section_path: Forms of fractions/Historical notions/Egyptian fraction
section_name: Egyptian fraction
paragraph_text: Egyptian fraction
  
  An Egyptian fraction is the sum of distinct positive unit fractions, for example 1 2 + 1 3 {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{3}}}. This definition derives from the fact that the ancient Egyptians expressed all fractions except 1 2 {\displaystyle {\tfrac {1}{2}}}, 2 3 {\displaystyle {\tfrac {2}{3}}} and 3 4 {\displaystyle {\tfrac {3}{4}}} in this manner. Every positive rational number can be expanded as an Egyptian fraction. For example, 5 7 {\displaystyle {\tfrac {5}{7}}} can be written as 1 2 + 1 6 + 1 21. {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{6}}+{\tfrac {1}{21}}.} Any positive rational number can be written as a sum of unit fractions in infinitely many ways. Two ways to write 13 17 {\displaystyle {\tfrac {13}{17}}} are 1 2 + 1 4 + 1 68 {\displaystyle {\tfrac {1}{2}}+{\tfrac {1}{4}}+{\tfrac {1}{68}}} and 1 3 + 1 4 + 1 6 + 1 68 {\displaystyle {\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{68}}}.