Unit 1 » Number Systems, Operations and Codes

1st Semester

Types of codes

BCD (Binary-Coded Decimal) code :

  • Four-bit code that represents one of the ten decimal digits from 0 to 9.
  • Example – (37)10 is represented as 0011 0111 using BCD code, rather than (100101)2 in straight binary code.
  • Thus BCD code requires more bits than straight binary code.
  • Still it is suitable for input and output operations in digital systems.

Note: 1010, 1011, 1100, 1101, 1110, and 1111 are INVALID CODE in BCD code.






ASCII (American Standard Code Information Interchange) code :

  • It is 7-bit or 8-bit alphanumeric code.
  • 7-bit code is standard ASCII supports 127 characters.
  • Standard ASCII series starts from 00h to 7Fh, where 00h-1Fh are used as control characters and 20h-7Fh as graphics symbols.
  • 8-bit code is extended ASCII supports 256 symbols where special graphics and math’s symbols are added.
  • Extended ASCII series starts from 80h to FFh.

EBCDIC (Extended Binary Coded Decimal Interchange Code) code

  • 8-bit alphanumeric code developed by IBM, supports 256 symbols.
  • It was mainly used in IBM mainframe computers.

Gray code





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  • Differs from leading and following number by a single bit.
  • Gray code for 2 is 0011 and for 3 is 0010.
  • No weights are assigned to the bit positions.
  • Extensively used in shaft encoders.

Excess-3 code

  • 4-bit code is obtained by adding binary 0011 to the natural BCD code of the digit.
  • Example – decimal 2 is coded as 0010 + 0011 = 0101 as Excess-3 code.
  • It not weighted code.
  • Its self-complimenting code, means 1’s complement of the coded number yields 9’s complement of the number itself.
  • Used in digital system for performing substraction operations.


 Table listing Binary, Gray, BCD and Express-3 Coding for Decimal Digits
Converting Binary to Gray Code –
Binary-To-Decimal Conversion
Any binary number can be converted to its decimal equivalent simply by summing together the weights of the various positions in the binary number which contain a 1.

Binary Decimal
110112
24 + 23+01+21+20 =16+8+0+2+1
Result 2710

And

Binary Decimal
101101012
27+06+25+24+03+22+01+20 =128+0+32+16+0+4+0+1
Result 18110





You should have noticed that the method is to find the weights (i.e., powers of 2) for each bit position that contains a 1, and then to add them up.
Decimal-To-Binary Conversion
There are 2 methods:
· Reverse of Binary-To-Decimal Method
· Repeat Division
Reverse of Binary-To-Decimal Method

Decimal Binary
4510 =32 + 0 + 8 + 4 +0 + 1
=25+0+23+22+0+20
Result = 1011012

Repeat Division-Convert decimal to binary
This method uses repeated division by 2.
Convert 2510 to binary

Division Remainder Binary
25/2 = 12+ remainder of 1 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 1 (Most Significant Bit)
Result 2510 = 110012

Binary-To-Octal / Octal-To-Binary Conversion

Octal 0 1 2 3 4 5 6 7
000 001 010 011 100 101 110 111

Each Octal digit is represented by three binary digits.





Example:
100 111 0102 = (100) (111) (010)2 = 4 7 28
Repeat Division-Convert decimal to octal
This method uses repeated division by 8.
Example: Convert 17710 to octal and binary

Division Result Binary
177/8 = 22+ remainder of 1 1 (Least Significant Bit)
22/8 = 2 + remainder of 6 6
2/8 = 0 + remainder of 2 2 (Most Significant Bit)
Result 17710 = 2618
Binary = 0101100012

Hexadecimal to Decimal/Decimal to Hexadecimal Conversion
Example:
2AF16 = 2 x (162) + 10 x (161) + 15 x (160) = 68710
Repeat Division- Convert decimal to hexadecimal
This method uses repeated division by 16.
Example: convert 37810 to hexadecimal and binary:

Division Result Hexadecimal
378/16 = 23+ remainder of 10 A (Least Significant Bit)23
23/16 = 1 + remainder of 7 7
1/16 = 0 + remainder of 1 1 (Most Significant Bit)
Result 37810 = 17A16
Binary = 0001 0111 10102

 
Binary-To-Hexadecimal /Hexadecimal-To-Binary Conversion

Hexadecimal Digit 0 1 2 3 4 5 6 7
Binary Equivalent 0000 0001 0010 0011 0100 0101 0110 0111
Hexadecimal Digit 8 9 A B C D E F
Binary Equivalent 1000 1001 1010 1011 1100 1101 1110 1111

Each Hexadecimal digit is represented by four bits of binary digit.
Example:
1011 0010 11112 = (1011) (0010) (1111)2 = B 2 F16
Octal-To-Hexadecimal Hexadecimal-To-Octal Conversion
· Convert Octal (Hexadecimal) to Binary first.
· Regroup the binary number by three bits per group starting from LSB if Octal is required.
· Regroup the binary number by four bits per group starting from LSB if Hexadecimal is required.
Example:
Convert 5A816 to Octal.

Hexadecimal Binary Octal
5A816 = 0101 1010 1000 (Binary)
= 010 110 101 000 (Binary)
Result = 2 6 5 0 (Octal)