STD :- XI AC 2. NUMBER SYSTEM NS-CS - lesson-2
Introduction :-
DATA :-
Data comes from the latin word ' Datum' which means 'something given'.
Data is a collection of raw facts and figures .
Data is used as input for the computer system.
It is meaningless
Data can be the following form 1.text [ letters]and 2.numbers[ 0-9] send
through keyboard ,3. Images through scanner 4.Audio through
microphone and 5.video through camera.
The amount of data that can be stored in storage unit that in which
storage capacity is expressed in terms of bytes.
DATA REPRESENTATION :-
The amount of data that can be stored in the storage unit that in which
storage capacity is expressed in terms of bytes.
BIT:-
Binary digit :0/1.A group of bits in a computer are used to represent
many different things.
It can represent a number
It can represent a character
It can represent an instruction
Byte is the basic unit of computer's memory.
Main memory and secondary memory is normally represented in terms
of kilobyte ,megabyte.
The speed of the memory depends on the number of bits it can process
at once.
1 Bit = 0 or 1
4 Bits = 1 NibblE
8 Bits = 1 Byte
1024 byte = 1 kilobyte
1024 kilobyte=1 Megabyte
1024 megabyte = 1Gigabyte
1024 gigabyte= 1Terabyte
1024 Terabyte = 1 Petabyte
1024 Petabyte = 1 Exabyte
1024 Exabyte= 1 Zettabyte
1024 Zettabyte = 1 Yottabyte
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NUMBER SYSTEM :-
A set of values used to represent different quantities is known as
number system. Eg; A number can be used to represent the number of
students in a class.
or A numbering system is a way of representing numbers.
TYPES OF NUMBER SYSTEM :-
1. Positional number system
2.Non-positional number system
* POSITIONAL NUMBER SYSTEM:-
It uses only a few symbols called digits.
These symbols represent different values depending on the position
they occupy in the number.
* NON-POSITIONAL NUMBER SYSTEM:-
It uses symbols such as I for 1,II for 2 etc
Each symbol represents the same value regardless of its position in
the number.
The symbols are simply added to find out the value of a particular
number.
DIFFERENT TYPES OF NUMBER SYSTEM:-
A numbering system is a way of representing numbers.each number
system is uniquely identified by its base value or radix.Base is the count of
number of digits in each number system.The most commonly used
numbering system in real life is decimal number system.
Decimal number system -Base 10
Binary number system - Base 2
Octal number system - Base 8
Hexa decimal number system -Base 16
I.DECIMAL NUMBER SYSTEM:-
Characteristics:-
It uses ten digits from 0 to 9.
Also known as base ten number system
Each position in decimal number represents a 0 power of the base 10.
example; 1897
It can be written as =1*1000+ 8 * 100+9*10+7
= 1000+800+90+7
= 1897
Structure:
Decimal no........ ....d3 d2 d1 d0 d-1 d-2.........
positional weights 103 102 101 100 10-1 10-2.........
2.BINARY NUMBER SYSTEM:-
Characteristics:-
It uses two digits 0 and 1
Also called as base 2 number sytem
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Each position in a binary represents a 0 power of the base 2.
Structure;
Base no ....................b3 b2 b1 b0 b-1 b-2
positional weights...23 22 21 20 2-1 2-2
Binary arithmetic:-
1.Binary addition:- RULES: 0+0 = 0
0+1 = 1
1+0 = 1
1+1 = 0 with carry 1
example: 0011010+001100
0011010
+ 0001100
0100110
2. Binary subtraction:- RULES: 0-0 = 0
1-0 = 1
1-1 = 0
0-1 = 1 borrow
example; - 0011010 - 001100
11 borrow
0011010
- 001100
0001110
3. Binary multiplication:- RULES ; 0*0 =0
0*1 = 0
1*0 =0
1*1 =1
example:- 0011010 * 001100
0011010
* 0001100
0000000
0000000*
0011010**
0011010***
0100111000
4.Binary division :
example: 101010/ 110
0001011
110 101010
-0
10
- 0
101
- 0
1010
- 110
1001
- 110
0110
- 110
0
Binary number representation:- There are two groups.Signed numbers and
unsigned numbers.
Signed numbers:-
signed numbers contain both sign and magnitude of the number.
generally the sign is placed infront of number.
so we have consider the positive sign for positive numbers and negative
sign for negative numbers
Unsigned numbers:-
It contains only magnitude of the number.
they don't have any sign that means all unsigned binary numbers are
positive.
Representation of signed binary numbers:-
0 0 1 1 1 0 1
The left most bit in the binary number is called as most significant bit.
It has the largest positional weight.
The most significant bit of signed binary number is used to indicate the
sign of the numbers.
Hence it is called as sign bit or parity bit.
The simplest method to present negative binary number is called signed
magnitude.
The right most bit is the least significant bit and has the smallest
positional weight.
Three types of representation of signed numbers:-
signed magnitude
1's compliment
2's complement
Representation of positive number in all these three form is same.but only
the representation of negative number will differ in each form.
Signed magnitude:-
In signed binary representation , the left most bit is considered as sign bit.
if this bit is 0,it is a positive number and if it is 1 , it is a negative
number.Therefore a signed binary number has 8 bits ,only 7 bits used for
storing values ( magnitude ) and the 1 bit is used for sign.
Eg: +43 or 43 is a positive number
- 43 is a negative number
2 43
2 21-1
2 10-1
2 5-0
2 2-1
1-0
+43 is represented in memory as
0 0 1 0 1 0 1 1
0's represents that the number is positive
-43 can be represented as in memory as
1 1 0 1 0 1 0 1
1' represent that the number is negative
1'S Complement form:
The 1's complement of a numberis obtained by complementing all the bits
of signed binary number .so 1's complement of positive number gives a
negative number .similarly 1's complement of negative number gives a
positive number.
steps for 1's complement:-
convert decimal number into binary number
check if the binary number contains 8 bits , if less add 0 at the left
most bit to make it as 8 bits
invert all bits( change 0 as 1 , 1 as 0 )
Eg:- ( -24)10
2 24
2 12-0
2 6-0
2 3-0
2 1-1
given number (- 24)10
binary number 00011000
1's complement 11100111
The 2's complement of a binary number is obtained by ading one to the 1's
complement of signed binary numbers.
steps for 2's complement:-
Invert all the bits in the binary sequence ( i.e change 0 to 1 and 1 to 0)(i.e
1's complement)
add 1 to the result to the least significant bit.
Eg:- 2's complement of ( - 24 )10
binary equivalent of + 24 = 11000
8 bit format = 00011000
1' s complement= 11100111
add 1 to LSB = + 1
2'S Complement of -24 = 11101000
DIFFERENCE BETWEEN 1'S COMPLEMENT AND 2'S COMPLEMENT:-
1'S complement 2's complement
It has two representations of 0 which is positive 0 and 1 which is negative 0
Only one representation for 0 because if we add 1 to 11111 (-1 ) we get 0000 (+0) which is the same as positive zero
While adding numbers using 1's complement we first do binary addition ,then add in an end around carry value
But 2's complement has only one value for 0 and doesn't require carry values.
Number system conversions:-
1. Decimal to binary conversion:-
two methods.
method :1 - repeated division by 2.
Any decimal number divided by 2 will leave a remainder of 0 or 1.
Repeated division by 2 will leave a sequence of 0s and 1s that
become the binary equivalent of the decimal number.
Eg: ( 65 )10
2 65
2 32-1
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2 16-0
2 8-0
2 4-0
2 2-0
1-0
( 65 ) 10 = ( 100001 )2
Method : 2 sum of powers of 2 Eg:- ( 36 ) 10
find the largest power of 2 that is smaller than or equal to 36
36 > 32
set the 32's bit to 1 and subtract 32 from the original number
36-34 = 4
16 is greater than the remaining total.therefore set the 16's
bit to 0
8 is greater than the remaining total .hence set the 8 'sbit to 0
as the remaining value is itself in powers 2. set 4's bit to 1 and
subtract 4.
4-4 = 0
conversion is complete when there is nothing to left to
subtract.
( 36 )10 = ( 100100 )2
32 16 8 4 2 1
1 36-34=4
32 16 8 4 2 1
1 0 0 1 4-4=0
32 16 8 4 2 1
1 0 0 1 0 0 36 = 100100
2. Fractional decimal to binary:-
multiply the decimal fraction by 2 and note the integer part.
the integer part is either 0 or 1
discard the integer part of previous product . multiply the
fractional part of the previous product by 2.
Repeat step 1 until the same fraction repeats or ends 0.
The resulting integer part forms a sequence of 0s and 1s that
become the binary equivalent of decimal fraction.
Eg :- ( 98. 46 )10
2 98
2 49-0
2 24-1
2 12-0
2 6-0
2 3-0
1-1
MULTIPLICATION[ INTEGER ] FRACTION
.46 * 2 = 0.92 0 .92
.92 * 2 = 1.84 1 .84
.84 *2 = 1.68 1 .68
.68 * 2 = 1.36 1 .36
.36 * 2 = 0.72 0 .72
.72 *2 = 1.44 1 .44
.44 * 2 = 0.88 0 .88
.88 * 2 = 1.76 1 .76
.76 *2 = 1.52 1 .52
( 98.46 )10 = ( 1100010.011101011........... )2
3. Binary to decimal conversion:-
steps:-
write down the binary digits and list the powers of 2 from right to
left
Eg ( 111011 )2 = 25 24 23 22 21 20
1 1 1 0 1 1
for each positional notation written for the digit, now write the
equivalent weight
25 24 23 22 21 20
32 16 8 4 2 1
multiply each digit with its corresponding weight
32 *1 + 16 *1 +8 *1+4*0 + 2* 1 +1 *1
32 + 16 +8 +0 + 2 +1 = 59
( 111011 )2 = ( 59 ) 10
4. FRACTIONAL BINARY TO DECIMAL:-
a. Convert integer part of binary to decimal equivalent using
positional notation method
b. to conv ert the fractional part of binary to its decimal equivalent
c. write down the binary digits in the fractional part
d. for all the digits write powers of 2 from left to right starting from
2-1 , 2-2, 2-3 ......... ,now write the equivalent weight
2-1= 21/2 = 0.5 , 2-2 = 21/4 =0.25 , 2-3= 21/8 = 0.125 ...........
e. multiply each digit with its corresponding weight
f. add all the values which you obtained in step e
g. to get final answer write the integral part folowed by a decimal
point 21202-12-22-3
Eg:- ( 11. 011 )2 1 1 . 0 1 1
1 * 21 + 1*20 = 3 . 0*0.5 + 1 *0.25 +1 *0.125 = 375
( 11.011)2 = ( 3.375 )10
iii. OCTAL NUMBER SYSTEM:-
Characteristics:
uses eight digits
also called as base 8 number system
each position in an octal number represents a 0 power of the
base 8.
a.Conversion of octal to binary:-
convert each octal digit to a 3-digit binary number
combine all the binary
Eg:- ( 25 ) 8 210 510
(010)2 ( 101 )2
( 25 ) 8 = ( 010101 )2
b. Conversion of binary to octal:-
divide the binary digits into groups of three ( starting from right )
convert each group of three binary digits to one octal digit
Eg:- 101110101
1 0 1 = 1 *20= 1, 0 *21=0, 1 *22= 4 (1 +0+4 ) = 5
1 1 0 = 0 *20=0, 1 *21 =2, 1 *24 = 4 ( 0+2+4) = 6
101 = 1 *20= 1,0 *21= 1, 1*22= 4 ( 1+0+4 ) = 5
( 101110101 )2 = (565)8
c. Conversion of decimal to octal:-
to convert decimal to octal repeated division by 8 method can be
used
Eg:- ( 65 ) 10
8 65
8 8-1
1-0
( 65 ) 10 = ( 101 )8
d. conversion of octal to decimal number:-
write down the octal digits and list the powers of 8 from right to left
for each positional notation of the digit write the equivalent weight
multiply each digit with its corresponding weight
add all the values
Eg:- ( 1265 ) 8
1 2 6 5
83 82 81 80
512 64 8
= 1*512 + 2 * 64 + 6 * 8 + 5 *1
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=512+128+48+5
=693
( 1265 )8 = ( 693 )10
IV. HEXADECIMAL NUMBER SYSTEM:-
Uses 10 digits and 6 letters i.e 0,1,2,.........9,A,B,C,D,E,F
Letters represent the numbers starting from 10
A=10,B=11,C=12,D=13,E=14,F=15
Also called as base 16 number system
each position in a hexadecimal number represents a 0 power of the base
16.
conversion of hexadecimal to binary:-
write 4 bits binary equivalent for each hexadecimal digit for the given
number using positional notation method.
Eg:- 8 BC
2 8 B=11 2 11 C= 12 2 12
2 4-0 2 5-1 2 6-0
2 2-0 2 2-1 2 3-0
1-0 1-0 1-1
8=1000 B= 1011 C= 1100
( 8BC )16 = ( 100010111100 )2
Conversion of Decimal to hexadecimal :-
To convert decimal to hexadecimal repeated division by 16 method can be
used
Eg:- ( 31 ) 10
16 31
1-15
Conversion of hexadecimal to decimal:-
write down the hexadecimal digits and list the powers of 16 from right
to left
for each positional notation written for the digit ,now write the equivalent
weight
multiply each digit with its corresponding weight
add all the values to get one final value
Eg:- ( 25 F )16
2 5 F=15
162 161 160
256 16 1
= 2*256 + 5 *16 + 15 * 1
=512+80+15
=607
(25F )16 = (607 )10
Advantages of number system:-
The biggest advantages of binary number system is its simplicity.As
the switches used in computer language or either ON or OFF ,They
can be easily read with little possibilities of errror.
The main advantage of hexadecimal number is that it is very
compact also it is quick and easy to convert between hexadecimal
number and binary.
Disadvantages of number system:-
The main disadvantages of binary number is that the binary string
equivalent of a large decimal base 10 number can be quite long.
when working with large digital system such as computer it is
common to find binary number consisting of 8.16 and 2 digits which
makes it difficult to both read and write without producing errors
especially when working with lot of 16 or 32 bits binary number.
Application of number system:-
The most common application for the binary number system can be
found in computer technology.
All computer language and programming is based on the 2- digit
number systeem used in digital encoding.
BINARY CODE:-or REPRESENTING CHARACTERS IN MEMORY:-
Combination of bits to represent numbers,letters,or symbols are called as
binary codes or digital codes.
Coding system:-
The transformation of data or information from human
understandable form to machine understandable form is called as
ENCODING.
The transformation of data or information from machine
understandable form to human understandable form is called as
DECODING.
Classification of binary codes:-
Binary codes
weighted non-weighted sequential reflective Alpha numeric Error coding
and correcting
binary BCD Excess-3 Gray five bit BCD 8421 Excess-3 parity Hamming
2421 5211 Excess-3 ASCII EBCDIC Hollerith:
BCD :- Binary coded decimal :-
This is one of the earliest memory code
each decimal digit is represented by four bits
modern computers do not use BCD code since theycannot process
non-numeric data
for example 6 is represented as 0110
The devices such as electronic calculators ,digital clocks work with
BCD numbers
ASCII - American standard code for information interchange:-
pronounced as Ask-ee
it is also an alphanumeric code
it is the standardised alphanumeric code mostly used by computer
manufacturers.
initially it used seven bits and later it was extended to 8 bits .
it can handle 256 characters
in ASCII code, the 8 bits are divided into two 4 bit groups called zone
and numeric group.
The ASCII code equivalent to the upper case letter A IS 65 AND THE
BINARY REPRESENTATION OF ITS 01000001.
EBCDIC- Extended binary coded decimal interchange code:
it is also an alphanumeric code
This is similar to ASCII code with 8 bit representation
it is formulated by international business machine
it can handle 256 characters
The 8 bits are divided into two 4 bit groups.The left part called as
zone and the right part called as digital bit group
ISCII- Indian standard code for information interchange:-
This is a 8 bit coding system
formulated by the department of electronics in India in 1986
Recognized by Bureau of Indian standards
it can handle the character of Indian local languages
UNICODE:-
This coding system is used in the modern computers
Unicode uses 16 bits and it can represent 65536 characters
This code facilitates to represent characters of other languages like
Tamil,Greek, Chinese And Japanese in computers.
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