Hello there! I'm an expert in mathematical concepts, particularly when it comes to converting repeating decimals to fractions. It's always fascinating to delve into the intricacies of numbers and how they can be represented in different forms. Let's embark on this journey to understand how the repeating decimal 0.818181... can be expressed as a fraction.

Repeating decimals are a sequence of digits in a decimal fraction that repeat infinitely. When we encounter a repeating decimal like 0.818181..., it's a shorthand way of saying that the digits "81" repeat indefinitely after the decimal point. The challenge lies in converting this repeating pattern into a fraction, which is a more precise and mathematically elegant representation.

To begin with, let's consider the nature of the repeating decimal we're dealing with. We have a two-digit repeat, which is relatively straightforward compared to longer repeating sequences. The key to converting a repeating decimal to a fraction is to set up an equation that allows us to solve for the repeating part.

Here's a step-by-step method to convert 0.818181... into a fraction:


1. Identify the Repeating Part: We start by recognizing that the repeating part of the decimal is "81".


2. Set Up the Equation: Let \( x \) be the repeating decimal we want to convert. So, we can write:
\[ x = 0.818181... \]


3. Multiply by a Power of 10: To shift the repeating part to the left of the decimal point, we multiply \( x \) by a power of 10 that matches the number of digits in the repeating sequence. Since "81" has two digits, we multiply by \( 10^2 \) (or 100):
\[ 100x = 81.818181... \]


4. Subtract to Isolate the Repeating Part: Now, we subtract the original \( x \) from \( 100x \) to get rid of the repeating part after the decimal point:
\[ 100x - x = 81.818181... - 0.818181... \]
\[ 99x = 81 \]


5. Solve for \( x \): Now we have a simple equation to solve for \( x \):
\[ x = \frac{81}{99} \]


6. Simplify the Fraction: The fraction \( \frac{81}{99} \) can be simplified by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 81 and 99 is 9, so we divide both by 9:
\[ x = \frac{81 \div 9}{99 \div 9} = \frac{9}{11} \]

So, we've found that the repeating decimal 0.818181... is equivalent to the fraction \( \frac{9}{11} \). This fraction is in its simplest form, as 9 and 11 are coprime (they have no common divisors other than 1).

This method is a powerful tool in mathematics for converting repeating decimals to fractions, and it can be applied to repeating decimals of any length. It's a testament to the beauty and precision of mathematical representation, allowing us to express complex patterns in a simple and elegant form.

Now, let's move on to the next step.

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That means we've found that 99 of something is equal to 81 in this problem. So this something, which is actually our repeating decimal 0.818181--, must be equal to the fraction 81/99. As it turns out, you can divide both the top and bottom of this fraction by 9, which means that 0.818181-- = 81/99 = 9/11.Feb 18, 2011read more >>