Programming

Encoding a BCD Number

Definition

BCD represents each of the digits of an unsigned decimal as the 4-bit binary equivalents.

Unpacked BCD

Unpacked BCD representation contains only one decimal digit per byte. The digit is stored in the least significant 4 bits; the most significant 4 bits are not relevant to the value of the represented number.

Packed BCD

Packed BCD representation packs two decimal digits into a single byte.

Invalid BCD Numbers

These binary numbers are not allowed in the BCD code:  1010, 1011, 1100, 1101, 1110, 1111

Packing a Two-Byte BCD

To pack a two-byte unpacked BCD number into a single byte creating a packed BCD number, shift the upper byte left four times, then OR the results with the lower byte.

For example,

Converting between Decimal and BCD

From Decimal to Unpacked BCD:   To convert a decimal number into an unpacked BCD number, assign each decimal digit its 8-bit binary equivalent.

For example,

From Decimal to Packed BCD:   To convert a decimal number into a packed BCD number, assign each digit of the decimal to its 4-bit equivalent, padding the upper nybble with zeroes if necessary.

For example,

From BCD to Decimal:   To convert a BCD into a decimal number, just reverse the appropriate process from above; beginning with the LSB, group the binary digits by either 4 or 8 bits for packed and unpacked, respectively, then convert each set into its decimal equivalent.

Converting between Binary and BCD

From Binary to Unpacked BCD:   To convert a binary number into an unpacked BCD, divide the binary number by decimal 10 and place the quotient in the most significant byte and the remainder in the least significant byte.

For example,

00110101(base 2) = 00000101 00000011(unpacked BCD)
0011 0101	=	53(base 2)
÷ 0000 1010	=	10(base 10)
0000 0101	=	5		5 0000 0101	3 0000 0011
with a remainder of
0000 0011	=	3		MSB	LSB

From Two-Byte Unpacked BCD to Binary:   To convert from a two-byte unpacked BCD to a binary number, multiply the most significant byte of the BCD by decimal ten, then add the product to the least significant byte.

For example,

00001001 00000010_{(unpacked BCD)} = 01011100_{(base 2)}

BCD Addition

Either packed or unpacked BCD numbers can be summed. BCD addition follows the same rules as binary addition. However, if the addition produces a carry and/or creates an invalid BCD number, an adjustment is required to correct the sum. The correction method is to add 6 to the sum in any digit position that has caused an error.

For example,

BCD Subtraction

Either packed or unpacked BCD numbers can be subtracted. BCD subtraction follows the same rules as binary subtraction. However, if the subtraction causes a borrow and/or creates an invalid BCD number, an adjustment is required to correct the answer. The correction method is to subtract 6 from the difference in any digit position that has caused an error.

For example,

Unpacked BCD Multiplication

Multiplication cannot be performed on packed BCD; the 4 most significant bits must be zeroed for the adjustment to work. Multiply the two unpacked BCD numbers using the rules for binary multiplication. To adjust the product divide it by decimal 10, then place the quotient in the most significant byte and the remainder in the least significant byte (convert the binary answer to unpacked BCD).

For example,

00001001 × 00000100 = 00000011 00000110
0000 1001	=	9
× 0000 0100	=	4
0010 0100	=	24	(error)
0010 0100	=	24	(error)
÷ 0000 1010	=	10(base 10)	(adjustment)
0000 0011	=	3		3 0000 0011	6 0000 0110
with a remainder of
0000 0110	=	6		MSB	LSB

Unpacked BCD Division

BCD division also cannot be performed on packed numbers. Before dividing an unpacked BCD number, the division adjustment is made by converting the BCD numbers to binary. Adjust the two-byte BCD number by multiplying the upper byte by decimal 10 and adding the product to the lower byte. After the adjustment, divide the two binary numbers using the rules of binary arithmetic. Finally, convert the binary quotient into an unpacked BCD number if necessary.

For example,

00000010 00001000 ÷ 00000111 = 00000100 (28 ÷ 7 = 4)
0000 0010	=	2
× 0000 1010	=	10(base 10)	(adjustment)
0001 0100	=	20(base 10)
0001 0100	=	20(base 10)
+ 0000 1000	=	8
0001 1100	=	28(base 10)
0001 1100	=	28(base 10)
÷ 0000 0111	=	7
0000 0100	=	4
00000101 00000010 ÷ 00000100 = 00000001 00000011 (52 ÷ 4 = 13)
0000 0101	=	5
× 0000 1010	=	10(base 10)	(adjustment)
0011 0010	=	50(base 10)
0011 0010	=	50(base 10)
+ 0000 0010	=	2
0110 0100	=	52(base 10)
0110 0100	=	52(base 10)
÷ 0000 0100	=	4
0000 1101	=	?	(invalid)
0000 1101	=	?	(invalid)
÷ 0000 1010	=	10(base 10)	(adjustment)
0000 0001	=	1		1 0000 0001	3 0000 0011
with a remainder of
0000 0011	=	3		MSB	LSB

Other Decimal Codes

ASCII Representation of Digits

ASCII digits are examples of unpacked binary coded decimals. A digit is represented as its 4-bit binary equivalent, which is stored in the lower nybble. The upper nybble contains 011 and has no bearing on the value of the represented number.

 

EBCDIC Representation of Digits

EBCDIC digits are similiar to ASCII digits. The lower nybble still contains the 4-bit binary equivalent, but the upper nybble is padded with 1111 instead of 011.

 

Excess-3 (XS3)

In excess-3 representation, each decimal digit is the 4-bit binary equivalent with 3 (0011) added.

 

4221

For 4221 coded-decimals, each digit also is represented as a 4-bit group. Instead of each bit representing a successive power of two (8, 4, 2, and 1, from the left) as in binary, the bits represent 4, 2, 2, and 1.

Other Resources

• BCD Arithmetic, A Tutorial
• BCD Number System (packing algorithms)
• Binary Arithmetic

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