Fraction Calculator

This tool operates across four dedicated mathematical pipelines. Each calculation is run purely client-side within your browser using precise integer mathematics to avoid the rounding errors common in standard floating-point computations.

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1. Arithmetic Operations Matrix

When adding, subtracting, multiplying, or dividing, inputs are automatically converted into standard improper forms ($\frac{\text{numerator}}{\text{denominator}}$):

• Addition & Subtraction: The engine extracts the Least Common Multiple (LCM) of both denominators to establish a Least Common Denominator (LCD). Each fraction is scaled up by the necessary factor before operations are run directly across the adjusted numerators.
• Multiplication: The tool multiplies numerators across the top and denominators across the bottom: $$\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$$
• Division: The tool takes the reciprocal of the divisor fraction and switches the calculation to direct multiplication.

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2. Simplification and Reduction

To simplify fractions to their absolute lowest terms, the engine utilizes the Euclidean Algorithm to calculate the Greatest Common Divisor (GCD) of the numerator and denominator. Both terms are then divided by this number. If the resulting fraction is top-heavy, a modulo division isolates the integer portion to output a clean mixed number.

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3. Comparison Logic

To check if fractions are greater than, less than, or equal to one another without relying on flawed decimal conversions, the engine scales both inputs to their shared LCD. With matching denominators established, a definitive inequality analysis is safely performed directly on the scaled numerators.

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4. Decimal-to-Fraction Convergence

For standard short decimals, the tool counts the active decimal positions ($n$) and converts the string into a baseline fraction over a matching power-of-10 base matrix ($10^n$).

For long or repeating decimal sequences, the engine executes a Continued Fraction Expansion Algorithm. This method calculates successive rational approximations (convergents) until it hits an exact algebraic match within a strict error tolerance limit of $1.0 \times 10^{-9}$, perfectly mapping recurring decimals back to their original fractional bounds.