NUMBER SYSTEM
NUMBER SYSTEM
collection of digit makes number.
unlke analog system signal say data are continouly changing such as amplitude, frequency with respective time,digital system have only two stage High (1) or Low (0) in +ve logic.
only series of 1’s and 0’s represent signal say data.
Generaly used number system are shown in table format.
Number may consist of integrail and fractional part as shown in table
number base Noof weight example
symbol asign
Binary 2 0,1 2’n 10110.10
Octal 8 0,1,..7 8’n 157.27
Decimal 10 0,1,..9 10’n 159.24
Hex 16 0,1,..9,
A,B,.F 16’n 2DF5.A7
BINARY NUMBER SYSTEM
Combinatin of 0’s and 1 form binary number.
Nibble: Group of 4 bit called nibble.
Byte: Group of 8 bitcalled byte.
binary to decimal conversion:
let we convert (10110)2 to decimal (??)10
10110 == IMAGE NS01
Decimal to binary conversion:
METHOD 1
Succesive operasion method
First subtract the largest possible POWER of TWO (for binary base) and then keeping subtraction largest number from remainder as shown in the table.
putting 1 for respective no.
let we convert (55)dec to binary Val
55
-32 ==> 1 MSB
—-
23
-16 ==> 1
—-
07
-04 ==> 1
—-
03
-02 ==> 1
—-
01 ==> 1 LSB
(if remainder 1 last bit 1
if remainder 0 last bit 0)
nth power .. 7 6 5 4 3 2 1 0
Lage_val .. 128 64 32 16 08 04 02 01
Bin_dig .. 0 0 1 1 0 1 1 1
thus (55)dec=(110111)bin
Method 2
division by base method
55/2 =27 remainder 1 lsb
27/2 =13 remainder 1
13/2 =6 remainder 1
6/2 =3 remainder 0
3/2 =1 remainer 1
1/2 =0 remainder 1 msb
Thus
(55)dec=(110111)bin
OCTAL NUMBER SYSTEM
Octal number system has base 8, means it can represent as sysbol consist of digit 0,1,2,..7.
Decimal to octal conversion
let we do with succesive division by BASE 8
(247)dec to (??)oct
247/8 =30 remainder (247-240) 7 lsb
30/8 =3 remainder (30-24) 6
3/8 =0 remainder (3-0) 3 msb
Thus
(247)dec = (367)oct
Octal to decimal conversion
(367)oct to (??)dec
here base is 8
(367)oct =3×8^2+6×8^1+7×8^0
=3×64+6×8+7×1
=(247)dec
Octal to binary conversion
octal number can be converted to binary number just replacing equivalent binary.
for example
(367)oct =(011 110 111)bin
=(011110111)bin
Binary to octal conversion
binary number can be converted into octal number just making group of 3 bit starting from lsb.
for example
(11000110101)bin =(??)oct
(011 000 110 101)bin =(3 0 6 5)oct (start from lsb)
=(3065)oct
Hexadecimal number system
These number system is very popular in computer lower langauges, Assembly langages are coaded in Hex format.
It has base 16.
It is form by 16 distist symbol (0,1,..9,A,..F).
Hex to decimal conversion
Let we convert (103A)hex to its equivalent value
(103A)hex =(??)dec
=(1×16^3+0x16^2+3×16^1+Ax16^0)
=(1×4096+0x256 +3×16 +10×1)
=(4154)dec
Decimal to hex conversion
Decimal to hex can be convert by seccesevely dividing with 16 and keeping track on remainder.
let we convert (4154)dec to its hex value.
(4154)dec =(??)hex
4154/16 = 259 remainder(4154-4144) 10 A lsb
259/16 = 1 remainder(259-256) 3 03
1/16 = 0 remainder(1- 1 01 msb
Thus
(4154)dec=(0103A)hex
Hex to binary conversion
hex to binary conversion is carried out by replacing hex bit of equivalent four digit binary bit.
let we convert (103A)hex to (??)bin value.
(103A)hex =(0001 0000 0011 1010)bin
=(1000000111010)bin