Number System and Codes

2’S Complement, BCD Code, XS-3 Code, Gray Code

1. N’s Complement

• For a number system with base n, n’s complement of a number is obtained by subtracting the number from nᵏ, where k = number of digits.
• Formula
  n’s complement of X = nᵏ − X
• Types
  • Binary (n = 2) → 2’s complement
  • Decimal (n = 10) → 10’s complement
  • Octal (n = 8) → 8’s complement
  • Hexadecimal (n = 16) → 16’s complement
• Relation with (n−1)’s complement ⭐ n’s complement = (n−1)’s complement + 1
• Use
  • Subtraction using addition
  • Representation of negative numbers
• Example (Decimal) Number = 275 10’s complement = 1000 − 275 = 725
• Example (Binary)
  Number = 1010
  2’s complement = 0101 + 1 = 0110

2’s Complement

Definition: Binary number representation for signed integers where negative numbers are obtained by taking the 1’s complement and adding 1. Use: Simplifies subtraction in digital systems. Steps (for negative numbers):

1. Write the number in binary.
2. Take 1’s complement (invert bits).
3. Add 1 to the result.

Example (–5 in 8-bit):

+5   = 00000101
1's  = 11111010
+1   = 11111011  → –5

Excess-3 Code

Excess-3 is mainly used for self-complementing decimal arithmetic, not primarily for error detection.

Definition: A self-complementary decimal code obtained by adding 3 to each decimal digit and converting to 4-bit binary.

Use: self-complementing decimal arithmetic.

Example table:

Decimal	Binary	Excess-3
0	0000	0011 (3)
1	0001	0100 (4)
2	0010	0101 (5)
3	0011	0110 (6)
4	0100	0111 (7)
5	0101	1000 (8)
6	0110	1001 (9)
7	0111	1010 (10)
8	1000	1011 (11)
9	1001	1100 (12)

BCD (Binary Coded Decimal)

Definition: Represents each decimal digit separately in binary (4 bits per digit).
Use: Digital displays, calculators.

Example:

                5    9
Decimal 59 = (0101 1001) in BCD

Gray Code

Definition: A binary code where consecutive values differ by only 1 bit (minimizes errors in transitions).
Use: Position encoders, error reduction.

Example (3-bit Gray code):

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

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Excess-3 Code (XS-3 Code)

Excess-3 code is a non-weighted, self-complementary decimal code obtained by adding 3 to each decimal digit and then converting the result into 4-bit binary.

It is called “Excess-3” because:

Code = Decimal Digit + 3

Formula

XS-3 Code = Binary of (Decimal Digit + 3)

Steps to Find Excess-3 Code :

1. Take the decimal digit
2. Add 3 to it
3. Convert the result into 4-bit binary

Excess-3 Code Table :

Decimal	Decimal + 3	Excess-3 Code
0	3	0011
1	4	0100
2	5	0101
3	6	0110
4	7	0111
5	8	1000
6	9	1001
7	10	1010
8	11	1011
9	12	1100

Examples

Example 1: Decimal 2

2 + 3 = 5
Binary(5) = 0101

Excess-3 Code:

0101

Example 2: Decimal 7

7 + 3 = 10
Binary(10) = 1010

Excess-3 Code:

1010

Example 3: Decimal Number 59

Separate each decimal digit: ⭐

5 → 5 + 3 = 8 → 1000
9 → 9 + 3 = 12 → 1100

So:

59 = 1000 1100 (XS-3)

Properties of Excess-3 Code

1. Non-Weighted Code

There are no fixed positional weights like:

8, 4, 2, 1

So XS-3 is a non-weighted code.

2. Self-Complementary Code

1’s complement of an XS-3 number gives XS-3 representation of the 9’s complement decimal digit.

Example:

Decimal 2 → XS-3 = 0101
1’s complement = 1010
1010 = XS-3 of 7

Since:

9 − 2 = 7

Hence XS-3 is self-complementary.

3. Uses 4 Bits

Each decimal digit is represented using 4 bits.

Why Excess-3 Code is Used

1. Simplifies Decimal Subtraction Due to self-complementary property, subtraction becomes easier.

2. Used in Decimal Arithmetic Useful in:

   • Adders
   • Subtractors
   • Digital arithmetic circuits

3. Reduces Transition Errors Some invalid combinations help identify errors in circuits.

4. Useful in Digital Electronics Used in:

   • Digital systems
   • Calculators
   • Arithmetic logic circuits

Difference Between BCD and Excess-3

Feature	BCD	Excess-3
Type	Weighted	Non-weighted
Representation	Actual binary	Binary after adding 3
Decimal 0	0000	0011
Decimal 5	0101	1000
Self-Complementary	No	Yes