Unit Fraction
A unit fraction is a fraction with a numerator of 1, representing exactly one equal part of a whole.
Formula
\frac{1}{n} \text{ (where } n \text{ is a positive integer)}
Definition
A unit fraction always has a $1$ on top, representing exactly one equal piece of something, like $1/2$, $1/3$, $1/4$, and so on. Formally, it is a rational number of the form $1/n$ where $n$ is a positive integer, and every positive fraction $a/b$ can be expressed as the sum of $a$ unit fractions each equal to $1/b$; ancient Egyptians used unit fractions as the primary way to represent all fractions. The Erdos-Straus conjecture (unproven) states that for every integer $n \ge 2$, the fraction $4/n$ can be written as the sum of three unit fractions, and Egyptian fraction representations, sums of distinct unit fractions, are studied in combinatorial number theory.
Example
$1/4$ means you cut something into $4$ equal pieces and take just one: one quarter of a dollar is $25$ cents, one piece out of four equal $25$-cent pieces. Any fraction can be thought of as a stack of unit fractions, $3/5 = 1/5 + 1/5 + 1/5$, though the ancient Egyptians insisted on distinct unit fractions, writing $2/5 = 1/3 + 1/15$ rather than repeating a fraction. The greedy algorithm (Fibonacci-Sylvester) finds such representations systematically: $5/7 \to$ subtract $1/2$ (since $1/\lceil 7/5 \rceil = 1/2$): $5/7 - 1/2 = 3/14 \to$ subtract $1/5$: $3/14 - 1/5 = 1/70$, so $5/7 = 1/2 + 1/5 + 1/70$.
Key Insight
Unit fractions are the building blocks of all fractions, and thinking of them that way reveals something surprising: the harmonic series $1/1 + 1/2 + 1/3 + 1/4 + \ldots$, the sum of all unit fractions, diverges despite each term getting smaller, meaning it has no finite sum. Every rational number in $(0,1)$ has infinitely many Egyptian fraction representations, and the question of which representations are "shortest" (fewest terms) or "smallest" (smallest largest denominator) remains an active research area in number theory.