ANCIENT EGYPTIAN MATHEMATICS

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1 ANCIENT EGYPTIAN MATHEMATICS Searching for the Origins of Math Nida Haider University of Houston-MATH 4388

2 ANCEINT EGYPTAIN MATHEMATICS Searching for the Origins of Math In the study of mathematics, the heritage of the concepts and theorems has always been of remarkable interest. Mathematical laws and theorems are often named after those to whom the origins can be traced back to. Here it is worth pondering what origin really means. How is a math concept attributed to a single person when it is just another building block resting on so many others before it? Titles like father of algebra or father of geometry have been given to mathematicians who have accumulated and built upon the discoveries of people past. The origins of most mathematical concepts are traced back to Ancient Greece. Euclid is often mentioned as the father of mathematics while Thales is referred to as the first mathematician. Is Greece really where it all started or were there a people who were rich in mathematical understanding even before the Greeks? Could these people be the harbingers of the concepts claimed as originating elsewhere? The Ancient Egyptian Empire from over 7000 years ago has made some of the greatest contributions to early mathematics. The Nile flooded every year, leaving behind it rich and fertile soil providing the stability and economic security that enabled the Egyptians to make these monumental contributions. Much evidence has been unearthed that clearly indicates the depth of the mathematical understanding and interest of the Ancient Egyptians. The initial problem faced by those who studied the details of these artifacts was that they were unable to decipher what was being said. For 3000 years the hieroglyphs and Egyptian scripts were dismissed as primitive until they were finally deciphered and translated by Jean-Francois Champollion in the late 19 th century. The Egyptians used three different types of writing systems. The famed hieroglyphics were used around 3400 BC and represented numbers using base of ten. A single stroke was used for numbers one to nine

3 while different symbols were used for ten, hundred, thousand, ten-thousand and one million. The symbol for millions was Heh the God of unending. 2 Story of Mathematics. Rooney 2015 To express, for example 3425 would mean three flower glyphs, four whirls, two arches and 5 strokes. Sometimes the hieroglyphs were not written any sort of order. Since each number had its own symbol they could be arranged for beauty and still be understood. The two ciphered systems were hieratic and demotic of which hieratic came first. It was used during the ear of the New Kingdom from BC. The system uses separate symbols for not only numbers one through nine but for each multiple of ten. It was a system developed for the priests and philosophers to maintain the secrecy and exclusiveness of a leisure club where ideas and concepts were discussed. The numbers became more compact but harder to read.

4 One of the richest source of Egyptian math is the Rhind papyrus purchased by Henri Rhind in 1858 after it was unearthed during an illegal excavation near the memorial temple of Pharaoh Ramesses II 1. The Rhind papyrus is also known as the Ahmes papyrus after the scribe who copied it from an original prototype from 2000 BC. The manuscript can be divided into three sections. The first on arithmetic and algebra; the second on geometry and the third on miscellaneous math. A major part of the document is made of 87 miscellaneous problems. Some are simple and were used in educating the youth. The first few problems talk about dividing different numbers of bread amongst different number of men. Some of the answers lead to fractions. The Ahmes papyrus was written in hieratic where fractions were written in a very peculiar and particular way. Mathematicians are still intrigued by the way each fraction was written, as a sum of non-repeated unitary fractions. The Leather roll that was discovered with the Rhind papyrus has tables of the sums of unitary fractions. How these sums were accomplished and how was it decided which unitary fraction to use for the sum is still a guess. Hyroglphs were also used to represent fractions. There were special symbols for half and two-thirds while others were drawn as sums of non-repeated unitary fractions. 2 Fractional parts of an eye glyph (Eye of Horus) were also used to represent fractions, where the required fraction was drawn as a sum of the parts of the eye. 2 5 = 6 15 = = Fractions as a sum of non-repeated unitary fractions

5 The other questions form the papyrus concern area. This was an extremely important math concept during that era. Early math was more of a problem-solving tool. Ancient mathematicians were mainly concerned with how to use math to make life easier. Every year as the Nile flooded the demarcated boundaries of land vanished. The King then sent surveyors that used rope to measure the land lost for recompense or to redraw the lost boundary. These surveyors were called rope stretchers and required a good understanding of the relationships between side length and area. Herodotus credited the Egyptians as the pioneers of geometry. He said Sesostris made a division of the soil amongst the inhabitants.if the river carried away any portion of a man s lot.the King sent persons to examine, and determine by measurement the exact extent of the loss By this practice, I think, geometry first came to be known in Egypt from whence, it passed into Greece 2 Measurements were first made standard by body parts where 1 palm was the breath of the hand and 1 cubit was seven palms or from elbow to fingertips. This caused ambiguity as some had bigger hands than others. Later on cubit sticks carved from wood were used to maintain standardization. Area was measured in different units. The setat was the basic unit of land measure and may have also varied in size. Later, it was equal to one square khet and 1 khet was equal to 100 cubits. The setat could be divided into strips one khet long and ten cubit wide where such a strip was called a kha. Ropes were knotted at equal intervals to facilitate the measuring of large land areas and recording large numbers was never a problem. There are records of numbers recorded in the millions, precise down to the ones place indicating the meticulousness of record keeping. 2

6 The Ahmes papyrus also has its share of mathematical puzzles and their solutions indicating that for this Ancient civilization math was not only about application but also taught for the sake of broadening the minds of the young. The advent of geometry is attributed to the Greek mathematicians while the Arabs are honored with the advent of algebra, however it is pretty clear that both the disciplines were pioneered by the Egyptians. Area of triangles and circles were being calculated and 3 was used for Pi 1. Right angles were ensured by using ropes knotted at units of 3, 4 and 5 forming a Pythagorean triple. This is why right-triangles with these unit lengths are also quite aptly referred to as the Egyptian. There are problems in the papyrus that asks to solve for an unknown value. Although the algebra and geometry of the Egyptians was not in the form we see today it is safe to say that the first building blocks of the disciplines can be found here. Other papyri like the Moscow papyrus and the Kahun papyrus from the 12 th dynasty contain complex calculations involving surface area and volume of irregular objects. There are also a close attempt to the squaring of the circle where Pi is set as The volume of irregular objects is calculated. The frustum of a square pyramid to be exact. The scribe directs to take the measure of the top and bottom base into account. Add the product of the measures to the square of the sums and finally multiply it with onethird of the height. If the top base is taken to be zero one arrives at the modern formula for the volume of a pyramid. The pyramids themselves have intrigued mathematicians, historians and philosophers alike. The precision of these megalithic monuments could not have been possible without the complex understating of numbers. The Great Pyramid has a perfect angle of with the horizon on all sides and exhibits a precise concavity on each face down the middle. The architects measured slope as run over rise. The word seqt meant the measure between the horizontal movement of an oblique line form the vertical axis as per change in height.

7 It is a well-known historical fact that trade brought the shores of Egypt and Greece into close and constant contact. Thales of Miletus himself is said to have been schooled in Egypt during his early years. His study of mathematics must have been greatly influenced by the mathematics passed down form the Ancient Egyptians. His return to Greece spread his ideas and learnings to the mathematicians of the future. Miletus is known as the first mathematician and was titled as one of the seven wise men by Plato. He is where the timeline of deductive mathematics starts, however it does not discredit him in the least to say that the Ancient Egyptians set the stage and that most deductions closely after were made on their knowledge. So astonishing is their mathematical accomplishment when viewed in reference to their ancient era that some enthusiasts have even suggested aliens teaching these ancestors. It would be foolish to redraw the beginning of the mathematical timeline to start form the Ancient Egyptians for surely there were people before them that must have contributed to the understandings of the Egyptians. However, it is safe to say that in Ancient times the example of the greatness of the Egyptian empire is unparalleled as is their contribution to early mathematics and that they were part of building the solid foundation on which future math stands. Bibliography For any astonishing fact the superscript indicates the number of the source in the list of citations the information was taken from.

8 Works Cited 1. Rooney, Anne. The Story of Mathematics. 1st ed., London, Arcturus, Merzbach, Uta C., and Carl B. Boyer. The History of Mathematics. 5th ed., Hoboken, NJ, John Wiley & Sons, Mastin, Luke. Egyptian Mathematics. Egyptian Mathematics - The Story of Mathematics, 2010, Accessed 17 Sept Rhind Mathematical Papyrus. Wikipedia, Wikimedia Foundation, 30 Aug. 2017, en.wikipedia.org/wiki/rhind_mathematical_papyrus. Accessed 17 Sept Class Lecture Notes 1-10

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