Real Numbers presentation

7/28/2019 Real Numbers presentation

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REAL NUMBERS

(as opposed to fake numbers?)


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Objective

TSW identify the parts of the RealNumber System

TSW define rational and irrationalnumbers

TSW classify numbers as rational or

irrational


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Real Numbers

Real Numbers are every number.

Therefore, any number that you canfind on the number line.

Real Numbers have two categories.


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What does it Mean?

The number line goes on forever.

Every point on the line is a REAL

number. There are no gaps on the number line.

Between the whole numbers and the

fractions there are numbers that aredecimals but they dont terminate andare not recurring decimals. They go onforever.


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Real Numbers

REAL NUMBERS

-8 -5,632.1010101256849765

61

49%

549.23789

154,769,852,354

1.333


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Two Kinds of Real Numbers

Rational Numbers

Irrational Numbers


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Rational Numbers

A rational number is a realnumber that can be written

as a fraction. A rational number written in

decimal form is terminatingor repeating.


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Examples of Rational

Numbers16

1/2

3.56

-8

1.3333

- 3/4


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Integers

One of the subsets of rational

numbers


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What are integers?

Integers are the whole numbers and theiropposites.

Examples of integers are

6

-12

0

186

-934


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Integers are rational numbersbecause they can be written as

fraction with 1 as the denominator.


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Types of Integers

Natural Numbers(N):Natural Numbers are counting numbersfrom 1,2,3,4,5,................

N = {1,2,3,4,5,................}

Whole Numbers (W):

Whole numbers are natural numbersincluding zero. They are0,1,2,3,4,5,...............W = {0,1,2,3,4,5,..............}

W = 0 + N


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WHOLE

Numbers

REAL NUMBERS

IRRATIONALNumbers

NATURALNumbers

RATIONALNumbers

INTEGERS


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Irrational Numbers

An irrational number is anumber that cannot be

written as a fraction of twointegers. Irrational numbers written as

decimals are non-terminatingand non-repeating.


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A repeating decimal may not appear torepeat on a calculator, becausecalculators show a finite number of digits.

Caution!

Irrational numberscan be written only as

decimals that do notterminate or repeat. They

cannot be written as the quotient of two

integers. If a whole number is not a perfect

square, then its square root is an irrational

number.


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Examples of Irrational

Numbers Pi


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Try this!

a) Irrational

b) Irrational

c) Rational

d) Rational

e) Irrational

66e)

d)

25c)12b)

2a)

115


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Additional Example 1: Classifying Real

Numbers

Write all classifications that apply to eachnumber.

5 is a whole number that isnot a perfect square.

5

irrational, real

12.75 is a terminating decimal.12.75

rational, real

162

whole, integer, rational, real

= = 242

162

A.

B.

C.


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A fraction with a denominator of 0 is

undefined because you cannot divide

by zero. So it is not a number at all.


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State if each number is rational,irrational, or not a real number.

21

irrational

03

rational

03

= 0

Additional Example 2: Determining the

Classification of All Numbers

A.

B.


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not a real number

Additional Example 2: Determining the

Classification of All Numbers

40C.

State if each number is rational,irrational, or not a real number.


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Objective

TSW compare rational and irrationalnumbers

TSW order rational and irrationalnumbers on a number line


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Comparing Rational andIrrational Numbers

When comparing different forms ofrational and irrational numbers,

convert the numbers to the sameform.

Compare -3 and -3.571(convert -3 to -3.428571

-3.428571 > -3.571

3

737


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Practice


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Ordering Rational andIrrational Numbers

To order rational and irrationalnumbers, convert all of the numbers

to the same form. You can also find the approximatelocations of rational and irrationalnumbers on a number line.


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Example Order these numbers from least to

greatest./, 75%, .04, 10%, /

/ becomes 0.2575% becomes 0.75

0.04 stays 0.04

10% becomes 0.10

/ becomes 1.2857142

Answer: 0.04, 10%, /, 75%, /


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Practice

Order these from least to greatest:


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Objectives

TSW identify the rules associatedcomputing with integers.

TSW compute with integers


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Examples: Use the number line

if necessary.

42) (-1) + (-3) =

-43) 5 + (-7) =

-2

0 5- 5

1) (-4) + 8 =


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Addition Rule1) When the signs are the same,

ADD and keep the sign.(-2) + (-4) = -6

2) When the signs are different,SUBTRACT and use the sign of the

larger number.

(-2) + 4 = 2

2 + (-4) = -2


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Karaoke Time!Addition Rule: Sung to the tune

of Row, row, row, your boatSame signs add and keep,

different signs subtract,keep the sign of the higher

number,then it will be exact!

Can your class do different

rounds?


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-1 + 3 = ?

1. -4

2. -2

3. 2

4. 4Answer Now


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-6 + (-3) = ?

1. -9

2. -3

3. 3

4. 9

Answer Now


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The additive inverses (oropposites) of two numbers add

to equal zero.

-3Proof: 3 + (-3) = 0

We will use the additiveinverses for subtraction

problems.

Example: The additive inverse of 3 is


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Whats the difference

between7 - 3 and 7 + (-3) ?7 - 3 = 4 and 7 + (-3) = 4

The only difference is that 7 - 3 is asubtraction problem and 7 + (-3) is anaddition problem.

SUBTRACTING IS THE SAME AS

ADDING THE OPPOSITE.

(Keep-change-change)

When subtracting change the


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When subtracting, change thesubtraction to adding the opposite (keep-

change-change) and then follow youraddition rule.Example #1: - 4 - (-7)

- 4+ (+7)Diff. Signs --> Subtract and use larger sign.3

Example #2: - 3 - 7

- 3+ (-7)Same Signs --> Add and keep the sign.

-10


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Which is equivalent to-12 (-3)?

Answer Now

1. 12 + 3

2. -12 + 3

3. -12 - 34. 12 - 3


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7 (-2) = ?

Answer Now

1. -9

2. -5

3. 5

4. 9


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1) If the problem is addition, followyour addition rule.

2) If the problem is subtraction,change subtraction to adding theopposite(keep-change-change) and thenfollowthe addition rule.

Review


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State the rule for multiplying and

dividing integers.

If the

signs

are the

same,

If the

signs are

different,

the

answer

will be

positive.

the

answer

will be

negative.


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1. -8 * 3 WhatsThe

Rule?

Different

Signs

NegativeAnswer

-24

2. -2 * -61

Same

Signs

Positive

Answer

122

3. (-3)(6)(1)

(-18)(1)

-18

4. 6 (-3)

-2

5. - (20/-5)

- (-4)4

6. 408

6

68

Start inside ( ) first


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7. At midnight the temperature is 8C.

If the temperature rises 4C per hour,

what is the temperature at 6 am?

How long

Is it from

Midnight

to 6 am?

How much

does the

temperature

rise eachhour?

6

hours

+4

degrees

(6 hours)(4 degrees per hour)

= 24 degrees

8 + 24 = 32C

Add this to

the original temp.


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8. A deep-sea diver must move up or down in

the water in short steps in order to avoid

getting a physical condition called the bends.Suppose a diver moves up to the surface in five

steps of 11 feet. Represent her total

movements as a product of integers, and find

the product.

Multiply

(5 steps) (11 feet)

(55 feet)

5 * 11 = 55

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Real Numbers presentation