Understanding Binary to Octal Conversion

This question requires converting a number from binary (base-2) format to octal (base-8) format. The given binary number is 11010001₂.

How to Convert Binary to Octal

The process involves grouping the binary digits (bits) into sets of three, starting from the rightmost digit (the least significant bit). Each group of three binary digits corresponds to one octal digit.

Step-by-Step Conversion Process

1. Write the binary number:

   The binary number is 11010001₂.

2. Group the bits:

   Group the binary number into sets of three bits, starting from the right. Add leading zeros if the leftmost group does not have three bits.

   11010001₂

   Grouping from the right:

   • 001 (Rightmost group)
   • 010 (Middle group)
   • 11 (Leftmost group)

   Since the leftmost group (11) has only two bits, we add a leading zero to make it a group of three: 011.

   The grouped binary number is: 011 010 001

3. Convert each group to octal:

   Now, convert each 3-bit binary group into its equivalent octal digit:

   • 011₂ = $(0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 0 + 2 + 1 = 3$₈
   • 010₂ = $(0 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 0 + 2 + 0 = 2$₈
   • 001₂ = $(0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 0 + 0 + 1 = 1$₈
4. Combine the octal digits:

   Combine the resulting octal digits from left to right to get the final octal number.

   The octal equivalent is 321₈.

Comparing with Options

The calculated octal number is 321₈. Let's compare this with the given options:

• Option 1: (321)₈
• Option 2: (123)₈
• Option 3: (641)₈
• Option 4: (146)₈

The calculated result matches Option 1.