Unit 1 » Number Systems, Operations and Codes

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

• 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)