NUMBER SYSTEM IN THE COMPUTER

NUMBER SYSTEMS

• When data is typed into a computer, the key board converts each key stroke
  into a binary character code. This code is then transmitted to the computer.
• When the computer transmits the data to the any device, each individual
  character is communicated in binary code. It is then converted back to the
  specific character while displaying or printing the data.
• Every character is represented by a string of “0s” and “1s” – the only digits
  found in the binary numbering system.
  “0” or “1” = bit (Binary Digit)
  8 bits = 1 Byte (1 Character)
  1024 Bytes = 1 KB (Kilo Bytes)
  1024 KB = 1 MB (Mega Byte)
  1024 MB = 1 GB (Giga Byte)
  1024 GB= 1 TB (Terra Byte)

TYPES OF NUMBER SYSTEM

DECIMAL NUMBER SYSTEM

• In the decimal system the successive positions to the left of the decimal point
  represent units, tens, hundreds, thousands etc.
• For example if we consider a number 7762, the digit 2 represents the number of units, 6 represents the number of tens, 7 the number of hundreds and 7 the number of thousands.
  (7 x 1000) + (7 x 100) + (6 x 10) + (2 x 1) = 7762
• Thus as we move one position to the left, the value of the digit increases by
  ten times. We can see that the position of the number affects its value.

BINARY NUMBER SYSTEM

• We now come to a different number system – the Binary number system.
  
• This binary number system has a base of two, and the symbols used are “0”
  And “1”.

DECIMAL TO BINARY CONVERSION

• In conversion from decimal to any other number system, the steps to be
  followed are:
• Divide the decimal number by the base of 2. o Note the remainder in one column and divide the quotient again with the base.
•  Repeat this process until the quotient is reduced to a zero.

BINARY TO DECIMAL CONVERSION

• The decimal number of 100001 is = (1*26)+(0*25)+(0*24)+(0*23)+(0*22)+(0*21)+(1*20)
  = (1*64) + (0*32) + (0*16) + (0*8) + (0*4) + (0*2) + (1 * 1)
  = 64 + 0 + 0 + 0 + 0 + 0 + 1
  = 65
  The decimal number of 1000001 is 65.

OCTAL NUMBER SYSTEM

• A commonly used positional system is the Octal System. The octal systemhas
  a base of 8.
  
• The values increase from left to right as 1, 8, 64, 512, 4096,….

DECIMAL TO OCTAL CONVERSION

• In conversion from decimal to any other number system, the steps to be
  followed are:
  Divide the decimal number by the base of the 8.

OCTAL TO DECIMAL CONVERSION

• The octal number is 340
  = (3*82)+(4*81)+(0*80)
  = (3*64) + (4*8) + (0*1)
  = 192 + 32 + 0
  = 224
  The decimal number of 340 is 224

HEXADECIMAL NUMBER SYSTEM

• There is another commonly used positional system, hexadecimal system.
  
• The hexadecimal system has a base of 16, so the value increases from left to right as 1, 16, 256, 65536,. . . .
• We need to keep a simple table in mind before we attempt any conversion from hexadecimal or vice-versa.

DECIMAL TO HEXADECIMAL CONVERSION

• In conversion from decimal to any other number system, the steps to be
  followed are:
  Divide the decimal number by the base of 16. 

BINARY TO HEXADECIMAL CONVERSION

• Each hexadecimal digit is represented by 4 binary digits.

• To convert a binary number to its hexadecimal equivalent we split the
  quantity into groups of four onwards, as before.
• Each of this group of four is directly converted into their hexadecimal
  equivalent.
  
• We may add zeros to the left of the number if necessary.
• For example:

            Binary -10101011000010
              = 0010 1010 1100 0010
              = 2 A C 2
           So, the hexadecimal equivalent of binary 10101011000010 will be 2AC2.

PARITY CHECK

• A parity check is a technique to detect the correctness of characters transmitted.
• For each character transmitted, a bit knows as the parity bit is added.
• In an Even Parity System, a parity bit is added such that the total number of ‘1’s,
  inclusive of the parity bit, is even.
• In an Odd Parity System, the total number of ‘1’ bits transmitted must be odd.
• The parity check is not fool proof. It will fail when an even number of bits were incorrectly received.
• This cannot be recognized by this system. That if they are incorrectly received or not.