Math Problem Statement

Convert i.010101 to fraction

Solution

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

  1. Let x=0.010101x = 0.\overline{010101}. This means x=0.010101010101x = 0.010101010101\ldots.

  2. To eliminate the repeating part, multiply xx by 1000000 (since the repeating block is 6 digits long):

1000000x=10101.0101011000000x = 10101.\overline{010101}

  1. Now, subtract the original xx from this equation:

1000000xx=10101.0101010.0101011000000x - x = 10101.\overline{010101} - 0.\overline{010101}

999999x=10101999999x = 10101

  1. Solving for xx:

x=10101999999x = \frac{10101}{999999}

Therefore,

i.010101=i+10101999999i.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:

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

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

10101÷3999999÷3=3367333333\frac{10101 \div 3}{999999 \div 3} = \frac{3367}{333333}

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

i+3367333333i + \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.

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Math Problem Analysis

Mathematical Concepts

Repeating Decimals
Fractions
Simplification
Greatest Common Divisor (GCD)

Formulas

x = 0.010101... → 1000000x = 10101.010101...
1000000x - x = 10101 → 999999x = 10101 → x = 10101 / 999999
Simplify by dividing numerator and denominator by GCD(10101, 999999)

Theorems

Euclidean Algorithm

Suitable Grade Level

Grades 9-12