Convert 1101.01102 to decimal.
The correct answer and explanation is :
The given binary number is 1101.01102. Let’s convert it to decimal step by step.
Step 1: Convert the integer part (1101) to decimal
Each digit in a binary number represents a power of 2, starting from the right (2⁰) and increasing to the left.
[
1101_2 = (1 × 2^3) + (1 × 2^2) + (0 × 2^1) + (1 × 2^0)
]
[
= (1 × 8) + (1 × 4) + (0 × 2) + (1 × 1)
]
[
= 8 + 4 + 0 + 1 = 13_{10}
]
Step 2: Convert the fractional part (0.0110) to decimal
The digits after the binary point represent negative powers of 2:
[
0.0110_2 = (0 × 2^{-1}) + (1 × 2^{-2}) + (1 × 2^{-3}) + (0 × 2^{-4})
]
[
= (0 × 0.5) + (1 × 0.25) + (1 × 0.125) + (0 × 0.0625)
]
[
= 0 + 0.25 + 0.125 + 0
]
[
= 0.375_{10}
]
Step 3: Add both parts
[
13 + 0.375 = 13.375_{10}
]
Thus, 1101.0110₂ = 13.375₁₀.
Explanation (300 Words)
Binary numbers use base 2, meaning each digit represents powers of 2. To convert a binary number to decimal, we break it into two parts: the integer part (left of the binary point) and the fractional part (right of the binary point).
For the integer part, we multiply each digit by increasing powers of 2, from right to left. The leftmost digit (most significant bit) has the highest power. In 1101₂, we multiply each digit by 2³, 2², 2¹, and 2⁰. This results in 13₁₀.
For the fractional part, each digit is multiplied by decreasing negative powers of 2. The first digit after the binary point is multiplied by 2⁻¹ (0.5), the second by 2⁻² (0.25), and so on. The sum of these values gives 0.375₁₀.
Finally, adding both parts results in 13.375₁₀.
This method works universally for any binary number, ensuring accurate conversion.