# Unit fraction

### Unit fraction

A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. A unit fraction is therefore the reciprocal of a positive integer, 1/n. Examples are 1/1, 1/2, 1/3, 1/4 ,1/5, etc.

## Contents

• Elementary arithmetic 1
• Modular arithmetic 2
• Finite sums of unit fractions 3
• Series of unit fractions 4
• Matrices of unit fractions 5
• Unit fractions in probability and statistics 7
• Unit fractions in physics 8
• References 9

## Elementary arithmetic

Multiplying any two unit fractions results in a product that is another unit fraction:

\frac1x \times \frac1y = \frac1{xy}.

However, adding, subtracting, or dividing two unit fractions produces a result that is generally not a unit fraction:

\frac1x + \frac1y = \frac{x+y}{xy}
\frac1x - \frac1y = \frac{y-x}{xy}
\frac1x \div \frac1y = \frac{y}{x}.

## Modular arithmetic

Unit fractions play an important role in modular arithmetic, as they may be used to reduce modular division to the calculation of greatest common divisors. Specifically, suppose that we wish to perform divisions by a value x, modulo y. In order for division by x to be well defined modulo y, x and y must be relatively prime. Then, by using the extended Euclidean algorithm for greatest common divisors we may find a and b such that

\displaystyle ax + by = 1,

from which it follows that

\displaystyle ax \equiv 1 \pmod y,

or equivalently

a \equiv \frac1x \pmod y.

Thus, to divide by x (modulo y) we need merely instead multiply by a.

## Finite sums of unit fractions

Any positive rational number can be written as the sum of unit fractions, in multiple ways. For example,

\frac45=\frac12+\frac14+\frac1{20}=\frac13+\frac15+\frac16+\frac1{10}.

The ancient Egyptian civilisations used sums of distinct unit fractions in their notation for more general rational numbers, and so such sums are often called Egyptian fractions. There is still interest today in analyzing the methods used by the ancients to choose among the possible representations for a fractional number, and to calculate with such representations. The topic of Egyptian fractions has also seen interest in modern number theory; for instance, the Erdős–Graham conjecture and the Erdős–Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.

In geometric group theory, triangle groups are classified into Euclidean, spherical, and hyperbolic cases according to whether an associated sum of unit fractions is equal to one, greater than one, or less than one respectively.

## Series of unit fractions

Many well-known infinite series have terms that are unit fractions. These include:

• The harmonic series, the sum of all positive unit fractions. This sum diverges, and its partial sums
\frac11+\frac12+\frac13+\cdots+\frac1n
closely approximate ln n + γ as n increases.
• The Basel problem concerns the sum of the square unit fractions, which converges to π2/6

## Matrices of unit fractions

The Hilbert matrix is the matrix with elements

B_{i,j} = \frac1{i+j-1}.

It has the unusual property that all elements in its inverse matrix are integers. Similarly, Richardson (2001) defined a matrix with elements

C_{i,j} = \frac1{F_{i+j-1}},

where Fi denotes the ith Fibonacci number. He calls this matrix the Filbert matrix and it has the same property of having an integer inverse.