What is number system of computer (decimal, octal,binary,hexadecimal number system) full tutorial|

what is number system of a computer 

 the computer can understand only binary  code or number .
whatever  machine language we use, the computer will understand binary language .
                       at first we read about computer number system , the computer number system has divided four parts, the names of some number system and their base are given bellow
             

                        Name of number system 

1. Decimal number system.
2. Octal number system.
3. Hexadecimal number system.
4. Binary number system.

                          Base of the number system

1. Decimal number system-10(base radix )
2. Octal number system-8(base radix)
3. Hexadecimal number system-16(base radix)
4. Binary number system-2(base radix)

1.Decimal number system

This number system popularly known as the international number system with which almost all the people of the world are accustomed. It was invented in india which has led to the rapid development of science and technology all over the world. 

Decimal system:The decimal system is composed of 10 numbers or symbols. these 10 number are 0,1,2,3,4,5,6,7,8,9 using these symbols as digits of a number ,we can express any quantity. the decimal system is also called the base-10 system because it has 10 digits. 

  An example for decimal number:decimals base/radix   number system is 10 , decimal number system is 10 in powers of which any can be expressed . thus 87 number can be expressed as  
                                                8 multiple with 10 power 1 add 7 equal to 80+7=87.





similarly we can be expressed with( point) means suppose 56.6 like this. 
Lets give a example with number  812.65 can be expressed as : 
                8 multiple with 10 power 2 add 1 multiple 10 add 2 multiple 10 power 0 add 6 multiple with 10 power -1 add 5 multiple 10 power -2 
which is equal to  800+10+2+.60+.05=812.65 

 `2-Octal number system

                  under this system a number is formed using any of the fundamental digits  0,1,2,3,4,5,6,7.hence, the radix or base of the system is 8 the example of octal system are: 8³ =512, 8² =64, 8¹ =8, 8⁰ =1 the most significant digit,  8^-1 =1/8,  8^-2 =1/64, 8^-3 =1/512 the least significant digit.
  some of the numbers of octal are:  0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,
......... it may be noted that a number under this system is always suffixed by the base . thus if the number  25,246,1547,are in octal number system they are to be presented as (25)8, (246)8, (1547)8 . the place value of each digit in the system is 8 times the place value of the digit to its right.

 3-Hexadecimal number system

Since hexa represents 6 and decimal represents 10, hexadecimal represents 16,. hence, under this system a number is constructed by using any of the 16 basic symbols 0,1,2,3,4,5,6,7,8,9,A, B,C,D,E,F The letters A,B,C,D,E,F, Represent the number 10,11,12, 13 ,14,15 respectively  since this system is based on the 16 fundamental symbols the radix or the base of the system is 16. the place value of each symbol is 16 times the place value of the symbol to its right.
             the hexadecimal system uses base 16. thus, it has 16 possible digits symbols it uses the digits through 9 plus the letters ,A,B,C,D,E, and F as digit symbols

4-binary number system

This number system is very useful for working of the computers because the computers can understand no other system except the binary one. under this number system a number is constituted with any of the two fundamental digits i.e.,0 and 1 which are frequently referred to as binary digits . the examples of such numbers are 1010,1111,0000,101.101.,1100.10 etc since this number system is based on 2 basic digits ,0 and 1 the radix or base of this system is 2. the place value of each digit in a binary number is twice the place value of the digit to its right. such numbers are usually written with the radix or base 2 indicated as its suffix. this base-2 system can be used to represent any quantity that can be represented in decimal or number system.the above binary number will be presented as (1010)2, (1111)2, (101.101)2,

       the special term of number 

the special term used in connection with the binary number system are explained.

BIT:the term bit is an abbreviation of the word binary digit.it takes the form of either of the binary digits 0 and 1 to represent exactly one character .

Binary digit

it is a logical 0 or 1 that represents a passive or an active state respectively of components in electric circuits. 

Nibble 

its a group of 4 bits. it is very often used in the binary system for each number. four bits can be arranged among themselves to represent 24(=16)different number as required in hexadecimal numbers system. 

Byte

it is a group of 8 bits. it is the smallest unit which can be represented  as data item or a character. 

Computers word 

It is a set of fixed number of bits varying from 8 bits to 64 bits and in multiples of 8 bits or a byte . however there are some small micro processors which use 4 bit or 16 bit words . the length of a computer word is measured in terms of number of bits accommodated therein . the size of such words, however , varies from computer to computer depending upon there size but it remains fixed for a particular form of computer words as computer stores the data and instructions in the form of word computer 

Conversion of number system

there are two methods of converting a decimal number to a number of any other system . they are: 

1. Remainder method or the method of division.

2. power method or the method of subtraction 

Binary to decimal conversion

any binary number can be converted to its decimal equivalent simple by summing together the weights of the various positions in the binary number.
                            the binary number  110112
                           24+23+0+21+20=16+8+0+2+1=2710(decimal)
                                                    and
                           10110101₂(binary)

2⁷ +0+2⁵+2⁴+0 +2² +0 +2⁰ =128+0+32+16+0+4+0+1=181₁₀(decimal)

you should notice that the method is find the weights for each bit position that contains a 1, and then to add them up.

Decimal to binary conversion

there are two methods 

reverse of binary-to-digital method

45₁₀ (decimal) =32+0+8+4+0+1

=2⁵+0+2³+0+2²+0+2⁰

= 101101(binary)

 next repeat division 

25/2 =12+remainder of 1(least significant bit)

12/2 =6+remainder of 0 (0)

6/2 = 3+remainder of 0 (0)

3/2 =1+ remainder of 1 (1)

1/2 =0 remainder of  1 ( most significant bit)

result 2510(decimal) =11001(binary)

octal to decimal conversion

24.68=2x(81)+4x(80)+6x(8-1)=20.7510

Binary-to-Octal/octal-to-binary conversion

octal digit 01234567

binary equivalent for 0(000) for 1 (001) for 2 (010) for 3 (011) for 4(100) for 5 (101) for 6 (110) for 7 (111)

each octal digit is represented  by three bits of binary digits

e.g.  100 111 0102 = (100) (111) (010)2=4728

Hexadecimal to decimal conversion

e.g.  2AF16=2x(162)+ 10x (161) +15x(160) =68710

Repeat division:convert decimal to hexadecimal

this method uses repeated division by 16 Eg. convert 37810 to hexadecimal and binary: 

378/16 =23 remainder of 10   A( least significant bit)

23/16 =1 remainder of 7   7

1/16 = 0 remainder of 1    1 (most significant bit)

result  

378₁₀ = 17A₈               

Convert to binary  =000101111010₂=0000 0001 0111 1010(16 bits)

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