbers are any numbers that are any numbers that cannot be expressed as . cannot be expressed as .

They are expressed as They are expressed as non-terminating, non-non-terminating, non-repeating decimalsrepeating decimals; decimals that go on ; decimals that go on forever without repeating a pattern.forever without repeating a pattern.

Examples of irrational numbers:Examples of irrational numbers:0.34334333433334…0.34334333433334…45.86745893…45.86745893… (pi)(pi)

b

a

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Other Vocabulary Other Vocabulary Associated with the Real Associated with the Real

Number SystemNumber System……(ellipsis)—continues without end(ellipsis)—continues without end

{ } (set)—a collection of objects or { } (set)—a collection of objects or numbers. Sets are notated by using numbers. Sets are notated by using braces { }.braces { }.

Finite—having bounds; limitedFinite—having bounds; limited

Infinite—having no boundaries or limitsInfinite—having no boundaries or limits

Venn diagram—a diagram consisting of Venn diagram—a diagram consisting of circles or squares to show relationships circles or squares to show relationships of a set of data.of a set of data.

ExampleExampleClassify all the following numbers as natural, Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that whole, integer, rational, or irrational. List all that apply.apply.a.a. 117117b.b. 00c.c. -12.64039…-12.64039…d.d. -½-½e.e. 6.366.36f.f. g.g. -3-3

To show how these number are classified, use To show how these number are classified, use the Venn diagram. Place the number where it the Venn diagram. Place the number where it

belongs on the Venn diagram.belongs on the Venn diagram.

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Rational Numbers

Integers

Whole Numbers

NaturalNumbers

Irrational Numbers

-12.64039…

117

0

6.36

-3

SolutionSolutionNow that all the numbers are placed where they Now that all the numbers are placed where they belong in the Venn diagram, you can classify belong in the Venn diagram, you can classify each number:each number:

117 is a natural number, a whole number, an 117 is a natural number, a whole number, an integer, and a rational number.integer, and a rational number. is a rational number.is a rational number.0 is a whole number, an integer, and a rational 0 is a whole number, an integer, and a rational number.number.-12.64039… is an irrational-12.64039… is an irrational number. number.-3 is an integer and a rational number.-3 is an integer and a rational number.6.36 is a rational number.6.36 is a rational number. is an irrational number.is an irrational number. is a rational number.is a rational number.9

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FYI…FYI…When taking the square root of any When taking the square root of any number that is not a perfect square, the number that is not a perfect square, the resulting decimal will be non-resulting decimal will be non-terminating and non-repeating. terminating and non-repeating. Therefore, those numbers are always Therefore, those numbers are always irrational.irrational.