✖️ Signed Multiplication

Signed multiplication in 2’s complement systems behaves like normal multiplication—with a few critical architectural quirks. These quirks arise from modular arithmetic, bit-width constraints, and the encoding of negative values. This notebook documents the four foundational principles that govern signed multiplication.

🔢 Multiplication with Quirks

• We multiply bitwise as usual: partial products, shifts, and accumulation
• But signed systems introduce:
  • Sign correction logic
  • Bit-width overflow planning
  • Negative weight propagation
• These quirks are not exceptions—they’re architectural features of 2’s complement arithmetic

📏 Maximum Bit-Length Is $m + n$

Derivation

• Max value of $m$-bit number: $r^m - 1$
• Max value of $n$-bit number: $r^n - 1$
• Max product: $(r^m - 1)(r^n - 1) = r^{m+n} - r^m - r^n + 1 < r^{m+n} \quad (\forall \space m,n \ge 0)$
• Numbers less than $r^{m+n}$ has $m+n$ bits

Thus, we always allocate $m + n$ bits to avoid overflow

This holds in any base, not just binary

🧮 Sign Extension Preserves Value via Modular Arithmetic

Formal Insight

• $x$ in $n$ bits → interpreted modulo $2^n$
• Extend to $m$ bits ($m > n$): replicate MSB
• Value remains congruent modulo $2^m$

Example

• $1101_4$ = $-3$
• Sign-extended: $11111101_8$ = still $-3$

⚙️ Signed Bit Triggers 2’s Complement Correction

Correction Strategy

• Use 2’s complement of multiplicand
• Shift appropriately
• Add/subtract based on sign logic (e.g., Booth’s transitions)

Semantic Insight

• 2’s complement lets us encode subtraction as addition
• This enables clean accumulation of negative partial products

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