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Computer Arithmetic: Multiplication and Division

Arithmetic operations are at the core of all computing. While addition and subtraction are handled by simple adders, multiplication and division require more complex algorithms to handle sign bits and efficiency.

1. Binary Multiplication

The simplest form of multiplication is the 'Shift and Add' method. However, for signed numbers (2's complement), we use more sophisticated methods like Booth's Algorithm.

Booth's Algorithm

Booth's algorithm is a multiplication algorithm that multiplies two signed binary numbers in 2's complement notation. It speeds up the process by skipping sequences of 1s or 0s.

  • **Look at current and previous bit ($Q_n, Q_{n+1}$):**
  • **01:** Add Multiplicand to partial product.
  • **10:** Subtract Multiplicand from partial product.
  • **00 or 11:** Do nothing, just shift.
  • **Arithmetic Right Shift:** Performed after every step.

2. Binary Division

Division is generally slower than multiplication. The two primary hardware algorithms are Restoring and Non-Restoring division.

  • **Restoring Division:** If the result of a subtraction is negative, the partial remainder is 'restored' by adding the divisor back.
  • **Non-Restoring Division:** Avoids the restoration step by performing an addition in the next step if the previous result was negative, making it slightly faster.

Comparison Table: Arithmetic Algorithms

OperationAlgorithmKey Characteristic
MultiplicationShift and AddSimplest, for unsigned numbers
MultiplicationBooth's AlgorithmEfficient for signed 2's complement
DivisionRestoringAdds back divisor if remainder is negative
DivisionNon-RestoringMore efficient, no explicit restoration

Hardware Implementation

Most modern CPUs contain a dedicated **Arithmetic Logic Unit (ALU)** with high-speed multipliers (like Wallace Tree multipliers) to perform these operations in just a few clock cycles.

TEXT
Registers used in Multiplication:
A: Accumulator (Partial Product)
Q: Multiplier
M: Multiplicand

Common Mistakes to Avoid

  • Forgetting to use 'Arithmetic' right shift in Booth's (which preserves the sign bit).
  • Assuming division by zero is handled by the algorithm (it must be caught by hardware/OS exceptions).
  • Confusing the Multiplier with the Multiplicand.
  • Ignoring the overflow bit in fixed-point arithmetic.

Advanced Concepts

  • Floating-point Multiplication/Division
  • Wallace Tree Multipliers
  • Carry-Save Adders (CSA)
  • Newton-Raphson Division
  • Array Multipliers

Practice Exercises

  • Multiply (7) x (-3) using Booth's Algorithm step-by-step.
  • What is the advantage of Non-Restoring division over Restoring division?
  • Explain why we need an n+1 bit register for the Accumulator in Booth's algorithm.
  • Perform a binary division of 1101 by 10 using the restoring method.

Conclusion

Efficient arithmetic algorithms are vital for performance in math-heavy applications like 3D graphics and AI. While addition is simple, the hardware's ability to quickly multiply and divide using Booth's and Non-Restoring methods defines the CPU's computational power.

Note: Note: In modern processors, multiplication is often performed in a single clock cycle due to massive parallel hardware arrays, whereas division still takes multiple cycles.