![]() ![]() Adders require extra logic to cause them to wrap and generate a carry early. ![]() Some operations are more complex to implement.Conversion from pure binary involves relatively complex logic that spans digits, and for large numbers no linear-time conversion algorithm is known (see Binary numeral system ยง Conversion to and from other numeral systems). Conversion to a character form or for display (e.g., to a text-based format such as XML, or to drive signals for a seven-segment display) is a simple per-digit mapping, and can be done in linear (O( n)) time.Alignment of two decimal numbers (for example 1.3 + 27.08) is a simple, exact, shift.Addition and subtraction in decimal does not require rounding. Rounding at a decimal digit boundary is simpler.This is useful in financial calculations. Consequently, a system based on binary-coded decimal representations of decimal fractions avoids errors representing and calculating such values. Many non-integral values, such as decimal 0.2, have an infinite place-value representation in binary (.001100110011.) but have a finite place-value in binary-coded decimal (0.0010). C STRING TO BCD SOFTWAREThis was an important case in determining the patentability of software and algorithms. Supreme Court overturned a lower court decision which had allowed a patent for converting BCD encoded numbers to binary on a computer. Both ASCII and EBCDIC character codes for the digits are examples of zoned BCD, and are also shown in the table. ![]() In the headers to the table, the ' 8 4 2 1', indicates the weight of each bit shown in the fifth column, 'BCD 8 4 โ2 โ1', two of the weights are negative. The method here can be referred to as Simple Binary-Coded Decimal ( SBCD) or BCD 8421.The following table represents decimal digits from 0 to 9 in various BCD systems. The binary-coded decimal scheme described in this article is the most common encoding, but there are many others. One important special case is digit 0, represented by a lone 0 punch in the card, and (8,2) in core memory. This allows the circuitry to convert between the punched card format and the internal storage format to be very simple with only a few special cases. The currency symbol $, (11,8,3) in the punched card, was encoded in memory as (B,8,2,1). Thus the letter A, which is (12,1) in the punched card format, is encoded (B,A,1). A '12 zone' punch set both B and A, an '11 zone' set B, and a '0 zone' (a 0 punch combined with any others) set A. For most other characters bits B and A are derived simply from the '12', '11', and '0' 'zone punches' in the punched card character code, and bits 8 through 1 from the 1 through 9 punches. For encoding digits 1 through 9, B and A are zero and the digit value represented by standard 4-bit BCD in bits 8 through 1. C STRING TO BCD PLUSThe IBM 1400 series are character-addressable machines, each location being six bits labeled B, A, 8, 4, 2 and 1, plus an odd parity check bit ( C) and a word mark bit ( M). Burroughs systems used 1101 (D) for negative, and any other value is considered a positive sign value (the processors will normalize a positive sign to 1100 (C)). These match the EBCDIC zone for digits without a sign overpunch. ![]() ILE RPG uses 1111 (F) for positive and 1101 (D) for negative. Most implementations also provide unsigned BCD values with a sign nibble of 1111 (F). IBM System/360 processors will use the 1010 (A) and 1011 (B) signs if the A bit is set in the PSW, for the ASCII-8 standard that never passed. Other allowed signs are 1010 (A) and 1110 (E) for positive and 1011 (B) for negative. This convention comes from the zone field for EBCDIC characters and the signed overpunch representation. Standard sign values are 1100 (hex C) for positive (+) and 1101 (D) for negative (โ). ![]()
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