1.3 Binary Arithmetic and Codes

Binary arithmetic operations form the foundation of digital computing. These operations, including addition, subtraction, multiplication, and division, enable computers to perform complex calculations using only 0s and 1s. Understanding these operations is crucial for designing efficient digital systems.

Overflow detection, binary codes, and error handling are essential concepts in digital design. These techniques ensure accurate data representation and processing, preventing errors that could lead to system failures. Mastering these concepts allows engineers to create robust and reliable digital systems.

Binary Arithmetic Operations

Binary arithmetic operations

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• Binary addition carries over 1 when sum exceeds 1, uses truth table for single-bit operations (0+0=0, 0+1=1, 1+1=10), extends to multi-bit addition (1101 + 1011 = 11000)

• Binary subtraction borrows 1 from next significant bit when necessary, often uses two's complement method for negative numbers, applies to single and multi-bit operations (1100 - 1010 = 0010)

• Binary multiplication shifts and adds partial products, similar to decimal multiplication but with only 0s and 1s (1101 x 1011 = 10001111)

• Binary division uses repeated subtraction or restoring/non-restoring algorithms, divides bits sequentially from most to least significant (1100 ÷ 11 = 100 remainder 0)

Overflow detection in binary

• Overflow occurs when result exceeds available bit width, often changing sign bit unexpectedly

• Detect overflow in addition when carry-in and carry-out of most significant bit differ, in subtraction when result sign is incorrect

• XOR gate can implement hardware overflow detection

• Overflow leads to incorrect results, potential system errors (calculator displaying wrong answer)

Purpose of binary codes

• Binary-Coded Decimal (BCD) represents decimal digits with 4-bit binary, useful in financial systems and calculators

• Gray code changes only one bit between adjacent values, applied in rotary encoders and error correction in digital communications

• Excess-3 code adds 3 to decimal before converting to binary, simplifies arithmetic operations

• ASCII uses 7 or 8 bits to represent characters, standardizes text encoding across systems

Binary code conversions

• Convert binary to BCD using Double Dabble algorithm, grouping binary digits into sets of four (10110 to BCD: 010110)

• BCD to binary conversion multiplies each BCD digit by appropriate power of 10, then sums results (BCD 0001 0110 to binary: 10110)

• Binary to Gray code applies XOR operation between adjacent bits, keeping MSB unchanged (1011 to Gray: 1110)

• Gray code to binary uses XOR operation with running XOR of previous bits (Gray 1110 to binary: 1011)

Error handling in binary data

• Common errors include carry/borrow propagation and overflow errors in arithmetic operations

• Error detection uses parity bits, checksums, or Cyclic Redundancy Check (CRC) to identify data corruption

• Error correction methods like Hamming code or Reed-Solomon code can recover original data from certain types of errors

• BCD errors detected by invalid combinations (>9), Gray code errors identified by multiple bit changes between adjacent values