The ancients in Egypt had a pragmatic mathematics without much abstraction or generalization. They needed only a few simple fractions to accomplish everything they needed to do with arithmetic, using the unit fractions (having 1 in numerator), together with a few memorizable rules for combining them, and fixed glyphs for several of the most common (natural) fractions.
Let’s do something like the Egyptians did and denote as . Further, the Egyptians had a special glyph for two thirds: . Mathematics waited for the classical Greeks to invent true multiplication and thus the concepts of ratios, where means .
The Egyptians had lists of simple rules for adding unit fractions, such as:
(3) , derived from (2) by division by 2.
The Egyptians combined unit fractions to effect multiplication by 2, simply by duplicating an existing expression. Thus:
This would soon get unwieldy, but they were clever to recognize simpler forms:
. Now that doesn’t seem like much improvement, but then
. That’s progress, doubling the prior expression. And then
, by observing again that .
Now we’re getting somewhere (don’t laugh; they built the pyramids with this arithmetic)*.
Looking at these methods, an inquisitive mind might ask: can any rational number be represented as sum of distinct unit fractions? With above, there is a duplicated fraction because . But substituting from (3) above, one then gets:
This is longer than the equivalent Egyptian formula, length being the price of uniqueness.
An observant mind might answer the prior question by noting a pattern in the rules (1) and (3) above. The pattern can be expressed as a general recursion formula:
This is easily verified by multiplying both sides by .
Any rational number begins in the form and is thus the sum of m unit fractions with m-1 duplications. To remove the duplications, one just expands each duplicate by the recursion formula enough times to produce unit fractions unique among all the sums.
We have shown that any rational number is the sum of unique terms from a subset of the harmonic series. But assuredly, the Egyptians had no use for such foolishness. They were clever arithmeticians, not mathematicians.
* A complete description of Egyptian arithmetic can be found in Science Awakening by Van Der Waerden, a book that is perhaps the best in class for describing pre-CE mathematics of Greece, Babylonia, and Egypt.