To find the product of the two binary numbers 011 and 110, we need to perform binary multiplication. This process is similar to decimal multiplication but uses binary arithmetic rules.

Binary Number Multiplication Explained

Binary numbers are the foundation of digital computing. Understanding how to perform basic arithmetic operations like multiplication with binary numbers is essential. Here, we will illustrate the multiplication of binary 011 by binary 110.

Understanding Binary Numbers

The binary system is a base-2 number system that uses only two digits: 0 and 1. Each digit in a binary number is called a bit, and its position determines its value based on powers of 2.

• The binary number 011 can be converted to its decimal equivalent: \( (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 0 + 2 + 1 = 3 \).
• The binary number 110 can be converted to its decimal equivalent: \( (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 4 + 2 + 0 = 6 \).

Therefore, finding the product of 011 and 110 in binary is equivalent to finding the product of $3$ and $6$ in decimal, which is $18$. We expect our binary result to convert to $18$ decimal.

Performing Binary Multiplication

Binary multiplication involves multiplying the multiplicand (011) by each digit of the multiplier (110). Each partial product is then shifted to the left according to the position of the multiplier's digit, and finally, all partial products are summed using binary addition rules.

Step-by-Step Multiplication Process

Let's set up the multiplication of 011 by 110:

      011   (Multiplicand)
    × 110   (Multiplier)
    -----
      000   (011 × 0, no shift)
     0110   (011 × 1, shifted one position left)
    01100   (011 × 1, shifted two positions left)
    -----
    

Here's a detailed breakdown of how each partial product is obtained:

• First Partial Product: Multiply the multiplicand (011) by the rightmost digit of the multiplier (0). The result is placed in the first row of partial products.

  \( 011 \times 0 = 000 \)

• Second Partial Product: Multiply the multiplicand (011) by the middle digit of the multiplier (1). The result is $011$. This result is then shifted one position to the left.

  \( 011 \times 1 = 011 \implies 0110 \)

• Third Partial Product: Multiply the multiplicand (011) by the leftmost digit of the multiplier (1). The result is $011$. This result is then shifted two positions to the left.

  \( 011 \times 1 = 011 \implies 01100 \)

Summing the Partial Products

Finally, we add these partial products together using binary addition. Remember the binary addition rules:

• \( 0 + 0 = 0 \)
• \( 0 + 1 = 1 \)
• \( 1 + 0 = 1 \)
• \( 1 + 1 = 0 \) (with a carry-over of 1)
• \( 1 + 1 + 1 = 1 \) (with a carry-over of 1)

Adding the partial products:

      000
     0110
    +01100
    -----
    10010
    

Verifying the Product

To ensure the correctness of our binary multiplication, we can convert the resulting binary product (10010) back into its decimal equivalent:

Binary \( 10010_{2} \) to Decimal:

\( (1 \times 2^4) + (0 \times 2^3) + (0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) \)

\( = (1 \times 16) + (0 \times 8) + (0 \times 4) + (1 \times 2) + (0 \times 1) \)

\( = 16 + 0 + 0 + 2 + 0 \)

\( = 18_{10} \)

This decimal value of $18$ matches the result we anticipated from multiplying the decimal equivalents of the original binary numbers ($3 \times 6 = 18$). This confirms that our binary multiplication is accurate.

Final Product of Binary Numbers

The product of the two binary numbers 011 and 110 is 10010.