What “oldest known” means here
The Egyptians did not invent calculation, and nobody can say who made the first. Counting, tallying and measuring are far older than any writing, and Mesopotamian scribes were recording sophisticated arithmetic on clay tablets in the same broad period. What Egypt provides is a handful of well-preserved papyri from the early second millennium BCE and later that show how trained scribes handled everyday problems. Those papyri are important surviving sources, not the first mathematics ever done.
Numbers and calculation
Egyptian numerals were written in hieroglyphs and, in everyday documents, in the faster hieratic script. The system was decimal, with separate signs for 1, 10, 100, 1,000 and higher powers of ten, but it had no place value, so a number was built by repeating signs. A scribe wrote, for instance, 243 with two hundred signs, four ten signs and three single strokes. The base of the system is similar to ours, but the way of writing it is not.
Multiplication was done by doubling and adding. To work out 12 times 13, a scribe would double 13 repeatedly: 13, 26, 52, 104. Since 12 is 8 plus 4, he would add the lines for 8 and 4, giving 104 plus 52, or 156. Division worked backwards in the same way. It is a method that needs only addition and doubling, and it works for any whole numbers.
Unit fractions
The most distinctive feature of Egyptian arithmetic is how it handles fractions. With one special sign for two thirds, the Egyptians wrote nearly every fraction as a sum of unit fractions, which have 1 on top. So two fifths was written as one third plus one fifteenth, and a check shows why: one third is five fifteenths, and adding one fifteenth gives six fifteenths, which reduces to two fifths. Likewise, two sevenths was written as one quarter plus one twenty-eighth, because seven twenty-eighths plus one twenty-eighth equals eight twenty-eighths, or two sevenths.
The Rhind papyrus opens with a long table giving such expansions for fractions with 2 on top and odd numbers below, up to 101. Scribes could look up values rather than work them out each time. Why the Egyptians preferred this approach is not fully understood. It may have suited the practical business of dividing rations, since shares like “a third plus a fifteenth” can be described in terms of objects that can actually be cut up.
A related set of fractions, halving down from one half to one sixty-fourth, is traditionally linked to the parts of the Eye of Horus, and was used for measures of grain. Their total is sixty-three sixty-fourths, not a whole, and scholars debate whether the mythological link was really the origin of these measures.
The Rhind papyrus
The Rhind papyrus is named after the Scottish collector Alexander Henry Rhind, who bought it in Luxor in the nineteenth century, and is now in the British Museum. The scribe Ahmes (Ahmose) wrote it around 1550 BCE and says he was copying an older work, which scholars place in the Middle Kingdom. It contains roughly 84 problems, depending on how they are counted.
Its problems cover dividing loaves of bread among a number of people, working out the capacity of granaries, the amount of grain needed to make beer or bread of a given strength, and finding the area of fields. One famous example takes the area of a circle to be that of a square whose side is eight ninths of the circle’s diameter. That amounts to a value for pi of about 3.16 (256 divided by 81 is just over 3.16), reasonably close to the true value of 3.14159. The scribe did not state it as pi, and there is no sign the Egyptians thought in terms of that constant.
The Moscow papyrus
The Moscow papyrus, now in the Pushkin Museum in Moscow, is somewhat older, probably from about 1850 BCE, and has about 25 problems. Its best-known problem is the volume of a truncated pyramid, a pyramid with the top cut off. The solution matches the correct modern formula: one third of the height times the sum of the two base areas and the product of the sides. That a scribe knew this is impressive, though how the rule was found is not recorded.
Geometry for land and building
Practical geometry was needed in a country where the Nile flood covered fields each year and boundaries had to be re-established afterwards. Land was measured in units based on the cubit, about half a metre, and a field area was calculated from its sides. The Rhind papyrus gives the areas of rectangles, triangles and the circle approximation described above, with rules that are correct for the first two and approximate for the others.
For building work, the Egyptians used the seked, a measure of slope given as how far a sloping face moves horizontally for each cubit of rise, counted in palms and fingers. The Rhind papyrus includes problems that calculate the seked of a pyramid from its dimensions. These show that the scribes had tools for the sort of figures needed in the Giza pyramid complex, although how the pyramids were actually laid out and built is still a matter of pyramid construction debates, and the papyri were not building manuals.
Weights, measures and accounts
Numbers also ran the state’s accounts. Scribes recorded rations, taxes and labour, and standard units of length, area and capacity were needed so that people on opposite sides of the country could agree. Egypt was one of several early states to standardise ancient weights measures, since fair taxation and trade depend on units everyone accepts.
What the sources do not show
Egyptian mathematics, as recorded, consists of worked examples and rules, with no theorems or proofs of the kind later Greek mathematicians developed. A few papyri are a small sample, and the pieces that survive were chosen by luck. Compared with the base-60 arithmetic of Mesopotamia, built on the sexagesimal number system, the Egyptian approach was less flexible for fractions but fully adequate for the tasks scribes performed.