The Lahun Mathematical Papyri (also known as the Kahun Mathematical Papyri) is an ancient Egyptian mathematical text. It forms part of the

Kahun Papyri The Kahun Papyri (KP; also Petrie Papyri or Lahun Papyri) are a collection of ancient Egyptian texts discussing administrative, mathematical and medical topics. Its many fragments were discovered by Flinders Petrie in 1889 and are kept at the U ...

, which was discovered at

El-Lahun El Lahun ( ar, اللاهون ''El Lāhūn,'' alt. Illahun, Lahun, or Kahun (the latter being a neologism coined by archaeologist William Matthew Flinders Petrie) is a workmen's village in Faiyum, Egypt. El Lahun is associated with the Pyramid of ...

(also known as Lahun, Kahun or Il-Lahun) by

Flinders Petrie Sir William Matthew Flinders Petrie ( – ), commonly known as simply Flinders Petrie, was a British Egyptologist and a pioneer of systematic methodology in archaeology and the preservation of artefacts. He held the first chair of Egypt ...

during excavations of a workers' town near the pyramid of the

12th dynasty The Twelfth Dynasty of ancient Egypt (Dynasty XII) is considered to be the apex of the Middle Kingdom by Egyptologists. It often is combined with the Eleventh, Thirteenth, and Fourteenth dynasties under the group title, Middle Kingdom. Some s ...

pharaoh Sesostris II. The Kahun Papyri are a collection of texts including administrative texts, medical texts, veterinarian texts and six fragments devoted to mathematics.

Fragments

The mathematical texts most commented on are usually named: * Lahun IV.2 (or Kahun IV.2) (UC 32159): This fragment contains a

table Table may refer to: * Table (furniture), a piece of furniture with a flat surface and one or more legs * Table (landform), a flat area of land * Table (information), a data arrangement with rows and columns * Table (database), how the table data ...

Egyptian fraction An Egyptian fraction is a finite sum of distinct unit fractions, such as \frac+\frac+\frac. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from each ...

representations of numbers of the form 2/''n''. A more complete version of this table of fractions is given in the

Rhind Mathematical Papyrus The Rhind Mathematical Papyrus (RMP; also designated as papyrus British Museum 10057 and pBM 10058) is one of the best known examples of ancient Egyptian mathematics. It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased ...

.Clagett, Marshall ''Ancient Egyptian Science, A Source Book''. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ;

Annette Imhausen Annette Imhausen (also known as Annette Warner, born June 12, 1970) is a German historian of mathematics known for her work on Ancient Egyptian mathematics. She is a professor in the Normative Orders Cluster of Excellence at Goethe University Fr ...

, Jim Ritter: ''Mathematical Fragments'', In: Marc Collier, Stephen Quirke: ''The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical'', Oxford 2004, , 92-93 * Lahun IV.3 (or Kahun IV.3) (UC 32160) contains numbers in

arithmetical progression An arithmetic progression or arithmetic sequence () is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common differ ...

and a problem very much like problem 40 of the Rhind Mathematical Papyrus. Another problem on this fragment computes the volume of a cylindrical granary. In this problem the scribe uses a formula which takes measurements in

cubit The cubit is an ancient unit of length based on the distance from the elbow to the tip of the middle finger. It was primarily associated with the Sumerians, Egyptians, and Israelites. The term ''cubit'' is found in the Bible regarding No ...

s and computes the volume and expresses it in terms of the unit ''khar''. Given the diameter (d) and height (h) of the cylindrical granary: :

V = ((1+1/3)d)^2 \ ((2/3) h)

. : In modern mathematical notation this is equal to :

V = \frac d^2\ h = \frac r^2\ h

(measured in khar). : This problem resembles problem 42 of the

. The formula is equivalent to

V =  \frac  r^2\ h

measured in cubic-cubits as used in the other problems. * Lahun XLV.1 (or Kahun XLV.1) (UC 32161) contains a group of very large numbers (hundreds of thousands). * Lahun LV.3 (or Kahun LV.3) (UC 32134A and UC 32134B) contains a so-called ''aha'' problem which asks one to solve for a certain quantity. The problem resembles ones from the Rhind Mathematical Papyrus (problems 24–29). * Lahun LV.4 (or Kahun LV.4) (UC 32162) contains what seems to be an area computation and a problem concerning the value of ducks, geese and cranes. The problem concerning fowl is a baku problem and most closely resembles problem 69 in the

and problems 11 and 21 in the

Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, geom ...

. * Unnamed fragment (UC 32118B). This is a fragmentary piece.

2/''n'' tables

The Lahun papyrus IV.2 reports a 2/''n'' table for odd ''n'', ''n'' = 1, , 21. The

reports an odd ''n'' table up to 101. Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp. 19–27 These fraction tables were related to multiplication problems and the use of unit fractions, namely n/p scaled by LCM m to mn/mp. With the exception of 2/3, all fractions were represented as sums of unit fractions (i.e. of the form 1/''n''), first in red numbers. Multiplication algorithms and scaling factors involved repeated doubling of numbers, and other operations. Doubling a unit fraction with an even denominator was simple, divided the denominator by 2. Doubling a fraction with an odd denominator however results in a fraction of the form 2/n. The

RMP 2/n table The Rhind Mathematical Papyrus, an ancient Egyptian mathematical work, includes a mathematical table for converting rational numbers of the form 2/''n'' into Egyptian fractions (sums of distinct unit fractions), the form the Egyptians used to write ...

and RMP 36 rules allowed scribes to find decompositions of 2/n into unit fractions for specific needs, most often to solve otherwise un-scalable rational numbers (i.e. 28/97 in RMP 31, and 30/53 n RMP 36 by substituting 26/97 + 2/97 and 28/53 + 2/53) and generally n/p by (n - 2) /p + 2/p. Decompositions were unique. Red auxiliary numbers selected divisors of denominators mp that best summed to numerator mn.

References

External links

John Legon: A Kahun Mathematical Fragment

History of Egyptian fractions
* *{{planetmath reference, urlname=EgyptianFraction, title=Egyptian fraction Egyptian mathematics Egyptian fractions Egyptian papyri Mathematics manuscripts de:Medizinische Papyri aus Lahun es:Papiro Kahun fr:Papyrus Kahun sv:Kahun-papyrusen

Fragments

2/''n'' tables

See also

References

External links