BCD

• Reference: Kumar
• Wikipedia is ok too
• Binary Coded Decimal
  • Used in early computer systems.
  • IE they did not use binary.
• This is also called 8-4-2-1 or 8421, just naming the bits.
• d₃d₂d₁d₀ = d₃ × 8 + d₂ × 4 + d₁ × 2 + d₀ ×1
• It is a way to represent decimal numbers in an easy to use binary format.
• Wikipedia says it is still used in some financial systems.
  • A round-off error in decimal is something we can "deal" with legally and socially.
  • A round-off error in binary is more complex socially.
• Idea: Use 4 bits to represent each digit, encoded in binary, to encode a decimal number.
  • 0100 1000 0111_BCD = 487
  • 3873 = 0011 1000 0111 0010_BCD
• We only use the digits 0-9 and corresponding binary codes.
  • This means we "waste" 1010 through 1111 or 6 representations per digit.
• Mathematical operations in BCD are possible
  • But initially the sum may be one of the unrepresentable characters.
  • So we add 6 to the result.
  • Example 5+7

     01012
    +01112
    -----
     11002

  • But 1100₂ is not a legal BCD digit, so add 6

     11002
    +01102
     ----
    100102   Which is 2 with a carry

  • Why 6? This skips the 6 unused digits.
• So addition of BCD involves ripply carry with detection of "bad representations"
  • i.e. we add three to each digit.
  • 0 is represented as 0011
  • 1 is represented as 0100
  • ... 9 is 1100
  • 0000, 0001, 0010 and 1101, 1110, 1111 are not used.
  • If a carry occurs, then we just add three to find the result
  • This is called excess-3
• So 5 + 7 becomes

    1000BCD-3
  + 1010BCD-3
  ------
   10010BCD-3?
     

  • so the carry is 1, but the sum (0010) is not a legal value
  • So we add three (0101_BCD-3)

  •  00102
    +00112
    -----
     0101BCD-3
       

• I expect
  • You can tell me what BCD means.
  • You can tell me it was used in old computer systems.
  • You can represent numbers in BCD
  • You can give the decimal equivalent for numbers in BCD