If (L) represents a number L in base-M number system, then identify the correct descending order of the following numbers A-D when converted to decimal number system.

(A) (10110.11)₂

(B) (110.2)₄

(C) (32.5)₈

(D) (14.6)₁₆

Choose the correct answer from the options given below:

The correct answer is - (C), (A), (B), (D)

Key Points

• Conversion to decimal number system
  • (A) (10110.11)₂
    • Convert binary to decimal: $(10110.11)_2 = 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 + 1 \times 2^{-1} + 1 \times 2^{-2}$
    • Decimal equivalent: $16 + 4 + 2 + 0.5 + 0.25 = 22.75$
  • (B) (110.2)₄
    • Convert quaternary to decimal: $(110.2)_4 = 1 \times 4^2 + 1 \times 4^1 + 0 \times 4^0 + 2 \times 4^{-1}$
    • Decimal equivalent: $16 + 4 + 0 + 0.5 = 20.5$
  • (C) (32.5)₈
    • Convert octal to decimal: $(32.5)_8 = 3 \times 8^1 + 2 \times 8^0 + 5 \times 8^{-1}$
    • Decimal equivalent: $24 + 2 + 0.625 = 26.625$
  • (D) (14.6)₁₆
    • Convert hexadecimal to decimal: $(14.6)_{16} = 1 \times 16^1 + 4 \times 16^0 + 6 \times 16^{-1}$
    • Decimal equivalent: $16 + 4 + 0.375 = 20.375$

Additional Information

• Base-N Number Systems
  • Each digit in a number represents a coefficient of a power of the base.
  • The base determines the number of different digits, including zero, that a positional numeral system uses to represent numbers.
  • Common bases include:
    • Binary (Base-2): Uses digits 0 and 1.
    • Quaternary (Base-4): Uses digits 0, 1, 2, and 3.
    • Octal (Base-8): Uses digits 0 through 7.
    • Hexadecimal (Base-16): Uses digits 0 through 9 and letters A through F.
• Conversion Methods
  • To convert from any base to decimal, multiply each digit by its base raised to the power of its position, and sum the results.
  • Fractional parts involve negative powers of the base.