# Lahun Mathematical Papyri

The Lahun Mathematical Papyri (also known as the Kahun Mathematical Papyri) are part of a collection of Kahun Papyri discovered at El-Lahun (also known as Lahun, Kahun or Il-Lahun) by Flinders Petrie during excavations of a worker's town near the pyramid of Sesostris II. The Kahun Papyrus 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): 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.[2]
• Lahun IV.3 (or Kahun IV.3) (UC 32160) contains numbers in arithmetical progression and a problem very much like problem 40 of the Rhind Mathematical Papyrus.[2][3][4] Another problem on this fragment computes the volume of a cylindrical granary.[5] 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:
$V = ((1+1/3)d)^2 \ ((2/3) h)$.
In modern mathematical notation this is equal to
$V = \frac{32}{27} d^2\ h = \frac{128}{27} r^2\ h$ (measured in khar).
This problem resembles problem 42 of the Rhind Mathematical Papyrus. The formula is equivalent to $V = \frac{256}{81} r^2\ h$ measured in cubic-cubits as used in the other problems.[6]
• Lahun XLV.1 (or Kahun XLV.1) (UC 32161) contains a group of very large numbers (hundreds of thousands).[2][7]
• 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).[2][8]

## The 2/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.[12] 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

1. ^ The Lahun Papyri at University College London
2. 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
3. ^ 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
4. ^ Legon, J., A Kahun mathematical fragment, retrieved from [1], based on Discussions in Egyptology 24 (1992), p.21-24
5. ^ Gay Robins and Charles Shute, "The Rhind Mathematical Papyrus", British Museum Press, Dover Reprint, 1987.
6. ^ 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
7. ^ 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
8. ^ 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
9. ^ 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
10. ^ UC 32162 Lahun LV.4
11. ^ 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
12. ^ Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp. 19–27