Binary Multiplication of 111₂ by 101₂

This explanation details the process of multiplying two binary numbers, specifically 111₂ by 101₂. Binary multiplication follows similar principles to decimal multiplication, involving multiplying the multiplicand by each digit of the multiplier and then summing the partial products.

Understanding Binary Multiplication

In binary, multiplication involves only 0s and 1s. The rules are simple:

• $1 \times 1 = 1$
• $1 \times 0 = 0$
• $0 \times 1 = 0$
• $0 \times 0 = 0$

When multiplying multi-digit binary numbers, we perform multiplication digit by digit, shifting the partial products to the left according to their place value, similar to decimal multiplication.

Step-by-Step Calculation

Let's multiply 111₂ by 101₂:

Step 1: Set up the multiplication

Align the numbers vertically:

Step 2: Multiply by the rightmost digit (1₂) of the multiplier (101₂)

Multiply 111₂ by 1:

Step 3: Multiply by the next digit (0₂) of the multiplier (101₂), shifted left

Multiply 111₂ by 0. Add a 0 as a placeholder for the shift:

Step 4: Multiply by the next digit (1₂) of the multiplier (101₂), shifted left twice

Multiply 111₂ by 1. Add two 0s as placeholders for the shifts:

Step 5: Add the partial products

Sum the results obtained in the previous steps:

To perform the addition column by column from right to left:

• Rightmost column: $1 + 0 + 0 = 1$
• Next column: $1 + 0 + 0 = 1$
• Next column: $1 + 0 + 1 = 10_2$. Write down 0, carry over 1.
• Next column: $1 (\text{carry}) + 0 + 1 = 10_2$. Write down 0, carry over 1.
• Leftmost column: $1 (\text{carry}) + 1 = 10_2$. Write down 10.

The final result is 100011₂.

Verification

To verify, we can convert the binary numbers to decimal:

• $111_2 = (1 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 4 + 2 + 1 = 7_{10}$
• $101_2 = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 4 + 0 + 1 = 5_{10}$
• The product in decimal is $7_{10} \times 5_{10} = 35_{10}$.

Now, convert the result 100011₂ back to decimal:

$100011_2 = (1 \times 2^5) + (0 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0)$

$= 32 + 0 + 0 + 0 + 2 + 1 = 35_{10}$

Since the decimal calculation matches the decimal conversion of the binary result, the multiplication is correct.