To convert the repeating decimal \( i.010101\ldots \) to a fraction, we can follow these steps:

1. Let \( x = 0.\overline{010101} \). This means \( x = 0.010101010101\ldots \).
   
2. To eliminate the repeating part, multiply \( x \) by 1000000 (since the repeating block is 6 digits long):

\[ 1000000x = 10101.\overline{010101} \]

3. Now, subtract the original \( x \) from this equation:

\[ 1000000x - x = 10101.\overline{010101} - 0.\overline{010101} \]

\[ 999999x = 10101 \]

4. Solving for \( x \):

\[ x = \frac{10101}{999999} \]

Therefore,

\[ i.010101\ldots = i + \frac{10101}{999999} \]

To simplify this fraction:

1. The greatest common divisor (GCD) of 10101 and 999999 needs to be determined. Using the Euclidean algorithm or a calculator, we find:

\[ \text{GCD}(10101, 999999) = 3 \]

2. Divide both the numerator and the denominator by their GCD:

\[ \frac{10101 \div 3}{999999 \div 3} = \frac{3367}{333333} \]

Thus, the simplified fraction form of \( i.010101\ldots \) is:

\[ i + \frac{3367}{333333} \]

If you have any questions or need further details, feel free to ask!

Here are 5 related questions:
1. How do you convert other repeating decimals to fractions?
2. What is the Euclidean algorithm for finding the greatest common divisor (GCD)?
3. How do you add fractions with different denominators?
4. What are some methods for simplifying fractions?
5. How do repeating decimals differ from terminating decimals in fraction form?

**Tip:** When converting repeating decimals to fractions, always determine the length of the repeating sequence and use it to form the appropriate equation.

This guide explains how to convert the repeating decimal i.010101 to a fraction using algebraic methods and simplification. It covers the application of the Euclidean algorithm to find the greatest common divisor (GCD) and shows step-by-step calculations. Suitable for Grades 9-12.

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