The Ancient Egyptian Number System and Unit Fractions

It is often said that our current Hindu/Arab written numerals,  first introduced into Europe by Leonardo of Pisa (Fibonacci) in the thirteenth century, are the most convenient possible. They certainly make for a very concise way of representing quantity and have an indefinitely extendable range in both directions (to the very large and the very small by way of decimals) which other systems, such as the ancient Greek and Babylonian systems, did not possess. Most authors likewise believe, or rather assume, that our ways of handling the basic four operations of arithmetic, addition, multiplication, division and subtraction,  are also the most advantageous, in short that we have the best possible Number System imaginable.
However, “You don’t get owt for nowt” as the Yorkshire saying goes. A serious disadvantage of our way of doing things is that, in pre-computer  days, our arithmetic involved much tedious rote learning such as multiplication tables — schoolboys in the Edwardian era learned the thirteen times table as well. And ‘long’ division is so complicated that, even as late as the Restoration period, we find an important State official in one of the most powerful countries in the world, Samuel Pepys, getting up at five o-clock in the morning at the age of 30 to have lessons in it.
A more serious disadvantage from the conceptual point of view is that the artifice of cipherisation obscures the underlying rationale of all numbering which is the pairing off of sets of discrete objects against a selected ‘standard’ set. But in our written system a single sign such as ‘5’ is associated with a plurality of objects and there is no way of deciding a priori whether a sign such as ‘5´ represents a larger or smaller collection of objects than ’7’. If we represent a single item by a chosen object such as a bean or by a single mark such as a scratch, and repeat this an appropriate amount of times, we end up with collections of beans or scratches that we can compare visually. True, we will still need a ‘base’ because human beings are unable to cope perceptually with even quite small quantities of objects — try guessing the number of beads in a necklace or the number of seagulls on a lawn and you will be amazed how way out you are. But in our ciphered system, not only is there no very marked difference between one level and the next but there is no quantitative difference (more or less symbols) within a single level.
The system of bases and consequent scaling up (or down) relies on the artifice of a ‘moving unit’ which becomes progressively larger (or smaller) but always maintains the same ratio as we move from one ‘level’ to the next  — in our system a ratio of one to ten. Such scaling is immediately apparent in systems such as those used by ‘primitive’ tribes (and sometimes quite advanced societies like the Yoruba or the Incas) where a collection of discrete objects, twigs or shells, is bound together, piled up or threaded together, to make the next ‘one’, then these bundles are themselves tied together as we proceed to the next stage, and so on. Our way of representing bases by moving a  digit to the left or right makes for a very compact style of notation but it obscures the fundamental principle involved and is one of the reasons why many people find mathematics, even ‘ordinary’  arithmetic, so peculiar : the operational principles seem to have little connection with what goes on in the real world and are essentially just ‘rules’ that you are obliged to learn if you want to get through your exams.
The Ancient Egyptian number system, despite its ‘poor cousin’ status compared to the Hindu or Greek systems, is actually  by far the clearest, the most logical and the most ‘user-friendly’ of all number systems. It is, like ours, a base ten system and numerals are written, like ours, in  descending order of size (largest quantity first) though this is not immediately apparent since the Egyptians, like the Hebrews and the Arabs to this day, wrote everything from right to left instead of left to right like us (Note 1).
The Egyptian sysytem does not use positional notation as such but has a different sign for the unit, the ten, the hundred &c.  Numerals less than ten are repeated upright strokes, as many strokes as there are objects, i.e.   ⁄   ⁄   ⁄   is our ‘3’ and if we have ‘three hundreds’ the hundred   sign will be repeated. A simple glance shows the approximate size of the quantity being   represented, since a collection in the thousands will have several thousand marks which are readily distinguishable from hundreds or ten thousands. With our positional notation, one has to look closely to distinguish between say 10000 and 1000 especially since it has become the fashion to leave out the comma for the thousand. Admittedly, Egyptian notation does make it difficult to distinguish between two numbers less than our ten (or the equivalent within the tens or thousands). Some (though by no means all) Egyptian texts get round this by arranging the units or other repeated signs in two rows so that there is a maximum of five in the top row, e.g. writing ‘seven’ as

  ⁄  ⁄  ⁄  ⁄  ⁄    This makes instantaneous assessment much easier and in effect means having a sub-base of five.
⁄  ⁄
There are no symbols for quantities beyond 1,000,000 but this maximum would have been amply sufficient for normal purposes — since in effect this means that a scribe would be able to represent in writing any quantity less than a billion (million million). An ancient Egyptian child only needed to learn seven different pictorial signs  : an upright (stick) for the unit, a bent stick for ten, a coiled rope for a hundred, a lotus for a thousand, a snake for ten thousand, a tadpole or frog for a hundred thousand and a seated man holding up his hands in amazement for a million. There was no sign for zero and, since the system, though employing positioning, did not rely on positioning alone, none was needed.
The disadvantage, of course, is that, compared to our system, rather more signs are required which slows up calculation : a quantity we record as 1967 would require no less than twenty-three Egyptian characters (though our 20,000 would only require two Egyptian signs). Scribes, who were a respected and well-paid body of men, closely associated with but distinct from priests, did not in practice always paint careful ‘number pictures’ as seen on tomb murals : for everyday calculation they used a rapid freehand so-called ‘hieratic’ script which runs different signs together and which, to some extent, differs from one scribe to another. I have experimented with hieratic Egyptian numerals and, with some fairly natural simplifications (natural to me) I find that writing down numbers the Egyptian way is hardly more cumbersome than our present system, if at all.
Multiplication is incredibly easy using Egyptian methods since it depends wholly on doubling and then adding rather like Russian multiplication (see earlier article). An Egyptian child would thus only need to learn off  by heart the two times table and would soon be able to situate any given quantity between two successive powers of two, e.g see that, in our reckoning, 416 comes between 2⁸ and 2⁹. Every scribe would know his powers of two backwards (which most people do not today) and very likely be able to reduce at sight any number to combinations of powers of two, at any rate approximately. As Gillings writes in his excellent Mathematics in the time of the Pharaohs,
“If Egyptian multiplication was so clumsy and difficult, how did it come about that these same techniques were still used in Coptic times, in Greek times, and even up to the Byzantine period, a thousand or more years later? No nation, over a period of more than a millennium, was able to improve on the Egyptian notation and methods.”
        Division for the Egyptians was simply multiplication in reverse. Instead of dividing our 134 by 7 the Egyptian scribe would lay out his powers of two on one side and 7 repeatedly doubled on the other.

                 1                                 7

                 2                               14

                 4                               28

                 8                             56

                16                            112

        He would stop here because he would see that the next doubling would take him well beyond his goal 134. He would then get as close as he could to 134 using the entries on the Right Hand Side, namely by adding together 112, 14 and 7 . He would then add the corresponding powers of two, namely 16, 2 and 1 giving as quotient 23 with remainder 1 since the nearest combination of 7s fell one short. This procedure is, once you have got the hang of it, no lengthier than our ‘long division’ which children at school (and beyond) often have a lot of difficulty with, indeed scarcely anyone bothers with any more. In the Egyptian system the powers of 2 form a framework within which every number can be situated and, with practice, one can juggle them around mentally to situate any number, at least to a fair degree of accuracy. The method works, of course, only because every number can be expressed as a combination of powers of 2 , in effect, as we have only recently rediscovered, can be written in  binary. The Egyptian scribes must have realized this though they do not say so specifically.

What of fractions ?  Here the Egyptians ran into difficulties because they did not have our stroke notation ¼, 5/6 &c. They got round the problem by using reciprocals of numbers, noting this by a bar placed over the top (which I cannot reproduce with the limited alphabet at my disposal here). In practice, this meant, in our terms, reducing every proper fraction to a series of unit fractions — with the important exception of 2/3 which has a special sign of its own. For some reason, a scribe doing a calculation would not write the sign for the reciprocal of 7 more than once (except in prepared tables). He would always express  our 3/7 as a series of unit fractions (i.e. all having 1 as numerator). It is not clear why the Egyptian scribes did this though one can guess certain reasons. One is that, if this reduction to unit fractions is done efficiently, the series tails off rapidly so that one can see at a glance what is significant and what is negligible. This is what we ourselves do since, although most people do not realize it, the decimal fraction notation is a sum of proper fractions with progressively decreasing denominators. For example, 0.2347 is a concise way of representing the sum
                               2/10   + 3/100 + 4/1,000 + 7/10,000 

Conceptually, the Egyptian scribes were apparently unable to make the giant leap in thought involved in extending the base-10 system backwards to represent quantities smaller than 1. This is understandable since a fraction like 4/9 is actually a rather complex beast. It is not a single but a double number : it indicates the number of pieces we have (four) and the  number of (supposedly equal) pieces that make up the whole (nine).
One advantage of using reciprocals was that it was not necessary to invent any new signs for quantities less than the unit, and this seems to have been an important consideration. The Greeks themselves, who did not have a positional number system either, for some reason also baulked at extending  their ciphered base-10 numerals to small quantities and astronomers like Ptolemy used normal base-10 numerals for quantities greater than one, but sexagesimal (60-base) fractions inherited from the Babylonians for quantities smaller than the unit. The great advantage here was that it was rarely necessary to express a quantity to more than two places, since 1/(60)² = 1/(3600) was already small and 1/(60)³ minute.
The Egyptians themselves needed fractions for eminently practical reasons : at an early stage in their development they, like most other  societies at the time, did not use currency for everyday payments, gold and silver being reserved for large scale State transactions. Temple personnel were paid mainly in bread and beer (with free lodging presumably thrown in). Since Ancient Egypt was a very hierarchical society, it became a matter of some importance to be able to divide up the bread and beer equitably.
“To divide 3 loaves equally among 5 men, each man would be given three separate portions, a 1/3, a 1/5 and a 1/15. One advantage of this division was that not only was justice done, but justice also appeared to have been done. In a modern distribution, three of the five men would get 3/5 of a loaf in one large piece, while the other two men would get two smaller pieces, 2/5 and 1/5 of a loaf, which division might be regarded as an injustice b y an ignorant workman.” 
Gillings, Mathematics in the Time of the Pharaohs 

The scribe would, seemingly, never write our 3/5 as 5*  5*  5*  (where I have used * instead of the Egyptian bar over the top of a number to indicate the reciprocal of 5). We have, then, the Egyptian ‘unit fraction equation’

3/5  =  1/3 + 1/5 + 1/15 

This at once raises several interesting mathematical questions:

(1) Can every proper fraction a/b (where a,b are positive integers)    be expressed as a sum of unit fractions?
(2) For any given a/b, how many ways are there of representing it as a sum of unit fractions, supposing it to be possible ?
(3) Is there a way to reduce the number of terms in the series of unit fractions to a minimum, supposing a/b can be so represented ?

Sebastian Hayes

Note 1 In a version of this article published in M500 magazine, I stated erroneously that the Egyptians “like the Arabs today” write numerals in ascending rather than descending order. A correspondent, Rakph Hancock, pointed out that present-day Arabs speak and record their numerals much as we do, i.e. in descending order, but write them, like everything else, from right to left.
There are no less than three distinct issues here (1) writing anything sentences, names, numbers &c. from right to left or from left to right, (2) writing numerals in ascending or descending order, (3) the order in which we deal with numerals when performing addition/subtraction and so on.
My main source, Gillings, writes “Today most nations write from left to right, and our numbers are so written also; but the values of the digits in our ‘Hindu-Arabic’ decimal system increase in place value from right to left. So, if we have to perform an addition or subtraction, we begin with the units column on the right, and work toward the left through the tens, hundreds and so on.  Conversely, the Egyptians wrote their words and numbers from right to left. Of necessity, however, the Egyptian mathematicians, like ourselves, had to start adding in the opposite direction to that in which they were accustomed to write, so the place value of the Egyptians’ digits increases from left to right, and the Egyptian system therefore runs widdershins to ours.”  Gillings, Mathematics in the Time of the Pharaohs

The confusion about left to right and right to left has been compounded because renderings of ancient Egyptian texts into English naturally reverse the orientation with respect to sentences but often print the numerals in imitation hieroglyphs in the order in which they appear in the papyrus with a modern numeral written underneath each one (so that people can see the hieroglyphs). This gives the erroneous impression that the Egyptians wrote their numerals in ascending order which, seemingly, they did not.
At first sight  it seems something of a mystery why most, if not all, societies write their numerals in descending order.   Presumably this is so because of the way in which large quantities were assessed by State officials. Confronted with, say, a confused mass of prisoners or pottery imports, an official would start by working out the thousands or hundreds, then pass to the tens and finally to the units. He would call out the amounts as he worked them out and the scribe would record the numerals in the order in which he heard them, i.e. largest amount first (but writing from right to left) . Moreover, in this way, a visiting Egyptian official could get a rough idea of the size of a batch of prisoners or pottery imports simply by glancing at the first hieroglyphic numeral (which, remember, is a different pictogram for each power of ten). But, when it came to actual operations with numbers, the scribes like everyone else had to proceed the other way (though some mental arithmetic experts say they add the higher columns first).
The confusion demonstrates a basic conflict between two very different functions of numbers : (1) as devices for the compact recording of data and (2) as a means of drawing original conclusions from given data. The first process is a movement from the unknown (or very roughly known) to the known, the second a movement from the known into the unknown which, if the reasoning is valid, eventually transforms it into part of the known.  Professional mathematicians tend to think that numbers were invented for the purpose of getting out precise solutions to mathematical equations, but the recording function of numbers was by far the more important for millennia and arguably still is.    Sebastian Hayes