Understanding Binary to Octal Conversion

Converting a binary number to its octal equivalent is a fundamental concept in digital systems and computer science. Both binary (base-2) and octal (base-8) are positional number systems.

The key to converting binary to octal is recognizing that \(8 = 2^3\). This means that every group of three binary digits corresponds directly to one octal digit.

Steps for Binary to Octal Conversion

To convert a binary number to octal, follow these steps:

• Start from the rightmost digit of the binary number.
• Group the binary digits into sets of three.
• If the leftmost group has fewer than three digits, pad it with leading zeros to make it a complete group of three.
• Convert each group of three binary digits into its corresponding single octal digit.
• Combine the resulting octal digits to get the final octal number.

Converting 1011101011 to Octal

Let's apply these steps to the given binary number: \(1011101011\)

Step 1: Group the binary digits into threes from the right.

We start from the right:

\(1011101011\)

Grouping:

\(1 \ 011 \ 101 \ 011\)

Step 2: Pad the leftmost group if necessary.

The leftmost group is just \(1\). It has only one digit. We need to pad it with two leading zeros to make it a group of three:

\(001 \ 011 \ 101 \ 011\)

Step 3: Convert each group of three binary digits to its octal equivalent.

We convert each group:

• Group 1 (leftmost): \(001_2\)
• Group 2: \(011_2\)
• Group 3: \(101_2\)
• Group 4 (rightmost): \(011_2\)

Let's convert each group to octal:

• \(001_2 = (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 0 + 0 + 1 = 1_{8}\)
• \(011_2 = (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 0 + 2 + 1 = 3_{8}\)
• \(101_2 = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 4 + 0 + 1 = 5_{8}\)
• \(011_2 = (0 \times 2^2) + (1 \times 2^1) + (1 \times 2^0) = 0 + 2 + 1 = 3_{8}\)

Here is a quick reference table for converting 3-digit binary to octal:

Using the table, we can see:

• \(001_2 = 1_8\)
• \(011_2 = 3_8\)
• \(101_2 = 5_8\)
• \(011_2 = 3_8\)

Step 4: Combine the octal digits.

Putting the octal digits together from left to right, we get:

\(1353_8\)

Therefore, the octal equivalent of the binary number \(1011101011_2\) is \(1353_8\).

Revision Table: Binary to Octal Conversion

Additional Information: Number Systems

Number systems are ways of representing numbers. Common ones include:

• Decimal (Base-10): Uses digits 0-9. Each position value is a power of 10.
• Binary (Base-2): Uses digits 0 and 1. Each position value is a power of 2. Used extensively in computers.
• Octal (Base-8): Uses digits 0-7. Each position value is a power of 8. Often used as a shorthand for binary because 3 binary digits map to 1 octal digit.
• Hexadecimal (Base-16): Uses digits 0-9 and letters A-F (A=10, B=11, ..., F=15). Each position value is a power of 16. Used as a shorthand for binary because 4 binary digits map to 1 hexadecimal digit (\(16 = 2^4\)).

Conversions between binary, octal, and hexadecimal are straightforward because their bases are powers of 2.