How to use the Booth's Multiplication Calculator
- 1. Enter M and QThe multiplicand and multiplier, positive or negative.
- 2. Pick the bitsEnough bits to hold both numbers in two's complement.
- 3. MultiplyGet the table and the 2n-bit product.
How Booth's algorithm works
Booth's algorithm multiplies signed two's-complement numbers using only addition, subtraction and shifts. It uses three registers: A (starts at 0), Q (the multiplier) and a single extra bit Q₋₁ (starts at 0). M holds the multiplicand. Then, n times:
- Look at the last bit of Q and Q₋₁.
- 10: A = A − M (add the two's complement of M).
- 01: A = A + M.
- 00 or 11: do nothing.
- Then do an arithmetic shift right of A, Q and Q₋₁ together, copying A's sign bit.
After n steps, A and Q together hold the 2n-bit product. Runs of 1s in the multiplier need just one subtraction and one addition, which is why the method was faster in early hardware.
Worked example
Multiply M = 7 by Q = −3 with 4 bits. M = 0111, −M = 1001 and Q = 1101.
- Q₀Q₋₁ = 10: A = 0000 − M = 1001, then shift: A 1100, Q 1110, Q₋₁ 1.
- 01: A = 1100 + 0111 = 0011, then shift: A 0001, Q 1111, Q₋₁ 0.
- 10: A = 0001 − M = 1010, then shift: A 1101, Q 0111, Q₋₁ 1.
- 11: no add, just shift: A 1110, Q 1011, Q₋₁ 1.
The result is A Q = 1110 1011, which is −21 in 8-bit two's complement, and 7 × −3 is indeed −21. The tool checks every answer against ordinary multiplication.
One edge case: the most negative number (−8 in 4 bits) can't be negated in the same width, so the tool asks for one more bit when M is that value.
Frequently asked questions
How many bits should I choose?
Enough that both numbers fit in two's complement: 4 bits covers −8 to 7, 8 bits covers −128 to 127.
Does it work for two positive numbers?
Yes. Booth's algorithm works for any signed values.
Why arithmetic and not logical shift?
The sign bit must be kept, so A's leftmost bit is copied as it shifts right.
Is the booth's multiplication calculator free to use?
Yes, the booth's multiplication calculator is free and costs nothing. Your first multiplication shows a short form for your name and WhatsApp number one time; once that's done, this tool and every other Uncodemy tool open with no form.