The Real Numbers

1.1 Sets

A set is a collection of objects, symbols, or numbers called elements.

Example 1 3,2,1 is a set containing the first three counting numbers.

1, 2, and 3 are elements of the set.

Example 2 uoiea ,,,, 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.

Class ExerciseLet D = { x / x is a day of the week }. What are the elements of D ?

Using the symbols

is used toAny object or symbol that is contained in a set is called an element, or a member, of the set. The symbol indicate that an object is an element of the set.

Example 1 3 Set A = { 1, 2, 3, 4 }.

Example 2 January D = { x / x is a day of the week }.

Class Excercise Complete each statement with the symbols or

If B = { 1, 3, 5, a, c }

i) a B 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 SetsIf 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 The set { a, b, c, …, z } represents the entire alphabet

Example 2The 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 SetsA 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

The Set W = { 0, 1, 2, 3, … } = Set of Whole numbers and it is infinite

Example 5

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 Complete each statement with the symbols

or

i) 0 N

ii) 0 Z

iii) 0 W

iv) 1 Z

v) -1 N

vi) -1 W

vii) -1 Z

viii) 7 Z

ix) Z

x) W

2

1

Note : =3.141828….

Venn Diagram 1

Set of Natural NumbersN = { 1,2,3,…}

Set of Whole Numbers

W = {0,1,2,3,…}

Set of Integers

Z = {…, -3, -2, -1, 0, 1, 2, 3,… }

Rational Numbers

are considered as Rational Numbers

Numbers as

½

0.34

5

1.333…

5.2323…

-1.5

¾

-3

5

12

Because we cannot list the rational numbers in any meaningful fashion, we define the elements of that set as:

NumbersRationalofSetThe

qZqpq

pQ

0,,/

Examples of Rational numbers

i) 1/2 Q

ii) 0 Q

iii) 0.34 Q

iv) -1 Q

iv) 1.333 Q

Venn Diagram 2

Set of Natural NumbersN = { 1,2,3,…}

Set of Whole Numbers

W = {0,1,2,3,…}

Set of Integers

Z = {…, -3, -2, -1, 0, 1, 2, 3,… }

Set of Rational Numbers

Q

Important Notes About Rational NumbersThe numbers

3.5

3.111…

2.6565…

3.141828….

Are decimal numbers

Not all Decimal numbers are rational numbers

3.5 Q 3.111… Q2.6565…… Q

3.141828… QTerminating

Decimal

Repeating Decimals

More Notes about Decimal Numbers

1.3...111.3 Repeating Decimal is 1

65.2...6565.2 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 No Yes Yes Yes

9 Yes Yes Yes Yes

-4 No No Yes Yes

3.8 No No No Yes

2.546 No No No Yes

8.222… No No No Yes

Yes Yes Yes Yes

NO NO NO NO2

9

More Class PracticeClass Exercise

From the set

8,21.0,5.4,3,3,

3

4,0,

3

5,4

List the elements in N,Z,Q

How about the elements ?33 and

Irrational Numbers

• If a number is not rational, then it is irrational

Q = Set of Rational Numbers

Q` = Set of Irrational Numbers

Example 1 Q2 Q`2

Q Q`

Class Exercise Check whether these numbers are rational Q, or Irrational Q`

...5353.2...,1418.3,14.3,2

1,0,5,3 8and2,25,

Real NumbersThe set of real numbers is the union of the sets of rational numbers and irrational numbers.

RationalNumbers

Q

Irrational Numbers

Q` ( Not Q )

Real Numbers

R

All Numbers in N, Z,Q, and Q` are real numbers.

Real Line (Numbered Line )

1 2 3 4

Numbered Line ( Real Line )

Class Exercise On the number line provided, graph the points named by each set

1 2 3 4

2,0,2) a

…

3

7,

4

5,

2

1)b

1 2 3 4

2,3) c

1 2 3 4

1/25/4

-7/3 =-2.333

Home Work Assignment

Do all the home work exercises in the syllabus