Fractions represent one of the oldest and most fundamental concepts in mathematics, emerging from the practical need to express parts of a whole. The ancient Egyptians used a distinctive system of unit fractions (fractions with numerator 1) as early as 1800 BCE, as documented in the Rhind Mathematical Papyrus. They expressed quantities like two-thirds as the sum of one-half plus one-sixth, developing elaborate tables of decompositions. The Babylonians used their base-60 system to express fractions, a tradition that survives in our measurement of angles and time. Modern fraction notation with a numerator above a denominator separated by a horizontal bar was popularized by Arabic mathematicians and introduced to Europe through Fibonacci's "Liber Abaci" in 1202.

In formal mathematics, fractions are elements of the rational numbers, denoted by Q (from the German "Quotient"). A rational number is any number that can be expressed as the ratio p/q where p and q are integers and q is not zero. The rational numbers form a field, meaning they are closed under addition, subtraction, multiplication, and division (except by zero), and satisfy the commutative, associative, and distributive properties. An important property of rational numbers is that their decimal expansions either terminate (like 1/4 = 0.25) or eventually repeat (like 1/3 = 0.333... or 1/7 = 0.142857142857...). Conversely, any terminating or repeating decimal represents a rational number.

Continued fractions offer an alternative representation that reveals deep number-theoretic properties. A continued fraction expresses a number as a whole number plus one divided by another whole number plus one divided by another, and so on. For rational numbers, continued fractions terminate after finitely many steps, while for irrational numbers like the square root of 2 or the golden ratio, they continue infinitely but often with beautiful patterns. The golden ratio has the simplest possible infinite continued fraction, consisting entirely of ones, which is why it is sometimes called the most irrational number. Continued fractions provide the best rational approximations to real numbers and appear in algorithms for solving Diophantine equations.

At the heart of fraction simplification lies Euclid's algorithm for finding the greatest common divisor (GCD), one of the oldest algorithms in mathematics, described in Euclid's "Elements" around 300 BCE. The algorithm works by repeatedly replacing the larger number with the remainder when dividing the larger by the smaller, until the remainder is zero. The last non-zero remainder is the GCD. To simplify a fraction, divide both numerator and denominator by their GCD. For finding common denominators when adding fractions, the least common multiple (LCM) of the denominators is used, which can be computed from the GCD using the relationship LCM(a,b) equals the absolute value of a times b divided by GCD(a,b). This elegant connection between GCD and LCM underlies all fraction arithmetic and demonstrates how a simple algorithm from ancient Greece remains essential in modern computation.