Number System

Number System

Contents:  Number  Number

System  Types of number system  Non-positional number system  Draw Backs  Solution

Cont..  Positional 

number system

Case Study

Binary Number System  Octal Number System  Hexadecimal Number System 

Cont..  Conversion  Decimal

to some other base  Some other base to decimal  Converting from base X to base Y where X and Y are not decimal  Fraction Number

Number:  “Number

is an idea of quantity and name that allow us to speak out value.”

Cont..  Origination

of number traced back to stone age where people use to equalize different shapes with different quantities.  Shapes are termed as digit.  Digits are like character of language.

Numbering System: 

“Formal

and systematic way of representing larger values with the help of some basic digits.”

Cont..  Number

system begins with defining the digits by a

value with symbol

 Name: selected for digit  Rules: specified to form and represent larger values.

Cont..  Larger

values are named as well , we should have a nomenclature of numbering system.

Types Of Numbering System: 

There are two types of numbering system. Non-positional

Numbering system Positional Numbering system

Non-Positional Numbering System:  Each

symbol represents some value regardless of its position in Numbering system.  Symbols are simply added to find out values of particular numbers.

Cont..  First

known system is Roman Number System.

 V VI IX X

=

5

= = =

6 9 10

Draw Backs:  Representing

ambiguity  For

larger values prone to

example XIIV can be interpreted as (11+4) or (12 + 5) Depending upon how we group.

Cont..  More

than one symbol can be used to form digit. V + I = 6  Same symbol can be used in forming more than one digit I + V = 4

Solution:  Problem

is solved by putting bars over digits.

Positional Numbering System: 

Each symbol represent different values depend upon position they occupy in numbering system.

Cont..  Value

determine by three consideration. Digit itself Position of digit in numbering system Base of numbering system

Case Study :  Start

with Base 10 as we are familiar with it. let us take it as case study.  History came with idea to give weight to position along with symbol.

Cont..  Suppose

X and Y are digits of numbering system.  Numbers can be formed and represented as follows. C1 X

C0 Y

Cont..  Digit

Y in column ‘C0’ has its original value due to occurrence in 1st position. let Y = 2 base is 10 then =2X100 =2X1 =2 So original value of Y = 2 and result is also 2.

Cont..  Digit

X in column C1 represent value and determine by Multiplying weight with original value of X.

Cont.. Let weight =W1 = 1 X=3 base = 10 =3X101 =30

Cont.. Now add in Y so General Form: W1 * X + Y  Weight : Power of base  Base : Total number of digits available in Number system.

Cont.. Weight in term of base

103

102

101

100

Weight 1000 age Columns C3

100

10

1

C2

C1

C0

Numbers 1

3

4

1

values

3*100

4*10

1*1

1*1000

Cont..  Total

number of digits in an alphabet is called base of numbering system.  In our case study base=10  As second column was required when we were count ten.

Cont..  It

implies that weight age of second column in decimal is 10.  So W1=10

Cont..  Weight

age of subsequent column are also in magnitude of base of numbering system

Binary Number System  It

is base two numbering system ,it must have two digits.  Alphabet = {0,1}

Octal Number System  Octal

means 8  As it is base 8 numbering system it must have 8 digits.  Alphabet = {0,1,2,3,4,5,6,7}

Hexadecimal Number System  It

is base 16 numbering system ,it must have 16 digits.  Alphabet = {0,1,2,3,4,5,6,7,8,9,A,B,C,D,E ,F}

Converting values To/from Decimal System  Decimal

to some other base  Some other base to decimal  Converting from base X to base Y where X and Y are not decimal

Decimal to some other base:  What

we have : A number in decimal

 What

we

 What

we

system. want : To find out an equivalent number in another base. do : Divide decimal number repeatedly with base value of required numbering system.

From Some Other Base To Decimal:  What

we have: A number in base X.  What we want: To find out an  What

we do:

equivalent number in base ten. Multiply unit value of digit in each column with weight age assigned to that column.

Converting from base X to base Y where X and Y are not decimal: 

 

There is no direct method ,what we do is that we use transition. First we convert base X -> base 10 Then convert base 10 -> base Y

Fractional Numbers:  In

binary number system , fractional numbers are formed in decimal number system.

Cont.. position

4

3 2 1 0 . -1

-2

-3

Position value

24

23 22 21 20

2-1

2-2

2-3 2-4

1/2

1/4 1/8 1/16

Quantity 16 8 4 2 1 Represente d

-4