A fraction became a recipe of parts
For an Egyptian scribe, a fraction such as 3/4 was not usually written as one number above another. It could be expressed as 1/2 + 1/4: two shares, each defined by its own denominator. These are called unit fractions because each has a numerator of 1.
That changed the practical question from “How do I write three quarters?” to “Which simple parts add up to the amount I need?” The same approach could describe dividing a quantity among people or calculating portions of grain. A scribe working out a share could build it from familiar pieces rather than rely on a notation for arbitrary numerators.
The method made arithmetic a search
A unit-fraction expression was not always unique. For example, 3/4 can also be written as 1/2 + 1/6 + 1/12. The scribe’s task was to find a combination that added up correctly, often choosing a convenient set of parts for the calculation at hand. Egyptian mathematical texts include tables that helped with such decompositions.
The system had exceptions: 2/3 had its own commonly used sign, rather than being treated like an ordinary unit fraction. And while breaking a fraction into parts could make quantities easier to handle, it could also make a written calculation longer. Egyptian fraction notation turned one compact fraction into a small arithmetic plan—a way to show not just the amount, but the pieces from which it was assembled.
