Egyptian mathematics

The astronomical chart in Senemut's tomb, 18th dynasty

The earliest attested examples of mathematical calculations date to the predynastic Naqada period and show a fully developed numeral system. The importance of mathematics to an educated Egyptian is suggested by a New Kingdom fictional letter in which the writer proposes a scholarly competition between himself and another scribe regarding everyday calculation tasks such as accounting of land, labor, and grain. Texts such as the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus show that the ancient Egyptians could perform the four basic mathematical operations—addition, subtraction, multiplication, and division—use fractions, compute the volumes of boxes and pyramids and calculate the surface areas of rectangles, triangles, and circles. They understood basic concepts of algebra and geometry and could solve simple sets of simultaneous equations.

D22
2⁄3
in hieroglyphs
Mathematical notation was decimal and based on hieroglyphic signs for each power of ten up to one million. Each of these could be written as many times as necessary to add up to the desired number; so to write the number eighty or eight hundred, the symbol for ten or one hundred was written eight times respectively. Because their methods of calculation could not handle most fractions with a numerator greater than one, they had to write fractions as the sum of several fractions. For example, they resolved the fraction two-fifths into the sum of one-third + one-fifteenth. Standard tables of values facilitated this. Some common fractions, however, were written with a special glyph—the equivalent of the modern two-thirds is shown on the right.

Ancient Egyptian mathematicians

Ancient Egyptian mathematicians had a grasp of the principles underlying the Pythagorean theorem, knowing, for example, that a triangle had a right angle opposite the hypotenuse when its sides were in a 3–4–5 ratio. They were able to estimate the area of a circle by subtracting one-ninth from its diameter and squaring the result:

Area ≈ [(8⁄9)D]2 = (256⁄81)r 2 ≈ 3.16r 2,
a reasonable approximation of the formula πr 2.

The golden ratio seems to be reflected in many Egyptian constructions, including the pyramids, but its use may have been an unintended consequence of the ancient Egyptian practice of combining the use of knotted ropes with an intuitive sense of proportion and harmony.

Faience and glass

Ancient Egyptian medical instruments depicted in a Ptolemaic period inscription on the temple at Kom Ombo.
Even before the Old Kingdom, the ancient Egyptians had developed a glassy material known as faience, which they treated as a type of artificial semi-precious stone. Faience is a non-clay ceramic made of silica, small amounts of lime and soda, and a colorant, typically copper. The material was used to make beads, tiles, figurines, and small wares. Several methods can be used to create faience, but typically production involved the application of the powdered materials in the form of a paste over a clay core, which was then fired. By a related technique, the ancient Egyptians produced a pigment known as Egyptian Blue, also called blue frit, which is produced by fusing (or sintering) silica, copper, lime, and an alkali such as natron. The product can be ground up and used as a pigment. The ancient Egyptians could fabricate a wide variety of objects from the glass with great skill, but it is not clear whether they developed the process independently. It is also unclear whether they made their own raw glass or merely imported pre-made ingots, which they melted and finished. However, they did have technical expertise in making objects, as well as adding trace elements to control the color of the finished glass. A range of colors could be produced, including yellow, red, green, blue, purple, and white, and the glass could be made either transparent or opaque.