Understanding Number Base Conversions: Binary and Octal

This question asks us to identify which pair of numbers, one in binary (base 2) and the other in octal (base 8), are not equal. To do this, we need to understand how to convert between binary and octal number systems. The key relationship is that \(8 = 2^3\), which means each digit in an octal number corresponds to exactly three digits in a binary number.

How to Convert Binary to Octal

To convert a binary number to an octal number:

1. For the integer part, group the binary digits into sets of three, starting from the rightmost digit and moving left. If the leftmost group has fewer than three digits, add leading zeros to complete the group of three.
2. For the fractional part (digits after the decimal point), group the binary digits into sets of three, starting from the leftmost digit and moving right. If the rightmost group has fewer than three digits, add trailing zeros to complete the group of three.
3. Convert each group of three binary digits into its equivalent single octal digit.

How to Convert Octal to Binary

To convert an octal number to a binary number:

1. Convert each octal digit into its equivalent three-digit binary representation.
2. Combine the resulting binary groups to form the complete binary number.

Analyzing Each Pair

Let's examine each given pair by converting the binary number to octal or vice versa.

Option 1: \( (111110111)_2 \) and \( (767)_8 \)

Convert the binary number \( (111110111)_2 \) to octal:

• Group the binary digits in threes from right to left: \( (111)(110)(111) \).
• Convert each group to octal:
• \( 111_2 = 7_8 \)
• \( 110_2 = 6_8 \)
• \( 111_2 = 7_8 \)
• Combining these gives \( (767)_8 \).

So, \( (111110111)_2 = (767)_8 \). This pair is equal.

Option 2: \( (110110101)_2 \) and \( (665)_8 \)

Convert the binary number \( (110110101)_2 \) to octal:

• Group the binary digits in threes from right to left: \( (110)(110)(101) \).
• Convert each group to octal:
• \( 110_2 = 6_8 \)
• \( 110_2 = 6_8 \)
• \( 101_2 = 5_8 \)
• Combining these gives \( (665)_8 \).

So, \( (110110101)_2 = (665)_8 \). This pair is equal.

Option 3: \( (10101.11)_2 \) and \( (25.6)_8 \)

Convert the binary number \( (10101.11)_2 \) to octal:

• Integer part \( (10101)_2 \): Group from right to left: \( (010)(101) \). (Added a leading zero to the first group).
  • \( 010_2 = 2_8 \)
  • \( 101_2 = 5_8 \)
  • The integer part is \( (25)_8 \).
• Fractional part \( (.11)_2 \): Group from left to right: \( (110) \). (Added a trailing zero to the last group).
  • \( 110_2 = 6_8 \)
  • The fractional part is \( (.6)_8 \).
• Combining these gives \( (25.6)_8 \).

So, \( (10101.11)_2 = (25.6)_8 \). This pair is equal.

Option 4: \( (11010)_2 \) and \( (62)_8 \)

Let's convert the binary number \( (11010)_2 \) to octal:

• Group the binary digits in threes from right to left: \( (011)(010) \). (Added a leading zero to the first group).
• Convert each group to octal:
• \( 011_2 = 3_8 \)
• \( 010_2 = 2_8 \)
• Combining these gives \( (32)_8 \).

So, \( (11010)_2 = (32)_8 \). The given octal number is \( (62)_8 \). Since \( (32)_8 \ne (62)_8 \), this pair is NOT equal.

Alternatively, we can convert \( (62)_8 \) to binary:

• Convert each octal digit to its 3-bit binary equivalent:
• \( 6_8 = 110_2 \)
• \( 2_8 = 010_2 \)
• Combining these gives \( (110010)_2 \).

So, \( (62)_8 = (110010)_2 \). The given binary number is \( (11010)_2 \). Since \( (11010)_2 \ne (110010)_2 \), this pair is NOT equal.

The analysis shows that the pair \( (11010)_2 \) and \( (62)_8 \) are not equal.

Conclusion on Binary and Octal Pair Equality

Based on the conversions, the pair \( (11010)_2 \) and \( (62)_8 \) are not equal. \( (11010)_2 \) is equal to \( (32)_8 \), while \( (62)_8 \) is equal to \( (110010)_2 \).

Revision Table: Binary and Octal Conversions

This table summarizes the conversion of 3-bit binary numbers to single octal digits.

Additional Information: Number Systems

Number systems are fundamental ways of representing quantities. The base or radix of a number system indicates the number of unique digits used, including zero.

• Decimal System (Base 10): Uses digits 0-9. Each position value is a power of 10.
• Binary System (Base 2): Uses digits 0 and 1. Each position value is a power of 2. This system is used extensively in computers.
• Octal System (Base 8): Uses digits 0-7. Each position value is a power of 8. It's often used as a compact way to represent binary numbers because 8 is a power of 2 (\(2^3\)).
• Hexadecimal System (Base 16): Uses digits 0-9 and letters A-F (representing 10-15). Each position value is a power of 16. It's also used as a compact way to represent binary numbers because 16 is a power of 2 (\(2^4\)).

Converting between binary, octal, and hexadecimal is straightforward because their bases are powers of 2. This makes these conversions particularly useful in computer science and digital electronics.