# Lahun Mathematical Papyri

The **Lahun Mathematical Papyri** (also known as the **Kahun Mathematical Papyri**) is an ancient Egyptian mathematical text. It forms part of the Kahun Papyri, which was discovered at El-Lahun (also known as Lahun, Kahun or Il-Lahun) by Flinders Petrie during excavations of a workers' town near the pyramid of the 12th dynasty pharaoh Sesostris II. The Kahun Papyri are a collection of texts including administrative texts, medical texts, veterinarian texts and six fragments devoted to mathematics.[1]

The mathematical texts most commented on are usually named:

**Lahun IV.2**(or**Kahun IV.2**) (UC 32159[2]): This fragment contains a table of Egyptian fraction representations of numbers of the form 2/*n*. A more complete version of this table of fractions is given in the Rhind Mathematical Papyrus.[3]**Lahun IV.3**(or**Kahun IV.3**) (UC 32160[4]) contains numbers in arithmetical progression and a problem very much like problem 40 of the Rhind Mathematical Papyrus.[3][5][6] Another problem on this fragment computes the volume of a cylindrical granary.[7] In this problem the scribe uses a formula which takes measurements in cubits and computes the volume and expresses it in terms of the unit*khar*. Given the diameter (d) and height (h) of the cylindrical granary:

- .
- In modern mathematical notation this is equal to
- (measured in khar).
- This problem resembles problem 42 of the Rhind Mathematical Papyrus. The formula is equivalent to measured in cubic-cubits as used in the other problems.[8]

**Lahun XLV.1**(or**Kahun XLV.1**) (UC 32161[9]) contains a group of very large numbers (hundreds of thousands).[3][10]**Lahun LV.3**(or**Kahun LV.3**) (UC 32134A[11] and UC 32134B[12]) 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).[3][13]**Lahun LV.4**(or**Kahun LV.4**) (UC 32162[14]) contains what seems to be an area computation and a problem concerning the value of ducks, geese and cranes.[3][15] The problem concerning fowl is a baku problem and most closely resembles problem 69 in the Rhind Mathematical Papyrus and problems 11 and 21 in the Moscow Mathematical Papyrus.[16]- Unnamed fragment (UC 32118B[17]). This is a fragmentary piece.[18]

## 2/*n* tables

*n*tables

The Lahun papyrus IV.2 reports a 2/*n* table for odd *n*, *n* = 1, , 21. The Rhind Mathematical Papyrus reports an odd *n* table up to 101.[19] 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 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

- The Lahun Papyri at University College London
- "Lahun Papyri: table texts". Retrieved 15 August 2016.
- Clagett, Marshall
*Ancient Egyptian Science, A Source Book*. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0-87169-232-0; Annette Imhausen, Jim Ritter:*Mathematical Fragments*, In: Marc Collier, Stephen Quirke:*The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical*, Oxford 2004, ISBN 1-84171-572-7, 92-93 - "Lahun Papyri: table texts". Retrieved 15 August 2016.
- Annette Imhausen, Jim Ritter:
*Mathematical Fragments*, In: Marc Collier, Stephen Quirke:*The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical*, Oxford 2004, ISBN 1-84171-572-7, 84–85 - Legon, J., A Kahun mathematical fragment, retrieved from , based on Discussions in Egyptology 24 (1992), p. 21–24
- Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", British Museum Press, Dover Reprint, 1987.
- Katz, Victor J. (editor), Imhausen, Annette et al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007 ISBN 978-0-691-11485-9
- "Lahun Papyri: table texts". Retrieved 15 August 2016.
- Annette Imhausen, Jim Ritter:
*Mathematical Fragments*, In: Marc Collier, Stephen Quirke:*The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical*, Oxford 2004, ISBN 1-84171-572-7, 94-95 - "Lahun Papyri: table texts". Retrieved 15 August 2016.
- "Lahun Papyri: table texts". Retrieved 15 August 2016.
- Annette Imhausen, Jim Ritter:
*Mathematical Fragments*, In: Marc Collier, Stephen Quirke:*The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical*, Oxford 2004, ISBN 1-84171-572-7, 74–77 - "Lahun Papyri: table texts". Retrieved 15 August 2016.
- Annette Imhausen, Jim Ritter:
*Mathematical Fragments*, In: Marc Collier, Stephen Quirke:*The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical*, Oxford 2004, ISBN 1-84171-572-7, 78–79 - "Lahun Papyri: table texts". Retrieved 15 August 2016.
- "Lahun Papyri: table texts". Retrieved 15 August 2016.
- Annette Imhausen, Jim Ritter:
*Mathematical Fragments*, In: Marc Collier, Stephen Quirke:*The UCL Lahun Papyri: Religious, Literary, Legal, Mathematical and Medical*, Oxford 2004, ISBN 1-84171-572-7, 90–91 - Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp. 19–27

## External links

- John Legon: A Kahun Mathematical Fragment
- https://web.archive.org/web/20100117021426/http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
- Math-History-List
- MathOrigins.com
- History of Egyptian fractions
- Medical Papyrus, UCL website
- The Kahun Gynaecological Papyrus
- "Kahun papyrus".
*PlanetMath*. - "Egyptian fraction".
*PlanetMath*.