Middle Kingdom of Egypt by extrapolating dates of ancient sites constructed along its course. In 2011, archaeologists from Italy, the United States, and Egypt excavating a dried-up lagoon known as Mersa Gawasis have unearthed traces of an ancient harbor that once launched early voyages like Hatshepsut's Punt expedition onto the open ocean. Some of the site's most evocative evidence for the ancient Egyptians' seafaring prowess include large ship timbers and hundreds of feet of ropes, made from papyrus, coiled in huge bundles. In 2013, a team of Franco-Egyptian archaeologists discovered what is believed to be the world's oldest port, dating back about 4500 years, from the time of King Cheops on the Red Sea coast near Wadi el-Jarf (about 110 miles south of Suez). Mathematics 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, calculate the areas of rectangles, triangles, and circles and compute the volumes of boxes, columns and pyramids. They understood basic concepts of algebra and geometry, and could solve simple sets of simultaneous equations. 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 knew the Pythagorean theorem as an empirical formula. They were aware, for example, that a triangle had a right angle opposite the hypotenuse when its