# Excess-3 Code

The **excess-3 code** is another important BCD code. It is particularly significant for arithmetic operations as it overcomes the shortcomings encountered while using the 8421 BCD code to add two decimal digits whose sum exceeds 9. The excess-3 code has no such limitation, and it considerably simplifies arithmetic operations. Table 2.2 lists the excess-3 code for the decimal numbers 0–9.

## Excess-3 Code

The excess-3 code for a given decimal number is determined by adding ‘3’ to each decimal digit in the given number and then replacing each digit of the newly found decimal number by its four-bit binary equivalent. It may be mentioned here that, if the addition of ‘3’ to a digit produces a carry, as is the case with the digits 7, 8 and 9, that carry should not be taken forward.

### Excess-3 Code Equivalent Of Decimal Numbers

*Excess-3 code equivalent of decimal numbers*

The result of addition should be taken as a single entity and subsequently replaced with its excess-3 code equivalent. As an example, let us find the excess-3 code for the decimal number 597:

- The addition of ‘3’ to each digit yields the three new digits/numbers ‘8’, ‘12’ and ‘10’.
- The corresponding four-bit binary equivalents are 1000, 1100 and 1010 respectively.
- The excess-3 code for 597 is therefore given by: 1000 1100 1010=100011001010.

Also, it is normal practice to represent a given decimal digit or number using the maximum number of digits that the digital system is capable of handling. For example, in four-digit decimal arithmetic, 5 and 37 would be written as 0005 and 0037 respectively. The corresponding 8421 BCD equivalents would be 0000000000000101 and 0000000000110111 and the excess-3 code equivalents would be 0011001100111000 and 0011001101101010.

Corresponding to a given excess-3 code, the equivalent decimal number can be determined by first splitting the number into four-bit groups, starting from the radix point, and then subtracting 0011 from each four-bit group. The new number is the 8421 BCD equivalent of the given excess-3 code, which can subsequently be converted into the equivalent decimal number. As an example, following these steps, the decimal equivalent of excess-3 number 01010110.10001010 would be 23.57.

Another significant feature that makes this code attractive for performing arithmetic operations is that the complement of the excess-3 code of a given decimal number yields the excess-3 code for 9’s complement of the decimal number. As adding 9’s complement of a decimal number B to a decimal number A achieves A – B, the excess-3 code can be used effectively for both addition and subtraction of decimal numbers.