The Excess-3 code of decimal 3 is:

Excess-3 Code (XS-3 Code)

Concept

Excess-3 code is a non-weighted Binary Coded Decimal (BCD) code obtained by adding 3 to each decimal digit and then converting that sum into 4-bit binary: XS-3 = binary(decimal digit + 3), equivalently XS-3 = BCD + 0011.

It is also a self-complementing, sequential code. A valid 4-bit Excess-3 code always falls in the binary range 0011 to 1100 (since digits 0-9 map to 3-12); any 4-bit pattern outside that range cannot be a valid Excess-3 code.

Excess-3 Code Table

Steps to Convert Decimal to Excess-3

1. Take the decimal digit.

2. Add 3 to it.

3. Convert the result into 4-bit binary.

Application: Excess-3 Code of Decimal 3

1. Add 3 to the digit: 3 + 3 = 6.

2. Convert 6 into 4-bit binary: 6 = 0110.

So the Excess-3 code of decimal 3 is 0110.

Cross-check: Decoding Each Offered Code

Reversing the rule (subtract 3 from the decimal value of the binary code) checks which offered code round-trips back to the original digit 3, and confirms which codes are even valid Excess-3 codes at all:

Reversing 0110 reproduces the digit 3, matching the forward calculation in the Application step above.

Other Properties of Excess-3 Code

• Non-weighted: unlike 8-4-2-1 BCD, its bit positions carry no fixed positional weight.

• Self-complementing: the 1's complement of a digit's Excess-3 code equals the Excess-3 code of that digit's 9's complement (e.g. Excess-3 of 2 is 0101; its 1's complement 1010 is the Excess-3 code of 9−2=7).

• Sequential: Excess-3 code values increase monotonically as the decimal digit increases from 0 to 9.

• Only decimal digits 0-9 are represented; codes outside 0011-1100 (0000, 0001, 0010, 1101, 1110, 1111) are invalid.