Ancient Egyptian mathematics is the mathematics, developed and used in Ancient Egypt c. 3000 to c. 300 BCE, from the Old Kingdom of Egypt until the beginning of Hellenistic Egypt. The ancient Egyptians utilized a numeral system for counting and solving written mathematical problems involving multiplication and fractions. Evidence for Egyptian mathematics is limited to a scarce amount of surviving sources written on papyrus. From these texts it is known that ancient Egyptians understood concepts of geometry, such as determining the surface area and volume of three-dimensional shapes useful for architectural engineering, algebra, such as the false position method and quadratic equations. Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found in Tomb U-j at Abydos; these labels appear to have been used as tags for grave goods and some are inscribed with numbers. Further evidence of the use of the base 10 number system can be found on the Narmer Macehead which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners.
The evidence of the use of mathematics in the Old Kingdom is scarce, but can be deduced from inscriptions on a wall near a mastaba in Meidum which gives guidelines for the slope of the mastaba. The lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement; the earliest true mathematical documents date to the 12th dynasty. The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period; the Rhind Mathematical Papyrus which dates to the Second Intermediate Period is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so called mathematical problem texts, they consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems.
An interesting feature of Ancient Egyptian mathematics is the use of unit fractions. The Egyptians used some special notation for fractions such as 1 2, 1 3 and 2 3 and in some texts for 3 4, but other fractions were all written as unit fractions of the form 1 n or sums of such unit fractions. Scribes used tables to help them work with these fractions; the Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain 2 n tables; these tables allowed the scribes to rewrite any fraction of the form 1 n as a sum of unit fractions. During the New Kingdom mathematical problems are mentioned in the literary Papyrus Anastasi I, the Papyrus Wilbour from the time of Ramesses III records land measurements. In the workers village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs. Current understanding of ancient Egyptian mathematics is impeded by the paucity of available sources.
The sources that do exist include the following texts: The Moscow Mathematical Papyrus The Egyptian Mathematical Leather Roll The Lahun Mathematical Papyri The Berlin Papyrus 6619, written around 1800 BC The Akhmim Wooden Tablet The Reisner Papyrus, dated to the early Twelfth dynasty of Egypt and found in Nag el-Deir, the ancient town of Thinis The Rhind Mathematical Papyrus, dated from the Second Intermediate Period, but its author, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text. From the New Kingdom there are a handful of mathematical texts and inscriptions related to computations: The Papyrus Anastasi I, a literary text written as a letter written by a scribe named Hori and addressed to a scribe named Amenemope. A segment of the letter describes several mathematical problems. Ostracon Senmut 153, a text written in hieratic Ostracon Turin 57170, a text written in hieratic Ostraca from Deir el-Medina contain computations. Ostracon IFAO 1206 for instance shows the calculation of volumes related to the quarrying of a tomb.
Ancient Egyptian texts could be written in either hieroglyphs or in hieratic. In either representation the number system was always given in base 10; the number 1 was depicted by a simple stroke, the number 2 was represented by etc.. The numbers 10, 100, 1000, 10,000 and 1,000,000 had their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog, a million was represented by a god with his hands raised in adoration. Egyptian numerals date back to the Predynastic period. Ivory labels from Abydos record the use of this number system, it is common to see the numerals in offering scenes to indicate the number of items offered. The king's daughter Neferetiabet is shown with an offering of 1000 oxen, beer, etc; the Egyptian number system was additive. Large numbers were represented by collections of the glyphs and the value was
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