Transcript The Real Numbers

The Real Numbers
1.1
Sets
A set is a collection of objects, symbols, or numbers called
elements.
Example 1
1,2,3
is a set containing the first three
counting numbers.
1, 2, and 3 are elements of the set.
a, e, i, o, u
is a set containing the the vowel
letters in English language
Question: What are the elements of this set?
Answer: The elements are: a, e, i, o, and u.
Let D = { x / x is a day of the week }.
Class Exercise
What are the elements of D ?
Example 2
Using the symbols

Any object or symbol that is contained in a set is called an
element, or a member, of the set. The symbol
is used to
indicate that an object is an element of the set.

Example 1
3
Example 2
January
Class Excercise

Set A = { 1, 2, 3, 4 }.

Complete each statement with the symbols
 or 
If B = { 1, 3, 5, a, c }
i) a
B
D = { x / x is a day of the week }.
ii) 2

B
iii) c

B
Equal Sets
Two sets are equal if they contain the same elements.
Example 1
Let A = { a, b, c, d } and B = { a, d, b, c }. Since
A and B have the same elements, then they are
equal.
We Write A = B
Ordered Sets
If the elements of a set can be ordered and we wish to indicate
that the set continues as described, we use an ellipses, three dots
that mean “ and so on”.
Example 1
Example 2
The set { a, b, c, …, z } represents the entire alphabet
The set { 1, 2, 3, … , 100 } represents the counting
numbers from 1 till 100.
More Examples
Example 3
The set{ 1, 3, 5, … } represents the positive odd numbers
Example 4
The set{ 2, 4, 6, … } represents the positive Even numbers
Finite or Infinite Sets
A set that has a specific number of elements is said to be finite,
otherwise, it is infinite.
Example 1
The set A = { 1, 2, 3} is finite.
Example 2
The set B = { 1, 2, 3,…, 10} is finite.
Example 3
The set C= {2, 3,4,…} is infinite.
More Examples
Example 4
The Set N = { 1, 2, 3, … } = Set of Natural numbers and it is infinite
Example 5
The Set W = { 0, 1, 2, 3, … } = Set of Whole numbers and it is
infinite
Example 6
The Set Z = { …,-3, -2, -1,0, 1, 2, 3, … } = Set of Integer numbers
and it is infinite
Important Notes
Every element in N is in W, and every element in W is in Z.
Class Exercise
i) 0
N
ii) 0
Z
iii) 0
iv) 1
W
Z
Complete each statement with the symbols
 or 
v) -1
N
ix)
vi) -1
W
vii) -1
Z
viii) 7
Z

Z
Note :
 =3.141828….
2
x) 1
W
Venn Diagram 1
Set of Integers
Z = {…, -3, -2, -1, 0, 1, 2, 3,… }
Set of Whole Numbers
W = {0,1,2,3,…}
Set of Natural Numbers
N = { 1,2,3,…}
Rational Numbers
Numbers as
½
0.34
5
1.333…
are considered as Rational Numbers
5.2323…
-1.5
¾
-3
2
1
5
Because we cannot list the rational numbers in
any meaningful fashion, we define the
elements of that set as:
p

Q   / p, q  Z , q  0
q

 The Set of RationalNumbers
Examples of Rational numbers
i) 1/2

Q
ii) 0

Q
iii) 0.34
iv) -1
iv) 1.333

Q

Q

Q
Venn Diagram 2
Set of Rational Numbers
Q
Set of Integers
Z = {…, -3, -2, -1, 0, 1, 2, 3,… }
Set of Whole Numbers
W = {0,1,2,3,…}
Set of Natural Numbers
N = { 1,2,3,…}
Important Notes About Rational Numbers
The numbers
3.5
3.111…
Are decimal numbers
2.6565…
3.141828….
Not all Decimal numbers are rational numbers
3.5

Terminating
Decimal
Q
3.111…

Q
2.6565……
Repeating Decimals
3.141828…

Q

Q
More Notes about Decimal Numbers
3.111...  3. 1
2.6565...  2.65
Repeating Decimal is 1
Repeating Decimal is 65
3.141828….
No Repeating Decimal
Class Participation About Rational
Numbers….
Class Exercise
Complete the following table with Yes or No
Number
N
W
Z
Q
0
9
-4
3.8
2.546
8.222…
No
Yes
No
No
No
No
Yes
NO
Yes
Yes
No
No
No
No
Yes
NO
Yes
Yes
Yes
No
No
No
Yes
NO
Yes
Yes
Yes
Yes
Yes
Yes
Yes
NO
9
2
More Class Practice
Class Exercise
5
4

From the set  4, ,0, , 3,3 ,4.5,0.21,8

3
3


List the elements in N,Z,Q
How about the elements
3 and 3 ?
Irrational Numbers
• If a number is not rational, then it is irrational
Q = Set of Rational Numbers
Q` = Set of Irrational Numbers
Example 1
Class Exercise
Q
2  Q`
 Q
  Q`
2
Check whether these numbers are rational
Q, or Irrational Q`
1
3,  5, 0, , 3.14, 3.1418..., 2.5353... 52 , ,2 8 dna
2
Real Numbers
The set of real numbers is the union of the sets of rational
numbers and irrational numbers.
Real Numbers
R
Rational
Numbers
Q
Irrational
Numbers
Q` ( Not Q )
All Numbers in N, Z,Q, and Q` are real numbers.
Real Line (Numbered Line )
1
3
2
4
Numbered Line ( Real Line )
Class Exercise
a)
On the number line provided, graph the
points named by each set
 2,0,2
1
2
3
4
…
1 5 7 
b)  , ,  
2 4 3
-7/3
=2.333
5/4
1/2

c)  3 , 2
3
2
1
4

1
2
3
4
Home Work Assignment
Do all the home work exercises in the syllabus