The binary number 0.101101101101101101…. is equal to the decimal number (without any approximation)
The correct answer and explanation is:
Let’s analyze the binary number:
0.101101101101101101… (repeating pattern “101” continuously)
Step 1: Identify the repeating pattern
The binary number is a repeating fractional binary with the repeating sequence:
101
This pattern repeats infinitely after the decimal point.
Step 2: Express as an infinite geometric series
The binary number can be written as: 0.101101101101…=0.101+0.000101+0.000000101+…0.101101101101… = 0.101 + 0.000101 + 0.000000101 + \dots
Each “101” block is shifted 3 bits further to the right each time.
Step 3: Convert one block “101” to decimal fraction
The first block “101” is the bits after the decimal point at positions 2−12^{-1}, 2−22^{-2}, and 2−32^{-3}:
- 1×2−1=12=0.51 \times 2^{-1} = \frac{1}{2} = 0.5
- 0×2−2=00 \times 2^{-2} = 0
- 1×2−3=18=0.1251 \times 2^{-3} = \frac{1}{8} = 0.125
Sum of one block “101”: 0.5+0+0.125=0.6250.5 + 0 + 0.125 = 0.625
Step 4: Write the full number as a geometric series
The first term a=0.625a = 0.625
Each subsequent term is shifted right by 3 bits, which means multiplied by 2−3=182^{-3} = \frac{1}{8}.
So the number is: x=0.625+0.625×18+0.625×(18)2+⋯x = 0.625 + 0.625 \times \frac{1}{8} + 0.625 \times \left(\frac{1}{8}\right)^2 + \cdots
Step 5: Sum the infinite geometric series
The sum of an infinite geometric series is: S=a1−rS = \frac{a}{1 – r}
where
- a=0.625a = 0.625
- r=18r = \frac{1}{8}
So x=0.6251−18=0.62578=0.625×87=58×87=57x = \frac{0.625}{1 – \frac{1}{8}} = \frac{0.625}{\frac{7}{8}} = 0.625 \times \frac{8}{7} = \frac{5}{8} \times \frac{8}{7} = \frac{5}{7}
Final answer:
0.101101101101…2=57(in decimal)\boxed{ 0.101101101101…_2 = \frac{5}{7} \quad \text{(in decimal)} }
Explanation (300 words):
The given binary number is a repeating binary fraction with the pattern “101” repeating indefinitely after the decimal point. To convert such a repeating binary fraction into a decimal fraction, we first express the number as an infinite geometric series.
Each group of three bits “101” corresponds to a decimal value when interpreted in binary fractional form. Specifically, the “1” in the first bit after the decimal point represents 2−1=0.52^{-1} = 0.5, the “0” in the next bit represents 0, and the last “1” represents 2−3=0.1252^{-3} = 0.125. Adding these gives 0.5+0+0.125=0.6250.5 + 0 + 0.125 = 0.625.
Because this pattern repeats every three bits, each subsequent “101” is shifted three places to the right, which in decimal terms means dividing by 23=82^3 = 8. This gives the ratio r=18r = \frac{1}{8} for the geometric series.
The number xx is thus the sum of the infinite series: x=0.625+0.625×18+0.625×(18)2+…x = 0.625 + 0.625 \times \frac{1}{8} + 0.625 \times \left(\frac{1}{8}\right)^2 + \dots
This is a geometric series with first term a=0.625a = 0.625 and common ratio r=18r = \frac{1}{8}. Using the sum formula for infinite geometric series: S=a1−rS = \frac{a}{1-r}
Plugging in values: x=0.6251−18=0.62578=0.625×87=57x = \frac{0.625}{1 – \frac{1}{8}} = \frac{0.625}{\frac{7}{8}} = 0.625 \times \frac{8}{7} = \frac{5}{7}
Hence, the repeating binary fraction 0.101101101…0.101101101… is exactly equal to the rational decimal number 57\frac{5}{7} without any approximation. This exact fraction shows how repeating patterns in binary fractions correspond to rational numbers in decimal form.