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The Algebraic Proof for Converting Repeating Decimals to Fractions
Converting a repeating decimal (like \(0.333...\)) to a fraction requires an algebraic proof that isolates the repeating decimal period. Let \(x\) equal the repeating decimal: $$x = 0.333...$$. Since the repeating pattern is 1 digit long, we multiply both sides of the equation by $$10^1 = 10$$:
$$10x = 3.333...$$
Next, we subtract the original equation from the multiplied equation: $$10x - x = 3.333... - 0.333... \implies 9x = 3$$. Solving for \(x\) gives: $$x = 3/9$$, which simplifies to \(1/3\). This algebraic method works for any repeating pattern length by multiplying by \(10^n\), where \(n\) is the number of repeating digits.