--- title: "Booth multiplication of 23 × -21 (6 bits)" subject: "Computer Organization and Architecture" chapter: 5 key: "semester_5/computer_organization_and_architecture:booth-23x-21" source: https://www.bctengineer.com/q/semester_5/computer_organization_and_architecture/booth-23x-21 site: BCT Engineering --- # Booth multiplication of 23 × -21 (6 bits) Computer Organization and Architecture · Chapter 5: Computer Arithmetic - Asked in: 2068 Chaitra, 8 marks - Video: [Booth's Algorithm | Multiplication ( Binary Arithmetic ) - Part 3](https://www.youtube.com/watch?v=MeD8iWb8wlo) — Semesters Simplified - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_5/computer_organization_and_architecture/booth-23x-21 - Whole chapter: https://www.bctengineer.com/md/semester_5/computer_organization_and_architecture/chapter-5 ## Question Explain Booth algorithm. Use the Booth algorithm to multiply 23 (multiplicand) by -21 (multiplier), where each number is represented using 6 bits. ## Short answer In Booth multiplication, inspect multiplier bit pair Q0,Q-1; subtract M on 10, add M on 01, else do nothing, then arithmetic-shift right. For 23 times -21 in 6 bits, the product A:Q is 111000011101, which equals -483. ## Answer #### Booth algorithm Booth's algorithm multiplies two signed numbers in 2's complement form. It looks at the multiplier bit pair (Q0, Q-1) and treats a run of 1s as (next higher power − lowest power), so a block of 1s needs only one subtraction and one addition. 1. Load M = multiplicand, Q = multiplier, A = 0, Q-1 = 0, SC = n (number of bits). 2. Look at (Q0, Q-1): - 00 or 11: no arithmetic. - 10 (start of a run of 1s): A = A − M. - 01 (end of a run of 1s): A = A + M. 3. Arithmetic shift right the combined A, Q, Q-1 by one bit (the sign bit of A is kept). 4. SC = SC − 1. If SC ≠ 0 go to step 2; otherwise stop. 5. The 2n-bit product is in A:Q. #### 23 × (−21), 6 bits Multiplicand M = 23 = 010111, multiplier Q = −21 = 101011 (2's complement of 21 = 010101), −M = 101001, n = 6 bits. ```text Step Action A Q Q-1 - Initial 000000 101011 0 1 Q0Q-1=10: A = A - M 101001 101011 0 ASR 110100 110101 1 2 Q0Q-1=11: no change 110100 110101 1 ASR 111010 011010 1 3 Q0Q-1=01: A = A + M 010001 011010 1 ASR 001000 101101 0 4 Q0Q-1=10: A = A - M 110001 101101 0 ASR 111000 110110 1 5 Q0Q-1=01: A = A + M 001111 110110 1 ASR 000111 111011 0 6 Q0Q-1=10: A = A - M 110000 111011 0 ASR 111000 011101 1 ``` Result in A,Q = 111000 011101 = -483. **Answer: 23 × -21 = -483** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In Booth's algorithm, what is done when (Q0, Q-1) = 10? A. A = A + M B. No arithmetic, only shift C. A = A - Q D. A = A - M Answer: D. A pair 10 marks the start of a run of 1s in the multiplier, so M is subtracted from A before the shift. (Key point: Booth pair 10) 2. What is -21 in 6-bit 2's complement? A. 101011 B. 110101 C. 010101 D. 101010 Answer: A. 21 = 010101; its 2's complement is 101011. (Key point: Multiplier form) 3. For M = 23 = 010111, -M is A. 101000 B. 101001 C. 010111 D. 110101 Answer: B. Invert 010111 to 101000 and add 1: 101001. (Key point: -M value) 4. What is 23 x -21 and the product width? A. 483, in 12 bits B. -483, in 6 bits C. -483, in 12 bits (A:Q) D. -441, in 12 bits Answer: C. A:Q = 111000 011101 = -483; 23 x 21 = 483. (Key point: Result) ## Related in Computer Organization and Architecture - [Booth multiplication algorithm and hardware](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Abooth-algorithm) — Computer Organization and Architecture, chapter 5, asked 8× - [Booth multiplication of -5 × -9](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Abooth-5x-9) — Computer Organization and Architecture, chapter 5, asked 5× - [Booth multiplication of 5 × -6](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Abooth-5x-6) — Computer Organization and Architecture, chapter 5, asked 5× - [Booth multiplication of -7 × -10](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Abooth-7x-10) — Computer Organization and Architecture, chapter 5, asked 5× - [Booth multiplication of 10 × (-5)](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Abooth-10x-5) — Computer Organization and Architecture, chapter 5, asked 5× - [Booth multiplication of -6 × -11](https://www.bctengineer.com/question.md?key=semester_5%2Fcomputer_organization_and_architecture%3Abooth-6x-11) — Computer Organization and Architecture, chapter 5, asked 5× ## Related in other subjects - [8085 ALP: multiplication table of a number](https://www.bctengineer.com/question.md?key=semester_4%2Fmicroprocessor%3Aalp-8085-multiplication-table) — Microprocessor, chapter 2 - [8086 ALP: multiplication table of a number](https://www.bctengineer.com/question.md?key=semester_4%2Fmicroprocessor%3Aalp-8086-multiplication-table) — Microprocessor, chapter 3 - [Complex number multiplication using structure](https://www.bctengineer.com/question.md?key=semester_1%2Fcomputer_programming%3Astruct-complex-multiply) — Computer Programming, chapter 8 - [Random number properties; KS test algorithm](https://www.bctengineer.com/question.md?key=semester_8%2Fsimulation_and_modeling%3Arn-properties-ks-algorithm) — Simulation and Modeling, chapter 6