Egyptian fraction

Summary

An Egyptian fraction is a finite sum of distinct unit fractions, such as

$$\frac{1}{2}+\frac{1}{3}+\frac{1}{16}.$$

That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each other. The value of an expression of this type is a positive rational number $\tfrac{a}{b}$; for instance the Egyptian fraction above sums to $\tfrac{43}{48}$. Every positive rational number can be represented by an Egyptian fraction. Sums of this type, and similar sums also including $\tfrac{2}{3}$ and $\tfrac{3}{4}$ as summands, were used as a serious notation for rational numbers by the ancient Egyptians, and continued to be used by other civilizations into medieval times. In modern mathematical notation, Egyptian fractions have been superseded by vulgar fractions and decimal notation. However, Egyptian fractions continue to be an object of study in modern number theory and recreational mathematics, as well as in modern historical studies of ancient mathematics.