Binary to Octal Conversion Explained

The question asks to convert the binary number 10101111001.0111$_{2}$ to its octal equivalent. Converting a binary number to an octal number involves grouping binary digits into sets of three, starting from the decimal point.

Converting the Integer Part of the Binary Number

For the integer part of the binary number (10101111001$_{2}$), we group the digits from right to left in sets of three. If the leftmost group has fewer than three digits, we add leading zeros to complete the group.

• Original integer binary part: 10101111001
• Grouped from right to left: 010 101 111 001
• (Note: A leading zero was added to the first group '10' to make it '010' for a complete set of three digits).

Now, we convert each group of three binary digits into its corresponding octal digit. The conversion table for binary to octal is provided below:

Applying this conversion to our grouped binary digits:

• $$\text{010}_{2} = 2_{8}$$
• $$\text{101}_{2} = 5_{8}$$
• $$\text{111}_{2} = 7_{8}$$
• $$\text{001}_{2} = 1_{8}$$

Combining these octal digits, the integer part of the binary number 10101111001$_{2}$ is 2571$_{8}$.

Converting the Fractional Part of the Binary Number

For the fractional part of the binary number (.0111$_{2}$), we group the digits from left to right in sets of three. If the rightmost group has fewer than three digits, we add trailing zeros to complete the group.

• Original fractional binary part: .0111
• Grouped from left to right: .011 100
• (Note: Two trailing zeros were added to '1' to make it '100' for a complete set of three digits).

Now, we convert each group of three binary digits into its corresponding octal digit using the same conversion table as above:

• $$\text{011}_{2} = 3_{8}$$
• $$\text{100}_{2} = 4_{8}$$

Combining these octal digits, the fractional part of the binary number .0111$_{2}$ is .34$_{8}$.

Combining the Converted Parts

Finally, we combine the converted integer part and the converted fractional part to get the complete octal number.

• Integer part: 2571$_{8}$
• Fractional part: .34$_{8}$

Therefore, the binary number 10101111001.0111$_{2}$ is equal to 2571.34$_{8}$ in octal.