Number System -Real Number (14.6.2012)

8/13/2019 Number System -Real Number (14.6.2012)

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NUMBER SYSTEM

Real number


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Learning Outcomes

(a) To define and understand natural, whole, integers, primenumbers, rational and irrational numbers.(b) To represent rational and irrational numbers in decimal

form,(c) To represent the relationship of number sets in a real

numbers

(d) To state clearly the properties of real numbers(e) To understand open, closed and half open interval and their

representations on the number line.(f) To understand that the end point of and open interval on

the number line are usually represented as empty circles, whereas the end points of a closed interval are

represented by dense circles, .(g) To simplify union, , and intersection, , of two or more

intervals with the aid of number line.


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

Prime numbers : the numbergreater than 1 and can bedivided by itself only.

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

Natural Numbers


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The natural numbers,together with the number 0

W= {0, 1, 2, 3 }.

Do you know what is the

whole numbers?


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The Integers

The whole numbers together with the

negative of counting numbers form the setof integers anddenoted by Z.

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

Integers

Positive integersZ+= {1, 2, 3 }

0Negative integersZ-= {, -3, -2, -1}.

Integers

Odd numbers{2k+ 1, kZ}

Even numbers{2k, kZ}


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The rational numbers

A rational numberis any numberthat can be represented as aratio (quotient) of two integers

and can be written as

0,,; bZbab

aQ

Rational number can be expressedas terminating or repeating

decimals


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Irrational number

The number cannot be writtenas a quotient and non repeating

decimal number.

Eg : 0.452138, ,3


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Rationalnumbers

IrrationalNumbers

14159265.3

414214.12

25.04

1

...363636.011

4

...333.03

1


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Numbers that eitherrational or irrational arecalled real number, R.


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Q

R

Q

Z

W

N

Relationship of Number Sets

From the diagram, we can see that : 1. N W Z Q R2. Q = RQ


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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 aquotient

(a) 1.555 (b) 5.45959

(a) Let x= 1.555 (1)

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

x=

914


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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 =

=

990

5405

198

1081


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Properties ofReal Numbers

Addition Multiplication

1. Closure a+ b= c, cR

6 + 7 = 13 R

ab= d, dR

6 7 = 42 R

2. Commutative a+ b= b+ a2 + 5 = 5 + 2

ab= ba2 5 = 5 2

3. Associative a+ b) + c= a+ (b+ c)(1 + 3) + 2 = 1 + (3 + 2)

(ab)c= a(bc)(4 3) 2 = 4 (3 2)

4. Distributive a(b+ c) = ab+ ac4 (2 + 3) = 4 2 + 4 3

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

a1 = 1 a= a3 1 = 1 3 = 3

6. Inverse a+ (a) = 0 = (a) + a7 + (7) = 0 = (7) + 7

5

1515

5

1

aaa

a

11

1


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ExampleWrite down the type of properties for this statement

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

Commutative

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

Distributive

(c) 3ab + 0= 3ab

Identity for Addition Operation

(d) 2(3n)= (2(3))n

Associative for Multiply Operation


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Example

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

b

1

.

Solution

Given ab = 1(ab)

b

1= 1

b

1

a

b

b 1

=

b

1 (associative and identity)

b

1 (Inverse)

b

1 (identity)

a1 =

a=


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The number line

The set of numbers that corresponds to all point on numberlines is called the set ofreal number.

The real numbers on the number line are ordered inincreasing magnitude 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 4W = {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


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ExampleList the number described and graph the numbers on anumber line.a) The integer between 3 and 9 :

Z = {4, 5, 6, 7, 8}

2 3 4 5 6 7 8 9

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

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


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ExampleRepresent the following interval on the real number line andstate the type of the interval.(a) [-1, 4] (b) {x: 2 x5}(c) [2, ) (d) {x: x 0,xR}

-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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ExampleSolve 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(b) (, 5) (1, 9)

1 5 9

(, 5) (1, 9) = (1, 5)


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

-

0

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


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Example

Given A = {x: -2 x5} and B = {x: 0 x7}.Show that A B = (0, 5].

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

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Number System -Real Number (14.6.2012)