NWZ

Q

IR

The Real Number SystemThe Real Number SystemPresented by Mr. LawsPresented by Mr. Laws

88thth Grade Math Grade MathJCMSJCMS

Goal/ObjectiveGoal/Objective

8.NS8.NS: Know that there are numbers that : Know that there are numbers that are not rational and approximate them are not rational and approximate them by rational numbersby rational numbers. .

8.NS.18.NS.1: Understand informally that every : Understand informally that every number has a decimal expansion; the number has a decimal expansion; the rational numbers are those with decimal rational numbers are those with decimal expansions that terminate in zeros or expansions that terminate in zeros or eventually repeat. Know that other numbers eventually repeat. Know that other numbers are called irrational numbers. are called irrational numbers.

Essential QuestionEssential Question

How do I understand and perform How do I understand and perform operations with the Real Number operations with the Real Number System?System?

NWZQ

IR

The Real Number SystemThe Real Number System The Real Number System is made up The Real Number System is made up

of a set of rational and irrational of a set of rational and irrational numbers.numbers.

It has at five subsets:It has at five subsets:1.1. Rational Numbers (Q)Rational Numbers (Q)2.2. Integers (Z)Integers (Z)3.3. Whole Numbers (W)Whole Numbers (W)4.4. Natural Numbers (N)Natural Numbers (N)5.5. Irrational Numbers (IR)Irrational Numbers (IR)

Real Numbers DefinitionsReal Numbers Definitions

Real NumbersReal Numbers – consists of all – consists of all rationalrational and and irrationalirrational numbers. numbers. It includes any number that can be It includes any number that can be

written as a fraction, mixed numbers, written as a fraction, mixed numbers, terminating and repeating decimals, terminating and repeating decimals, whole numbers, integers. whole numbers, integers.

41

2

354

1.5

2.3333

O -6

2

Rational NumbersRational Numbers

Rational NumbersRational Numbers – consists of – consists of integers, terminating, and repeating integers, terminating, and repeating decimals. decimals.

It can also be expressed as a It can also be expressed as a fraction.fraction.

1.5

2 .9

7.5

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

16 458 8.833336

Rational NumbersRational Numbers

IntegersIntegers – consist of natural – consist of natural numbers, their opposites (numbers, their opposites (negative negative ## ’’ss), and zero. ), and zero.

It It does not include fractions or does not include fractions or decimals.decimals.

All whole numbers are integers. All whole numbers are integers. For example:For example:

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

IntegersIntegers Whole numbersWhole numbers – consist of natural – consist of natural

numbers and zero. numbers and zero. {0, 1, 2, 3, 4,…}{0, 1, 2, 3, 4,…}

Natural numbersNatural numbers – are all the – are all the counting numberscounting numbers. . {1, 2, 3, 4…}{1, 2, 3, 4…}

Negative numbersNegative numbers ={…-4, -3, -2, -1}={…-4, -3, -2, -1}

Rational NumbersRational Numbers

Terminating DecimalsTerminating Decimals are rational are rational numbers that stops before or after numbers that stops before or after the decimal point. the decimal point. For example: For example: 5.0, 2.75, .40, .0001…etc5.0, 2.75, .40, .0001…etc..

Repeating DecimalsRepeating Decimals are rational are rational numbers that repeats after the numbers that repeats after the decimal point.decimal point. For example: For example: .3333…,.3333…, , ,

.75 10.635

Irrational NumbersIrrational Numbers Irrational numbersIrrational numbers consist of numbers consist of numbers

that that are non-terminating and non-are non-terminating and non-repeating decimalsrepeating decimals..

They cannot be express as a fraction!They cannot be express as a fraction! Pi is an great example of an irrational Pi is an great example of an irrational

numbernumber http://www.joyofpi.com/pi.html

pi .001, .0011, .00111, .001111…etc

47 4.25837547984...

2

Real Number System Tree Real Number System Tree DiagramDiagram

Real Numbers

IntegersTerminating

Decimals Repeating Decimals

WholeNumbers

Rational Numbers

Irrational Numbers

Negative #’s

Natural #’s Zero

Non-TerminatingAnd

Non-RepeatingDecimals

Your TurnYour Turn

1. How are the natural and whole numbers different?

2. How are the integers and rational numbers different?

3. How are the integers and rational numbers the same?

4. How are integers and whole numbers the same?

5. Can a number be both rational and irrational? Use the diagram to explain your answer.

Your TurnYour TurnAnswer True or False to the statements below. If the statement is False, explain why.

6. −5 is a rational number. _______

7. is rational. _______

8. is a natural number __________

9. is an integer. _______

10. 2.434434443… is a rational number.____________

16

3.25

8

SummarySummary

What did you learn in this lesson?What did you learn in this lesson? What are some important facts to What are some important facts to

remember about the real number remember about the real number system?system?

Is there something within the lesson Is there something within the lesson that you need help on?that you need help on?