Advanced Fraction Engine

Mathematical Principles & Theorems

Rational numbers \(\mathbb{Q} = \{a/b : a, b \in \mathbb{Z}, b \ne 0\}\). Reduced to canonical lowest terms via the Euclidean algorithm \(\gcd(a,b)\). Continued fractions represent reals as \(x = a_0 + \frac{1}{a_1 + \frac{1}{a_2 + \dots}} = [a_0; a_1, a_2, \dots]\), providing optimal rational approximations (best convergents \(p_k/q_k\)). Egyptian fractions express rationals as sums of distinct unit fractions \(\frac{a}{b} = \sum \frac{1}{d_i}\) via greedy Fibonacci-Sylvester expansion.

Operating Instructions

• Enter numerators and denominators for operands \(A\) and \(B\).
• Select rational operation: Addition (+), Subtraction (−), Multiplication (×), or Division (÷).
• Click Calculate to view canonical reduced fractions and step-by-step arithmetic factoring.
• Explore the Continued Fractions and Egyptian Unit Fraction decomposition sub-panels.