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Types of Numbers

There are many “types” of numbers.

Each type can be grouped into a

collection called a SET.

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Sets

In general, any collection of objects is called a SET. A set can be defined in several ways:

English: A description in words

Set Builder: A mathematical rule

Roster: A list of the objects or numbers inside braces

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Sets: Example 1

Consider the set of even numbers: 0,2,4,6,…

English: “The Even Numbers”

Set Builder: {x| x is divisible by 2}

Roster: {0, 2, 4, 6, 8, …}

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Sets: Example 2

Consider the set of digits: 0,1,2,3,4,5,6,7,8,9

English: “Digits”

Set Builder: {x| x is a digit}

Roster: {0,1,2,3,4,5,6,7,8,9}

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C

The Number Line

We use a Number Line to graph sets of

Real Numbers.

Zero is in the

center.

Positive numbers are on

the right.

Negative numbers are on

the left.

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The Natural Numbers

Natural numbers are usually the first set that we learn. They are also

called Counting numbers.

{1, 2, 3, 4, 5, …}

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The Natural Numbers

Here are the Natural numbers graphed on the

number line:

{1, 2, 3, 4, 5, …}

…

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The Whole Numbers

The set of Whole numbers is the set of

Natural numbers along with zero.

{0, 1, 2, 3, 4, 5, …}

…

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The Opposite

Each Natural number to the right of zero has an Opposite to the left of

zero.

-1 and 1 are Opposites.-2 and 2 are Opposites.-3 and 3 are Opposites.and so on...

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Opposite Numbers

Opposite numbers are the same distance from

zero, but they are on opposite sides of zero.

-a and a are opposites.

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What about Zero?

Two numbers are opposite if their sum is

zero.-1 + 1 = 0

Zero is it’s own opposite.

-2 + 2 = 0-3 + 3 = 0Since 0 + 0 =

0

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The Integers

The Integers are the Whole numbers

together with their Opposites.

{…,-3,-2,-1,0,1,2,3,…}

……

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The Rational Numbers

The set ofRational Numbers

consists of all quotients of Integers with non-zero

denominators.

ba

0b a and b are integers,

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Convert: Rational Number to Decimal

To convert a Rational Number into Decimal form,divide the numerator by the

denominator.

ba ab

A Rational number can always be converted to

a Terminating Decimal

or aRepeating Decimal.

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Conversion Example 1

41 25.0

00.14

41

25.0Terminating Decimal

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Conversion Example 2

31 ...333.0

0000.13

...3.0

Repeating Decimal

31

Ellipsis show the3 repeats.

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Conversion Example 3

52 4.0

0.25

52

4.0Terminating Decimal

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Conversion Example 4

74

...285714285714.00000000000000.47

...571428.0Repeating Decimal

74

Ellipsis show the571428 repeats.

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Conversion Example 5

40 0

04

0Terminating Decimal

40

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Conversion Example 6

04 ?

40

The denominator can never equal

zero!

04

undefined

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Conversion Example 7

811 375.1

000.118

811

375.1Terminating Decimal

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Conversion Example 4

625 ...1666.4

00000.256

...61.4Repeating Decimal

625

Ellipsis show the6 repeats.

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What about Negatives?

ba

ba

b

a

The negative sign can be in front of the ratio or in the numerator or in the denominator. Usually, it is best to place it in the

front.

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What about Negatives?Example 1

43

“Negative three-fourths”

43

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What about Negatives?Example 2

25

“Negative two and one-half”

25

21

2 5.2

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Irrational Numbers

Any Real number that is not a rational number is

called Irrational.

Irrational numbers cannot be written as the ratio of integers. The

decimal approximation for an irrational number will not

terminate or repeat.

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Irrational Numbers

Here are a few examples of numbers that are

Irrational.

3.14159…

e 2.71828…

1.41421…2

3.6055512…

13

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The REAL Numbers

REAL NUMBERS

The set of numbers that correspond to points on the

number line.The REAL NUMBERS include the following:

Natural, Whole, Integers, Rational, and

Irrational

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REAL NUMBERS

Rational Numbers:a/b with b0

Integers:…-2,-1,0,1,2,…

Whole Numbers:0,1,2,3,…

Natural Numbers:1,2,3,…

A Map of theNumber Sets

Irrationals:pi,e,3,…

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Order: Small to Large

The Real Numbers are named on the number line from small to large. If we choose any two numbers on the number line,

the number on the left is smaller and the number on the

right is larger.

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Order: Small to Large

The Real Numbers are named on the number line from small to large. If we choose any two numbers on the number line,

the number on the left is smaller and the number on the

right is larger.

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An Example:

“Negative three is less than one”-3 < 1

“One is greater than negative three”1 > -3

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> or <

How do these numbers compare?

-5 2<

11 -13>

0 6<

-5 0<

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Absolute Value

The ABSOLUTE VALUE of a number, |x|, is its distance from

zero on the number line.

|-5|= 5

|5|= 5

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|x| Examples

|-9| = 9

|20| = 20

|0| = 0

-|-9| = -1|-9|

= -19

= -9

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That’s All for Now!That’s All for Now!

That’s All for Now!