1.2 SKM & PP 1 Types of Numbers There are many “types” of numbers. Each type can be grouped into...
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Transcript of 1.2 SKM & PP 1 Types of Numbers There are many “types” of numbers. Each type can be grouped into...
1.2 S KM & PP 1
Types of Numbers
There are many “types” of numbers.
Each type can be grouped into a
collection called a SET.
1.2 S KM & PP 2
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
1.2 S KM & PP 3
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, …}
1.2 S KM & PP 4
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}
1.2 S KM & PP 5
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.
1.2 S KM & PP 6
The Natural Numbers
Natural numbers are usually the first set that we learn. They are also
called Counting numbers.
{1, 2, 3, 4, 5, …}
1.2 S KM & PP 7
The Natural Numbers
Here are the Natural numbers graphed on the
number line:
{1, 2, 3, 4, 5, …}
…
1.2 S KM & PP 8
The Whole Numbers
The set of Whole numbers is the set of
Natural numbers along with zero.
{0, 1, 2, 3, 4, 5, …}
…
1.2 S KM & PP 9
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...
1.2 S KM & PP 10
Opposite Numbers
Opposite numbers are the same distance from
zero, but they are on opposite sides of zero.
-a and a are opposites.
1.2 S KM & PP 11
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
1.2 S KM & PP 12
The Integers
The Integers are the Whole numbers
together with their Opposites.
{…,-3,-2,-1,0,1,2,3,…}
……
1.2 S KM & PP 13
The Rational Numbers
The set ofRational Numbers
consists of all quotients of Integers with non-zero
denominators.
ba
0b a and b are integers,
1.2 S KM & PP 14
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.
1.2 S KM & PP 15
Conversion Example 1
41 25.0
00.14
41
25.0Terminating Decimal
1.2 S KM & PP 16
Conversion Example 2
31 ...333.0
0000.13
...3.0
Repeating Decimal
31
Ellipsis show the3 repeats.
1.2 S KM & PP 17
Conversion Example 3
52 4.0
0.25
52
4.0Terminating Decimal
1.2 S KM & PP 18
Conversion Example 4
74
...285714285714.00000000000000.47
...571428.0Repeating Decimal
74
Ellipsis show the571428 repeats.
1.2 S KM & PP 19
Conversion Example 5
40 0
04
0Terminating Decimal
40
1.2 S KM & PP 20
Conversion Example 6
04 ?
40
The denominator can never equal
zero!
04
undefined
1.2 S KM & PP 21
Conversion Example 7
811 375.1
000.118
811
375.1Terminating Decimal
1.2 S KM & PP 22
Conversion Example 4
625 ...1666.4
00000.256
...61.4Repeating Decimal
625
Ellipsis show the6 repeats.
1.2 S KM & PP 23
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.
1.2 S KM & PP 24
What about Negatives?Example 1
43
“Negative three-fourths”
43
1.2 S KM & PP 25
What about Negatives?Example 2
25
“Negative two and one-half”
25
21
2 5.2
1.2 S KM & PP 26
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.
1.2 S KM & PP 27
Irrational Numbers
Here are a few examples of numbers that are
Irrational.
3.14159…
e 2.71828…
1.41421…2
3.6055512…
13
1.2 S KM & PP 28
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
1.2 S KM & PP 29
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,…
1.2 S KM & PP 30
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.
1.2 S KM & PP 31
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.
1.2 S KM & PP 32
An Example:
“Negative three is less than one”-3 < 1
“One is greater than negative three”1 > -3
1.2 S KM & PP 33
> or <
How do these numbers compare?
-5 2<
11 -13>
0 6<
-5 0<
1.2 S KM & PP 34
Absolute Value
The ABSOLUTE VALUE of a number, |x|, is its distance from
zero on the number line.
|-5|= 5
|5|= 5
1.2 S KM & PP 35
|x| Examples
|-9| = 9
|20| = 20
|0| = 0
-|-9| = -1|-9|
= -19
= -9
1.2 S KM & PP 36
That’s All for Now!That’s All for Now!
That’s All for Now!