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## Egyptian Mathematics

The most we know about Egyptian mathematics is from the papyrus Rhind, which contained 85 problems, and the Moscow papyrus, containing 25 problems. The papyri were dated about 1700 BC to 1550 BC. The Rhind Papyrus was the largest and best preserved mathematical papyrus from ancient Egypt. It is a copy of papyrus from around 1850 BC. (a portion of the Rhind Papyrus, Wikimedia: Unknown – British Museum, EA10057)

The Egyptians had a decimal system with specific signs, but it was different from the modern place value system.

The mathematics was mostly additive. For instance, in order to multiply 11 and 13, the number 13 was written as a sum of powers of $2$, i.e. $1,4,8$, then the number $11$ was multiplied by those numbers, i.e. $11,44,88$ and the numbers were added $11+44+88=143.$

The most prominent feature of Egyptian mathematics was its fractional arithmetics, which was based on the so-called unit fractions, i.e. fractions with the nominator $1.$ The Rhind Papyrus contains a table with the decomposition of the fractions $\frac 2n$ for $n=5,\ldots,331$, for instance, $$\frac 27=\frac 14+\frac 1{28},\quad \frac 2{97}=\frac 1{56}+\frac 1{679}+\frac 1{776}.$$
This kind of fractional arithmetics was cumbersome and prevented a substantial development of mathematics in Ancient Egypt.

The problems dealt with contents of bread and beer, with the feeding of animals or with the volumes of granaries. Some of them were more theoretical, for instance, containing geometrical progressions, like the problem dealing with $7$ houses, in each of which there were $7$ cats, each which chased $7$ mice, etc. The area of the circle with the diameter $d$ was given as $(d-\frac d9)^2,$ which corresponds to the value $\frac{256}{11}\approx 3.1605$ for the number $\pi.$ The most remarkable result in the papyri is the formula for the volume of a pyramid trunk with a square profile: $V=\frac h3(a^2+ab+b^2)$ with $a,b$ being the lengths of the sides of the profile and $h$ being its height, because the Pythagorean theorem was not yet known.