1. What is Binary Multiplication?

Binary multiplication is the process of multiplying two binary numbers using the same method as decimal multiplication, but with simpler rules since there are only two digits (0 and 1).

2. How Does Shift-and-Add Work?

The shift-and-add algorithm is based on:

Where:

• \( a \) — First binary number
• \( b \) — Second binary number
• \( a_i \) — i-th bit of number a
• \( 2^i \) — Left shift by i positions

Explanation: For each bit in the multiplier that's 1, add a shifted copy of the multiplicand to the result.

3. Binary Multiplication Steps

Example: Multiply 101 (5) by 110 (6)

1. Take the first number (101) and multiply by each bit of the second number (110)
2. For each 1 in the second number, write the first number shifted left by the bit position
3. Add all the partial products together
4. Result: 101 × 110 = 1010 + 10100 = 11110 (30)

4. Using the Calculator

Tips: Enter valid binary strings (containing only 0s and 1s). The calculator will show both the binary result and the decimal equivalent with calculation steps.

5. Frequently Asked Questions (FAQ)

Q1: Why is binary multiplication simpler than decimal?
A: Because you only need to multiply by 0 (result is 0) or 1 (result is the original number).

Q2: What's the maximum length of binary numbers I can multiply?
A: This calculator can handle numbers up to 32 bits, but extremely large numbers may cause display issues.

Q3: How does this relate to computer processors?
A: Processors use optimized versions of shift-and-add algorithms in their ALUs for multiplication.

Q4: What about signed binary numbers?
A: This calculator handles unsigned numbers only. For signed numbers, two's complement conversion is needed first.

Q5: Can I see the intermediate steps?
A: Yes, the calculator shows all the partial products and their decimal equivalents.