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A unit fraction is a fraction with numerator 1 such as 1/2, 1/3, 1/4 and so on. An Egyptian fraction is a sum of (usually distinct) unit fractions. The Rhind papyrus contains an extensive list of

2/n

fractions expressed as Egyptian fractions consisting of two, three or four unit fractions. Select from the following list of fractions appearing on the papyrus to visualize their Egyptian fraction decompositions as bar models together with the hieratic representation for each fraction:

Out[]=

​
2/n fraction into unit fractions 2/101 Show denominators in hieratic

Computational Explanation

The Rhind papyrus was written around 1550 BCE by the scribe Ahmes, the first writer of mathematics whose name is known. However, aspects of the document may have been copied from an unknown earlier text posited to date from around 1850 BCE.

The 2/n Egyptian Fraction Table

Egyptians used sums of distinct unit fractions, now known as Egyptian fractions, to write fractional numbers. For example, the fraction 2/13 would have been written as:

In[]:=

1

8

+

1

52

+

1

104

Out[]=

2

13

The first part of the Rhind papyrus consists of a mathematical table converting rational numbers of the form 2/n into Egyptian fractions, where the expansions span nine sheets of papyrus. (Clearly, the requirement that the unit fractions be distinct is needed in this case to avoid the trivial decomposition

2/n1/n+1/n

.)

Since the discovery of the Rhind papyrus, mathematicians have struggled to understand the method used by the Egyptians to calculate Egyptian expansions. This is difficult to determine because any rational number has not one, but in fact infinitely many possible expansions as a sum of unit fractions. In particular, any rational number has a representation as an Egyptian fraction with arbitrarily many terms and with arbitrarily large denominators, although for a given fixed number of terms, there are only finitely many. For example, 2/15 can be decomposed in two ways:

In[]:=



1

9

+

1

45

,

1

10

+

1

30



Out[]=



2

15

,

2

15



The Rhind papyrus gives Egyptian fraction representations for the following 50 rational numbers:

In[]:=

RhindPapyrus2OverNTable=

2/n table

;

In[]:=

TextGridPartition[RhindPapyrus2OverNTable[[All,1]],UpTo[10]],

opts



Out[]=

2 3	2 5	2 7	2 9	2 11	2 13	2 15	2 17	2 19	2 21
2 23	2 25	2 27	2 29	2 31	2 33	2 35	2 37	2 39	2 41
2 43	2 45	2 47	2 49	2 51	2 53	2 55	2 57	2 59	2 61
2 63	2 65	2 67	2 69	2 71	2 73	2 75	2 77	2 79	2 81
2 83	2 85	2 87	2 89	2 91	2 93	2 95	2 97	2 99	2 101

Twenty-nine of these are expressed as a sum of two unit fractions:

In[]:=

TextGridPartition[Row[Flatten[{#1,"",Riffle[#2,"+"]}]," "]&@@@Cases[RhindPapyrus2OverNTable,{_,{_,_}}],UpTo[3]],

opts



Out[]=

2 3  1 2 + 1 6	2 5  1 3 + 1 15	2 7  1 4 + 1 28
2 9  1 6 + 1 18	2 11  1 6 + 1 66	2 15  1 10 + 1 30
2 21  1 14 + 1 42	2 23  1 12 + 1 276	2 25  1 15 + 1 75
2 27  1 18 + 1 54	2 33  1 22 + 1 66	2 35  1 30 + 1 42
2 39  1 26 + 1 78	2 45  1 30 + 1 90	2 49  1 28 + 1 196
2 51  1 34 + 1 102	2 55  1 30 + 1 330	2 57  1 38 + 1 114
2 63  1 42 + 1 126	2 65  1 39 + 1 195	2 69  1 46 + 1 138
2 75  1 50 + 1 150	2 77  1 44 + 1 308	2 81  1 54 + 1 162
2 85  1 51 + 1 255	2 87  1 58 + 1 174	2 91  1 70 + 1 130
2 93  1 62 + 1 186	2 99  1 66 + 1 198	​

Thirteen are expressed as sums of three unit fractions:

In[]:=

TextGridPartition[Row[Flatten[{#1,"",Riffle[#2,"+"]}]," "]&@@@Cases[RhindPapyrus2OverNTable,{_,{_,_,_}}],UpTo[3]],

opts



Out[]=

2 13  1 8 + 1 52 + 1 104	2 17  1 12 + 1 51 + 1 68	2 19  1 12 + 1 76 + 1 114
2 31  1 20 + 1 124 + 1 155	2 37  1 24 + 1 111 + 1 296	2 41  1 24 + 1 246 + 1 328
2 47  1 30 + 1 141 + 1 470	2 53  1 30 + 1 318 + 1 795	2 59  1 36 + 1 236 + 1 531
2 67  1 40 + 1 335 + 1 536	2 71  1 40 + 1 568 + 1 710	2 95  1 60 + 1 380 + 1 570
2 97  1 56 + 1 679 + 1 776	​	​

… where interestingly, all the denominators in this list are prime with the exception of 2/95:

In[]:=

Cases[RhindPapyrus2OverNTable,{Rational[2,_?(!PrimeQ[#]&)],{_,_,_}}]

Out[]=



2

95

,

1

60

,

1

380

,

1

570



The remaining eight numbers are expressed as sums of four unit fractions:

In[]:=

TextGridPartition[Row[Flatten[{#1,"",Riffle[#2,"+"]}]," "]&@@@Cases[RhindPapyrus2OverNTable,{_,{_,_,_,_}}],UpTo[3]],



Out[]=

2 29  1 24 + 1 58 + 1 174 + 1 232	2 43  1 42 + 1 86 + 1 129 + 1 301	2 61  1 40 + 1 244 + 1 488 + 1 610
2 73  1 60 + 1 219 + 1 292 + 1 365	2 79  1 60 + 1 237 + 1 316 + 1 790	2 83  1 60 + 1 332 + 1 415 + 1 498
2 89  1 60 + 1 356 + 1 534 + 1 890	2 101  1 101 + 1 202 + 1 303 + 1 606	​

Problem 61 in the Rhind papyrus gives a formula to express

2/(3k)

as a sum of two unit fractions via:

In[]:=

1

2k

+

1

6k

Out[]=

2

3k

In[]:=

TableRow

2

3n

,"  ",

1

2n

," + ",

1

6n

,{n,1,33,2}//Column[#,BaseStyle12]&

Out[]=

2 3  1 2 + 1 6

2 9  1 6 + 1 18

2 15  1 10 + 1 30

2 21  1 14 + 1 42

2 27  1 18 + 1 54

2 33  1 22 + 1 66

2 39  1 26 + 1 78

2 45  1 30 + 1 90

2 51  1 34 + 1 102

2 57  1 38 + 1 114

2 63  1 42 + 1 126

2 69  1 46 + 1 138

2 75  1 50 + 1 150

2 81  1 54 + 1 162

2 87  1 58 + 1 174

2 93  1 62 + 1 186

2 99  1 66 + 1 198

Similarly,

2/(5k)

can be written as:

In[]:=

1

3k

+

1

15k

Out[]=

2

5k

… giving:

In[]:=

TableRow

2

5n

,"  ",

1

3n

," + ",

1

15n

,{n,1,19,2}//Column[#,BaseStyle12]&

Out[]=

2 5  1 3 + 1 15

2 15  1 9 + 1 45

2 25  1 15 + 1 75

2 35  1 21 + 1 105

2 45  1 27 + 1 135

2 55  1 33 + 1 165

2 65  1 39 + 1 195

2 75  1 45 + 1 225

2 85  1 51 + 1 255

2 95  1 57 + 1 285

However, these give expansions for 2/15, 2/45 and 2/75, which are different from those given in the papyrus. Moreover, the papyrus expresses

2/95

as a sum of three unit fractions as

1/60+1/380+1/570

even though the preceding formula produces the "simpler" two-term decomposition

1/57+1/285

.

Division by 10 Table

The famous list of

2/n

fractions of the Rhind papyrus is followed by a tiny table of fractional expressions for the numbers 1 through 9 divided by 10. For instance, the division of 7 by 10 is recorded as 7 divided by 10 yields 2/3 + 1/30.

Problem Concerning Arithmetic, Elementary Algebra and Units

Following the division by 10 table, the Rhind papyrus records 91 problems, which have been designated by Egyptologists with numbers 1–87, 7B, 59B, 61B and 82B.

Problems 1–7, 7B and 8–40 are concerned with arithmetic and elementary algebra. Problems 1–6 compute divisions of a certain number of loaves of bread by 10 men and record the outcome in unit fractions. Problems 7–20 show how to multiply the expressions

1+1/2+1/47/4

and

1+2/3+1/32

by different fractions. Problems 21–23 are problems in completion, which in modern notation are simply subtraction problems.

Problems 24–34 are linear equations known as aha problems. For example, problem 26 corresponds to:

In[]:=

Reduce[x+x/415,x]

Out[]=

x12

Problems 35–38 involve divisions of the hekat, which is an ancient Egyptian unit of volume. Beginning at this point, assorted units of measurement become much more important throughout the remainder of the papyrus, and indeed a major consideration throughout the rest of the papyrus is dimensional analysis.

Problems 39 and 40 compute the division of loaves and use arithmetic progressions. Problems 41–46 show how to find the volume of both cylindrical and rectangular granaries. Problem 47 is a table with fractional equalities, which represent 10 situations where the volume quantity of 100 quadruple hekats is divided by each of the multiples of 10, from 10 through 100.

Problems Concerning Areas

Problems 48–55 show how to compute an assortment of areas such as rectangles, triangles and trapezoids. Problem 48 is notable in that it succinctly computes the area of a circle by adopting the convention that a circle's area stands to that of its circumscribing square in the ratio

64/81

.

Out[]=

Equivalently, the papyrus approximates π as 3.16049…:

In[]:=

Reduce[64pi(9/2)^2,pi]

Out[]=

pi

256

81

Problems Concerning Pyramids, Algebra, Bread and Beer

Problems 56–62 are related to the slopes of pyramids. Problems 62–68 are general problems of an algebraic nature. Problems 69–78 are all pefsu problems in some form or another. They involve computations regarding the strength of bread and beer, with respect to certain raw materials used in their production.

Ahmes's St. Ives Problem

Problem 79 is among the most interesting in the papyrus since one of Ahmes's methods of solution for the following sum suggests an understanding of finite geometric series:

"An estate's inventory consists of 7 houses, 49 cats, 343 mice, 2401 spelt plants [a type of wheat] and 16807 units of hekat (of whatever substance). List the items in the estate's inventory as a table, and include their total."

In fact, this is a version of the famous St. Ives problem but with an additional multiple of 7 beyond "kits."

By brute force addition, the answer is:

In[]:=

7+49+343+2401+16807

Out[]=

19607

However, Ahmes pursued the simpler route of adding the first four terms of the sequence (7 + 49 + 343 + 2401  2800) to produce a partial sum, added one (2801) and then multiplied by 7 to produce the correct answer:

In[]:=

19607==7(1+2800)==7(1+7+

2

7

+

3

7

+

4

7

)==71+

4

∑

k=1

k

7

==

5

∑

k=1

k

7

Out[]=

True

Notice that Ahmes's method generalizes to any finite geometric series

x+

2

x

+

3

x

+⋯+

n

x

:

In[]:=

n

∑

k=1

k

x

x1+

n-1

∑

k=1

k

x

//Simplify

Out[]=

True

The general solution to a sum-of-powers problem such as this can be given in closed form for arbitrary base b and number of levels n:

In[]:=

Sum[b^k,{k,n}]

Out[]=

b(-1+

n

b

)

-1+b

For example, plugging in the values from problem 79 immediately gives:

In[]:=

b(-1+

n

b

)

-1+b

/.{b7,n5}

Out[]=

19607

Problems Concerning Volume and Animal Feed

Problems 80 and 81 involve converting different volume units (hekats, hinu and ro) used to measure grain, bread and beer. One hekat equals

4.8

litres, one hinu is 1/10 hekat and one ro is 1/320 hekat.

Finally, the last four mathematical items, problems 82, 82B and 83–84, compute the amount of feed necessary for various animals, such as fowl and oxen.

Other Resources

Additional Reading

• Aboulfotouh, H. M. K. "The Geometric Grids of the Hieratic Numeral Signs." Mediterranean Archaeology and Archaeometry, Vol. 12, No 1, pp. 129–138, 2012.
• Cajori, F. A History of Mathematical Notations: Two Volumes Bound as One, Vol. 1. New York: Dover, pp. 14–15, 1993.
• Chace, A. B. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translation. 1927–1929.
• Imhausen, A. "Egyptian Mathematics." Ch. 1 in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Ed. V. J. Katz). Princeton, NJ: Princeton University Press, pp. 7–56 and table on p. 20, 2007.
• Imhausen, A. Mathematics in Ancient Egypt. A Contextual History. Princeton, NJ: Princeton University Press, 2016.
• Mankiewicz, R. The Story of Mathematics. Princeton, NJ: Princeton University Press, pp. 12–13, 2004.
• Peet, T. E. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited, 1923.
• Pickover, C. The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. New York: Sterling, pp. 36–37, 2012.
• Robins, R. G. and Shute, C. C. D. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited, 1987.

Additional Links

• British Museum Collection Online: The Rhind Mathematical Papyrus: EA10057
• British Museum Collection Online: The Rhind Mathematical Papyrus: EA10058
• Scott W. Williams: Egyptian Mathematical Papyri: The Rhind Papyrus
• Wikipedia: Ahmes
• Wikipedia: Ancient Egyptian mathematics
• Wikipedia: Egyptian fraction
• Wikipedia: Rhind Mathematical Papyrus
• Wikipedia: Rhind Mathematical Papyrus 2/n table

Image Credits

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