Rhind Mathematical Papyrus

Contributed by British Museum

Title of the Rhind Mathematical Papyrus from the tomb of an ancient Egyptian scribe. © Trustees of the British Museum

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This is the most famous mathematical papyrus to have survived from Ancient Egypt. It contains 84 different mathematical problems - such as how to distribute 100 loaves of bread among a workforce in different ratios. Such skills were constantly needed by scribes - the administrators of the Egyptian society. This papyrus was placed in its owner's tomb as a sign of his highly educated status. Papyrus documents are made from the papyrus reed, which is common in the Nile valley but relatively expensive.

Why did maths first develop?

Complex societies like Egypt required maths to create buildings, manage food supplies and compute the flood levels of the Nile. The Egyptians were renowned by the Greeks as masters of geometry. However, very little evidence of complex Egyptian mathematics survives. Mathematics also developed in Mesopotamia, including the system of dividing time into units of 60 that is still used today for minutes and seconds. Higher maths, such as deductive reasoning and logic, developed in Greece from 500 BC onwards.

As well as paper, ancient Egyptians used papyrus to make boats and mattresses

How has papyrus survived?

Papyrus was used to write on in ancient Egypt and was used until the eighth century AD when it was replaced by the paper we are familiar with today. A remarkably large number of papyri from antiquity have survived, including the Rhind Mathematical Papyrus which is almost 4,000 years old.

There are several reasons why papyrus documents have survived so well.

Firstly, the way papyrus ‘paper’ was made was relatively simple which helped retain the purity of the plant material. It was made from strips of pith from inside the papyrus plant. A layer of horizontal strips was laid on top of a layer of vertical ones and the two layers were bonded together by pressing. When dry this made a sheet on which to write.

Also the dry climate in Egypt and the tradition of placing papyri in tombs helped to preserve them. Tombs especially have protected the documents by acting as a kind of ‘buffer’ against the fluctuations of temperature and the relative humidity outside, even if they could not protect them from insects or tomb robbers.

Despite their remarkable durability, today many of these ancient documents are very fragile. A conservator has to work out how to make the object available to the public while keeping it in a safe and stable environment. The Rhind Papyrus presents such a problem.

It is made up of two parts, each kept separately between two pieces of glass. The Rhind papyrus is especially delicate as it was a working document 4,000 years ago and had a lot of use. It was also re-written on which meant that the first text had to be rubbed out, making the surface extra fragile today.

When the Rhind Papyrus first came to the British Museum in 1865, one of the pieces was displayed in the Egyptian galleries in direct light. This was before it was understood how damaging light can be to plant and animal materials. Today, this part of the papyrus is too fragile to be on display. Fortunately, the other part of this remarkable manuscript is in better condition and can be displayed under suitable lighting conditions in the Museum.

Bridget Leach, Conservator, British Museum

Reading the texture of ancient lives

Practical mathematics underpinned not only the vast construction works of Ancient Egypt but also the bureaucracy that supported the whole civilisation for thousands of years.

Such everyday things were not always preserved, but the scribe Ahmose had this mathematical papyrus buried with him as a sign of his elite status and education.

It is very rare for a scribe to sign and date a papyrus at this period, as he did. Ahmose proudly claims that he copied these maths problems out ‘according to the writings of old made in the time of the Dual King Nimaatre’, some 250 years earlier. Such claims were often made (or invented) to stress how authoritative a text was, but I hope it is true here - it looks back to a golden age that Ahmose would have known from the old classic poems that he had to copy out as a student, and that was very different from his own war-torn world of 1530 BC.

What fascinates me about this papyrus is that we can still see the moments when Ahmose re-dipped his pen - especially when he was drawing ruled lines to keep his layout of the problems neat and straight. And here, we can see that his mind wandered; and here, he left part of the papyrus blank.

When you see the papyrus in front of you, you can realise the texture and reality of ancient lives. Details like these give us a sense of a real individual who was trying to be neat rather than a standard ‘ancient Egyptian’.

Such material artefacts are a common ground where the ancient and modern viewers can almost meet – as you look at these signs, it seems as if the hand that wrote them on the papyrus has just left them and moved on a few moments before.

Richard Parkinson, Curator, British Museum

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Comments

  • 9 comments
  • 1. At 20:08 on 14 March 2010, tyersome wrote:

    There is an error in the annotation of the 'St. Ives problem'. The answer given is for the problem as described in the program - that is, how many objects are described?. The annotation, however, states that the problem is to determine how many gallons of grain are saved - the correct answer for which is 7 to the power of 5 (16807).

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  • 2. At 11:41 on 26 March 2010, Richard Parkinson wrote:

    The 'St Ives problem' is entitled ?household inventory? and goes: houses 7, cats 49, mice 343, barley 2401, gallons 16,807, but the full ?total? is obtained by adding the various sums together. This represents the full total of the contents of the various houses (grain + mice + cats + houses) and not just the number of gallons of grain.

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  • 3. At 12:45 on 26 April 2010, Paul Sargeant wrote:

    Hi. Thanks for pointing this out. We have corrected the transcript and it now has the correct question on it.

    Paul
    A History of the World team

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  • 4. At 18:03 on 26 May 2010, Milo Gardner wrote:

    The RMP begins with a 2/n table that takes up 1/3 of the text. To decode the 2/n table's construction method the Egyptian Mathematical Leather Roll:

    http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll
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    and RMP 36 and RMP 37 must be parsed as recorded by Middle Kingdom scribes.

    The BBC radio broadcast of Feb 2010 muddled the 2/n table and the RMP 87 problems that applied 2/n table unit fraction arithmetic operations.

    Thank you very much for considering to update your web page and a future radio broadcast to correct BBC program errors.

    Best Regards,

    Milo Gardner
    Sacramento, CA, USA

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  • 5. At 18:09 on 26 May 2010, Milo Gardner wrote:

    Dear Richard Parkinson, and BBC radio staff

    The RMP began with a 2/n table that took up 1/3 of the text. To decode the 2/n table's construction method the Egyptian Mathematical Leather Roll:

    http://en.wikipedia.org/wiki/Egyptian_Mathematical_Leather_Roll
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    and RMP 36 and RMP 37 must be parsed as recorded by Middle Kingdom scribes.

    The BBC radio broadcast of Feb 2010 muddled the 2/n table and the RMP 87 problems that applied 2/n table unit fraction arithmetic operations by oddly accepting the 1927 additive version of the EMLR rather than the EMLR's 2002 and 2008 updated contents.

    Thank you very much for considering to update your web page and a future radio broadcasts concerning the RMP to correct Feb. 2020 BBC program errors.

    Best Regards,

    Milo Gardner
    Sacramento, CA, USA
    http://en.wikipedia.org/wiki/User:Milogardner
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  • 6. At 19:39 on 2 June 2010, Milo Gardner wrote:

    Dear British Museum and BBC radio staff:

    The Rhind Mathematical Papyrus is also titled the Ahmes Papyrus. Ahmes was the scribe the wrote the text as discussed by:

    http://ahmespapyrus.blogspot.com/2009/01/ahmes-papyrus-new-and-old.html
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    The finite arithmetic that Ahmes scaled rational numbers n/p by LCM m to mn/mp and divisors of denominator mp that summed to numerator mn was explained by Ahmes in RMP 36 and RMP 37. Ahmes detailed the 2/n table and n/p conversion methods with quotients and exact remainders, finite arithmetic. Oddly, 2010 British Museum scholars continue ignore the finite arithmetic ... like an emu with its head in the sand ... and accept the 1920s additive proposals for reasons that only they can explain.

    Please explain the BMs outdated and incorrect 1920s views of the RMP arithmetic on BBC ... the Feb. 12 2010 broadcast did not do it!

    Best Regards,

    Milo Gardner
    Sacamento,CA USA

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  • 7. At 14:13 on 3 June 2010, Milo Gardner wrote:

    Dear BM and BBC staff:

    Please note that 20th century scholars minimally added back missing scribal data and (therefore) did not report the RMP's finite math. Only the practical additive side was partly reported. The theoretical side of the finite arithmetic was censored!

    Many well-meaning Egyptology scholars continue to accept the 1920s additive premise that did not allow the two-sides of 1650 BCE finite arithmetic to be exposed.

    Ahmes doubled check the practical side of his calculations against the theoretical side of the ancient finite arithmetic. For example, 10 hekat of grain recorded as 3200 ro, meant 3200/320 hekat. 3200 ro was divided by 365 with 8 + 280/365 ro (by a unit fraction series) was the answer. Ahmes proved 8 + 280/365 ro correct by inverting the divisor 1/365 to 365 and multiplied -returning 3200/320, 10 hekat.

    If the inverse relationship of Ahmes' division and multiplication operations look familiar: you are correct. Beneath Ahmes' unit fraction arithmetic was a modern-like division operation. Ahmes did not use 'single false position' as a division operation as reported by the 1920s 'additive' scholars.

    In RMP 38 320 ro, one hekat, was multiplied by 7/22 obtaining 101 9/11 ro. Ahmes' proof inverted 7/22 to 22/7 such that:

    (101 9/11) times 22/7 = 320/320 = 1 hekat

    confirmed scribal division operation as closer to modern division's invert the divisor and multiply! "Single false position" was a scholarly guess. Ahmes ' finite arithmetic never guessed.

    I wonder how many years it will take for the Egyptology community to correctly decode the Ahmes Papyrus,and like Middle Kingdom texts (the EMLR also gathers dust in the British Museum .. read only as an additive texdt ... as the ancient texts were written?

    Modern guess work of the 1920s need to be thrown away! Start over ... adding back the correct set of arithmetic operations ... maybe 2015 will be the year of awakening ... unless the BM and BBC put their heads together before that time ... and update their misleading Feb. 12, 2010 info on the RMP (and the EMLR).

    Best Regards,

    Milo Gardner

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  • 8. At 20:55 on 28 June 2010, Milo Gardner wrote:

    THE BRITISH MUSEUM'S POSTION " Why did maths first develop? IS ANSWERED BY A DECENTRALIZED ECONOMY THAT USED EXACT RATIONAL NUMBER BASED WEIGHTS AND MEASURES.

    Complex societies like Egypt required maths to create (A DECENTRALIZED ECONOMY BASED ON EXACT WEIGHTS AND MEASURE) AND NOT buildings, manage food supplies and compute the flood levels of the Nile. The Egyptians were renowned by the Greeks as masters of geometry (FALSE ... THERE WAS NO ADVANCE MATH IN THE GEOMETRY SECTION OF THE RMP OR THE MMP). However, very little evidence of complex Egyptian mathematics survives. (FALSE ... ADVANCED MATH EXISTS IN THE 2/N TABLE ... READ BY RMP 36 .... AND OTHER PROBLEMS ... ANOTHER ADVANCED MATH DIVIDED ONE HEKAT UNITY (64/64)/n = Q/64 + (5R/n)1/320, Q = QUOTIENT AND R = REMAINDER ... EGYPTIANS CREATED FINITE mathematics ... THAT DID NOT develop in Mesopotamia, including the system of dividing time into units of 60 that is still used today for minutes and seconds. Higher maths, such as deductive reasoning and logic, developed in Greece from 500 BC onwards. ... FALSE ... EGYPTIAN LOGIC PUBLISHED 1,500 YEARS EARLIER IN PESU, INVERSE PROPORTION, AND ARITHMETIC PROGRESSIONS IN THE KAHUN PAPYRUS AND RMP 39,40 AND 64!!! THIS IS THE LOGIC THAT GREEKS COPIED ... GREEKS CREATED LITTLE ... ONLY ARCHIMEDES CREATED CALCULUS BY SOLVING A 1/4 GEOMETRIC SERIES AREA OF A PARABOLA:

    4A/3 = A+ A/4 + A/16 + A/64 + ...

    BY WRITING AN EXACT UNIT FRACTION SERIES

    4A/3 = A + A/4 + A/12

    (READ E.J DIJKSTERHUIS "ARCHIMEDES", Princeton U. Press, 1987 (page 129)

    BEST REGARDS TO AHMES AND THE EGYPTIAN SCRIBES ...GREAT NUMBER THEORY WORK ... SORRY THAT BRITS WEAR ALGORITHMIC AND GEOMETRIC BLINDERS. TAKE THEM OFF!!! THINK ANCIENT NUMBER THEORY ... Best Regards, Milo Gardner

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  • 9. At 13:46 on 30 June 2010, Milo Gardner wrote:

    Hi all:

    As an alternative to BBC and the British Museum's misleading 15 minute RMP broadcast ... aired on Feb. 9, 2010 ... a one hour Math 2.0 Webinar is being prepared: http://mathfuture.wikispaces.com/Egyptian+math
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    I'll let you folks know how the event is received.

    In summary, Fibonacci and Ahmes' rational number method that converted 4/13 to unit fraction series will be discussed. Both scribes began with 1/4. Fibonacci selected 1/18 for a second partition. The two calculations follow:

    A. Fibonacci:

    1. (4/13 - 1/4) = (16 - 13)/52

    2. (3/52 - 1/18)= (54 - 52)/936 = 1/468

    3. 4/13 = 1/4 + 1/18 + 1/468

    recorded left to right, following Ahmes' direction.

    B. Ahmes

    1. 4/13 x (4/4) = 16/52

    2. finding the divisors of 52, 13, 4, 2 and 1

    3. (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52

    recorded from right to left, absent the + sign.

    Best Regards,

    Milo Gardner

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