A unit fraction is a positive fraction with one as its numerator, 1/n. It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole. Multiplying two unit fractions produces another unit fraction, but other arithmetic operations do not preserve unit fractions. In modular arithmetic, unit fractions can be converted into equivalent whole numbers, allowing modular division to be transformed into multiplication. Every rational number can be represented as a sum of distinct unit fractions; these representations are called Egyptian fractions based on their use in ancient Egyptian mathematics. Many infinite sums of unit fractions are meaningful mathematically. In geometry, unit fractions can be used to characterize the curvature of triangle groups and the tangencies of Ford circles. Unit fractions are commonly used in fair division, and this familiar application is used in mathematics education as an early step toward the understanding of other fractions. Unit fractions are common in probability theory due to the principle of indifference. They also have applications in combinatorial optimization and in analyzing the pattern of frequencies in the hydrogen spectral series.
Arithmetic The unit fractions are the rational numbers that can be written in the form 1 n , {\displaystyle {\frac {1}{n}},} where n {\displaystyle n} can be any positive natural number. They are thus the multiplicative inverses of the positive integers. When something is divided into n {\displaystyle n} equal parts, each part is a 1 / n {\displaystyle 1/n} fraction of the whole.
Elementary arithmetic Multiplying any two unit fractions results in a product that is another unit fraction:
1 x × 1 y = 1 x y . {\displaystyle {\frac {1}{x}}\times {\frac {1}{y}}={\frac {1}{xy}}.}
However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction:
1 x + 1 y = x + y x y {\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {x+y}{xy}}}
1 x − 1 y = y − x x y {\displaystyle {\frac {1}{x}}-{\frac {1}{y}}={\frac {y-x}{xy}}}
1 x ÷ 1 y = y x . {\displaystyle {\frac {1}{x}}\div {\frac {1}{y}}={\frac {y}{x}}.}
As the last of these formulas shows, every fraction can be expressed as a quotient of two unit fractions.
Modular arithmetic In modular arithmetic, any unit fraction can be converted into an equivalent whole number using the extended Euclidean algorithm. This conversion can be used to perform modular division: dividing by a number x {\displaystyle x} , modulo y {\displaystyle y} , can be performed by converting the unit fraction 1 / x {\displaystyle 1/x} into an equivalent whole number modulo y {\displaystyle y} , and then multiplying by that number. In more detail, suppose that x {\displaystyle x} is relatively prime to y {\displaystyle y} (otherwise, division by x {\displaystyle x} is not defined modulo y {\displaystyle y} ). The extended Euclidean algorithm for the greatest common divisor can be used to find integers a {\displaystyle a} and b {\displaystyle b} such that Bézout's identity is satisfied:
a x + b y = gcd ( x , y ) = 1. {\displaystyle \displaystyle ax+by=\gcd(x,y)=1.}
In modulo- y {\displaystyle y} arithmetic, the term b y {\displaystyle by} can be eliminated as it is zero modulo y {\displaystyle y} . This leaves
a x ≡ 1 ( mod y ) . {\displaystyle \displaystyle ax\equiv 1{\pmod {y}}.}
That is, a {\displaystyle a} is the modular inverse of x {\displaystyle x} , the number that when multiplied by x {\displaystyle x} produces one. Equivalently,
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