# Binary Coded Decimal

The **binary coded decimal** (BCD) is a type of binary code used to represent a given decimal number in an equivalent binary form. BCD-to-decimal and decimal-to-BCD conversions are very easy and straightforward. It is also far less cumbersome an exercise to represent a given decimal number in an equivalent BCD code than to represent it in the equivalent straight binary form discussed in the previous chapter.

## Binary Coded Decimal

The BCD equivalent of a decimal number is written by replacing each decimal digit in the integer and fractional parts with its four-bit binary equivalent. As an example, the BCD equivalent of (23.15)_{10} is written as (0010 0011.0001 0101)_{BCD}. The BCD code described above is more precisely known as the 8421 BCD code, with 8, 4, 2 and 1 representing the weights of different bits in the four-bit groups, starting from MSB and proceeding towards LSB. This feature makes it a weighted code, which means that each bit in the four-bit group representing a given decimal digit has an assigned weight. Other weighted BCD codes include the 4221 BCD and 5421 BCD codes. Again, 4, 2, 2 and 1 in the 4221 BCD code and 5, 4, 2 and 1 in the 5421 BCD code represent weights of the relevant bits. Table above shows a comparison of 8421, 4221 and 5421 BCD codes. As an example, (98.16)_{10} will be written as 1111 1110.0001 1100 in 4221 BCD code and 1100 1011.0001 1001 in 5421 BCD code. Since the 8421 code is the most popular of all the BCD codes, it is simply referred to as the BCD code.

### BCD-to-Binary Conversion

A given BCD number can be converted into an equivalent binary number by first writing its decimal equivalent and then converting it into its binary equivalent. The first step is straightforward, and the second step was explained in the previous chapter. As an example, we will find the binary equivalent of the BCD number 0010 1001.0111 0101:

- BCD number: 0010 1001.0111 0101.
- Corresponding decimal number: 29.75.
- The binary equivalent of 29.75 can be determined to be 11101 for the integer part and .11 for the fractional part.
- Therefore, (0010 1001.0111 0101)
_{BCD}= (11101.11)_{2}.

### Binary-to-BCD Conversion

The process of binary-to-BCD conversion is the same as the process of BCD-to-binary conversion executed in reverse order. A given binary number can be converted into an equivalent BCD number by first determining its decimal equivalent and then writing the corresponding BCD equivalent. As an example, we will find the BCD equivalent of the binary number 10101011.101:

- The decimal equivalent of this binary number can be determined to be 171.625.
- The BCD equivalent can then be written as 0001 0111 0001.0110 0010 0101.

### Higher-Density BCD Encoding

In the regular BCD encoding of decimal numbers, the number of bits needed to represent a given decimal number is always greater than the number of bits required for straight binary encoding of the same. For example, a three-digit decimal number requires 12 bits for representation in conventional BCD format. However, since 2^{10} > 10^{3}, if these three decimal digits are encoded together, only 10 bits would be needed to do that. Two such encoding schemes are Chen-Ho encoding and the densely packed decimal. The latter has the advantage that subsets of the encoding encode two digits in the optimal seven bits and one digit in four bits like regular BCD.

### Packed and Unpacked BCD Numbers

In the case of unpacked BCD numbers, each four-bit BCD group corresponding to a decimal digit is stored in a separate register inside the machine. In such a case, if the registers are eight bits or wider, the register space is wasted.

In the case of packed BCD numbers, two BCD digits are stored in a single eight-bit register. The process of combining two BCD digits so that they are stored in one eight-bit register involves shifting the number in the upper register to the left 4 times and then adding the numbers in the upper and lower registers. The process is illustrated by showing the storage of decimal digits ‘5’ and ‘7’:

- Decimal digit 5 is initially stored in the eight-bit register as: 0000 0101.
- Decimal digit 7 is initially stored in the eight-bit register as: 0000 0111.
- After shifting to the left 4 times, the digit 5 register reads: 0101 0000.
- The addition of the contents of the digit 5 and digit 7 registers now reads: 0101 0111.