1.1 Real Number

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1.1 Learning Outcomes

To define and understand natural, whole, integers, primenumbers, rational and irrational numbers.

To represent rational and irrational numbers in decimalform,

To represent the relationship of number setsandproperties in a real numbers

To understand open, closed and half open intervalrepresentations on the number line.

To simplify union, , and intersection, , of two or moreintervals with the aid of number line.


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the set of counting numbersN= {1, 2, 3 }

Prime numbers : the number greaterthan 1 and can be divided by itselfonly.

Prime number = {2, 3, 5, 7 }

1.1.1 Natural Numbers

1.1.2 Prime Numbers


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The natural numbers, togetherwith the number 0

W= {0, 1, 2, 3 }

Do you know what is the wholenumbers?

1.1.3 Whole Numbers


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1.1.4 Integers

The whole numbers together withthe negative of counting numbersform the set of integers anddenoted by Z.

Z= {, -3, -2, -1, 0, 1, 2, 3 }


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1.1.5 Rational numbers

A rational number is any number thatcan be represented as a ratio(quotient) of two integers and can be

written as

,,; bZbab

aQ

Rational number can be expressedas terminating or repeatingdecimals eg. 0.3333,0.5


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1.1.6 Irrational number

The number cannot be written as aquotient and non repeating decimal

number.

Eg : 0.452138, ,3


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Rational numbers Irrational Numbers

1415.3

414.12

25.04

1

...363636.011

4

...333.03

1

Eg.


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Numbers that either

rational or irrational arecalled real number, R.

1.1.7 Real Numbers


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Exercise:

For the set of {-5, -3, -1, 0, 3, 8},identify the set of

(a) natural numbers :

N ={3,8}

(b) whole numbersW = {0,3,8}

(c) prime numbersprime number={3}


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(d) even numbers

even number = {0,8}

(e) negative integers= {-5, -3,-1}

(f) odd numbersodd number ={-5, -3, -1, 3}

Z


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Express each of this number as a quotient(a) 1.555 (b) 5.45959

(a) Let x= 1.555 (1)

(1) 10 10x= 15.555 (2)thus, (2) (1), 9x= 14

x= 9

14


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Let x= 5.45959 (1)

(1) 10 10x= 54.5959 (2)

(2) 100 100x= 5459.5959 (3)

therefore, (3) (2), 990x= 5405

x =

x =

990

5405

198

1081


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Properties of Real

NumbersAddition Multiplication

1. Closure a+ b= c, cR

6 + 7 = 13 R

ab= d, dR

6 7 = 42 R

2. Commutative a+ b= b+ a

2 + 5 = 5 + 2

ab= ba

2 5 = 5 2

3. Associativea+ b) + c= a+ (b+ c)

(1 + 3) + 2 = 1 + (3 + 2)

(ab)c= a(bc)

(4 3) 2 = 4 (3 2)

4. Distributivea(b+ c) = ab+ ac

4 (2 + 3) = 4 2 + 4 3

5. Identity a+ 0 = 0 + a= a5 + 0 = 0 + 5 = 5

a 1 = 1 a= a3 1 = 1 3 = 3

6. Inversea+ (a) = 0 = (a) + a

7 + (7) = 0 = (7) + 7515

5

1

aa

a

11


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Example

Write down the type of properties for this statement

(a) 3 + 4x =4x +3Commutative

(b) x(y + z) = xy + xzDistributive

(c) 3ab +0 =3ab

Identity for Addition Operation

(d) 2(3n) =(2(3)) nAssociative for Multiply Operation


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ExampleGiven a, bR, ab = 1. Prove that a =

b

1

.

Solution

Given ab =1(ab)

b

1= 1

b

1

a

b

b1

=

b

1 (associative and identity)

b

1 (Inverse)

b

1 (identity)

a 1 =

a=


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1.1.8 Number Line

The set of numbers that corresponds to all point on number lines iscalled the set ofreal number.

The real numbers on the number line are ordered in increasingmagnitude from the left to the right

3

2

4 -3 -2 -1 0 1 2 3 4

3.5


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Type of intervala) Open interval (a, b) or

{x : a < x < b}

b) Closed interval [a, b] or{x: a x b}

b) Half closed interval,(i) (a, b] or {x : a < x b}

(ii) [a, b) or {x : a x < b}

a b

a b

a b

a b

i)

ii)


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ExampleList the number described and graph the numbers on anumber line.a)The whole number less than 4

W = {0 , 1, 2 , 3}

-3 -2 -1 0 1 2 3 4

b) The integer between 3 and 9Z = {4, 5, 6, 7, 8}

2 3 4 5 6 7 8 9

-5 -4 -3 -2 -1 0 1 2

c) The integers greater than -3Z = {-2, -1, 0 }


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Example

Represent the following interval on the real number line andstate the type of the interval.(a) [-1, 4] (b) {x: 2 x 5}(c) [2, ) (d) {x: x 0)

-1 4

a) closed interval

2 5open intervalb)

c)

2

half-open interval

d)0

half-open interval


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Intersection and union operation

Example :

Given A = [1 , 6) and B = (2, 4),

-2 -1 0 1 2 3 4 5 6

Union ()

Intersection ()


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Example:Solve the following using the number line [0, 5) (4, 7)

0 4 5 7

[0, 5) (4, 7) = [0, 7)


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Solve the following using the number line(, 0] [0, )

-

0

(, 0] [0, ) = (, ) = R


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Solve the following using the number line(4, 2) (0, 4] [2, 2)

Consider(4, 2) (0, 4]

-4 0 2 4

(4, 2) (0, 4] = (4, 4]

-4 -2 0 2 4

(4, 4] [2, 2) = [2, 2)


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Example

Given A = {x: -2 x 5} and B = {x: 0 x 7}.

Show that A B = (0, 5].

2 0 5 7

(2, 5] (0, 7] = (0, 5]


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So today, you learn about...

IntersectionOr Union

Number line

Types OfReal Numbers

Real Numbers


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Documents

1.1 Real Number