# Mathematics

## Mathematics

The earliest attested examples of mathematical calculations date to the predynastic Naqada period, and show a fully developed number 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, circles and even spheres[citation needed]. They understood basic concepts of algebra and geometry, and could solve simple sets of simultaneous equations.
2⁄3
in hieroglyphs
D22

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.[165] Because their methods of calculation could not handle most fractions with a numerator greater than one, ancient Egyptian fractions had to be written as the sum of several fractions. For example, the fraction two-fifths was resolved into the sum of one-third + one-fifteenth; this was facilitated by standard tables of values.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 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.[168] 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)r2 ≈ 3.16r2,

a reasonable approximation of the formula πr2.

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.

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