Unit Fraction
Fractions & DecimalsA 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 is a fraction that always has a $1$ on top. It represents exactly one equal piece of something. Examples are $1/2$, $1/3$, $1/4$, $1/5$, and so on.
Example
$1/4$ means you cut something into $4$ equal pieces and you take just one of them. One quarter of a dollar is $25$ cents - one piece out of four equal $25$-cent pieces.
Key Insight
Unit fractions are the building blocks of all fractions. Any fraction can be thought of as a stack of unit fractions: $3/5 = 1/5 + 1/5 + 1/5$.
Definition
A unit fraction is a rational number of the form $1/n$ where $n$ is a positive integer. Every positive fraction $a/b$ can be expressed as the sum of $a$ unit fractions each equal to $1/b$. Unit fractions were the primary way ancient Egyptians represented all fractions.
Example
$3/7 = 1/7 + 1/7 + 1/7$. The ancient Egyptians would write any fraction as a sum of distinct unit fractions: $2/5 = 1/3 + 1/15$, because they avoided repeating the same unit fraction.
Key Insight
The harmonic series $1/1 + 1/2 + 1/3 + 1/4 + \ldots$ is the sum of all unit fractions. Despite each term getting smaller, the series diverges - it has no finite sum. This surprises most people and reveals deep properties of infinity.
Definition
A unit fraction $1/n$ is an element of $\mathbb{Q}$ with numerator $1$ in reduced form. 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. Egyptian fraction representations - sums of distinct unit fractions - are studied in combinatorial number theory.
Example
The greedy algorithm (Fibonacci-Sylvester) finds Egyptian fraction representations: $5/7 \to$ 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
Every rational number in $(0,1)$ has infinitely many Egyptian fraction representations. The question of which representations are "shortest" (fewest terms) or "smallest" (smallest largest denominator) remains an active research area in number theory.