What is 0.1 repeating as a fraction?

Understand the Problem

The question is asking how to express the repeating decimal 0.1 (which can also be represented as 0.111...) as a fraction. The high-level approach involves using algebra to convert the repeating decimal into a fraction.

Answer

The repeating decimal $0.111...$ is expressed as the fraction $\frac{1}{9}$.

Steps to Solve

Let x equal the repeating decimal

First, we will define the repeating decimal. Let

$$ x = 0.111... $$

Multiply by 10 to shift the decimal point

To eliminate the decimal, we multiply both sides of the equation by 10:

$$ 10x = 1.111... $$

Set up a subtraction equation

Now, we can subtract the original equation from this new equation:

$$ 10x - x = 1.111... - 0.111... $$

This simplifies to:

$$ 9x = 1 $$

Solve for x

Now, divide both sides by 9 to isolate $x$:

$$ x = \frac{1}{9} $$

The repeating decimal $0.111...$ can be expressed as the fraction $\frac{1}{9}$.

More Information

The fraction $\frac{1}{9}$ is an important representation in mathematics for the repeating decimal $0.111...$. Interestingly, repeating decimals can often be expressed as fractions, and understanding this conversion plays a crucial role in working with rational numbers.

Tips

A common mistake is to assume that $0.111...$ equals exactly $0.1$. It's important to understand that $0.111...$ is a slightly larger value.
Not properly setting up the subtraction step can lead to confusion. Ensure both versions of the equation are written correctly before subtracting.

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