A decimal system without place value
Ancient Egyptians counted in tens, and their writing reflects that. Unlike a modern decimal notation, however, the hieroglyphic system did not use position. It gave a separate picture-sign to each power of ten and wrote numbers by repeating those signs. The result is a decimal additive system: the base is ten, but every digit is expressed by repetition rather than by a sign that changes value with its place.
The system is attested from very early in Egyptian history, long before the pyramids of the Old Kingdom. Among the best-known early examples is a ceremonial macehead from the time of the first kings, often cited because it records large counts of cattle, goats and captives. The exact figures and their interpretation are discussed by scholars, so it is best to treat it as evidence that large numbers were already being written in the earliest period.
The unit signs
The standard set of signs, as usually listed, is the following.
| Value | Sign (as usually described) |
|---|---|
| 1 | a single vertical stroke |
| 10 | a shape often called a heel bone or hobble |
| 100 | a coil of rope |
| 1,000 | a lotus plant |
| 10,000 | a raised finger |
| 100,000 | a tadpole or frog |
| 1,000,000 | a kneeling figure with raised arms |
The names are modern descriptions of what the pictures look like. Why the Egyptians chose these particular pictures is not recorded, and some of the ideas given in handbooks are guesses. The kneeling figure is the sign that also appears in writing for a word often rendered as “million” or “countless”, which suggests it conveyed a very great amount more than an exact one.
Combining the signs
To write a number, a scribe took the largest sign needed and repeated it as many times as necessary, then did the same for each smaller sign down to the units. No sign was repeated more than nine times, because ten of a kind could be replaced by the next sign up. Signs of the same kind were commonly arranged in tidy groups, often two or three rows high, to make them quick to count.
Take 2,413. It is 2,000 plus 400 plus 10 plus 3, so the scribe wrote:
- two lotus plants
- four coils of rope
- one heel-bone sign
- three strokes
Add them up, and 2,000 + 400 + 10 + 3 gives 2,413. The order in which the groups were placed was a matter of style and direction of writing. Hieroglyphs could run left to right, right to left or in columns, and the signs faced the direction the reader should start from. Numbers usually followed the same direction as the surrounding text, normally with the largest group first.
This is the same logic behind other additive notations, such as Roman numerals. Compared to a system with position, it is easy to learn and read, because each sign always means the same thing. It is clumsy for very large figures, as 999 needs 27 separate signs, three groups of nine.
Hieratic: a faster hand
The signs described above are the monumental hieroglyphic ones, carved on stone or painted on walls. For everyday administration, scribes wrote in a cursive script known as hieratic on papyrus and pottery. In hieratic, the numbers did not simply become quick copies of the pictures. Instead, each number up to nine, each multiple of ten and so on received its own compact sign, so a number like 7 was one stroke, not seven. This made records shorter, though the learner had to memorise more shapes. The system was still decimal and still without place value.
Fractions
Egyptian fractions are a subject of their own, but a few points belong here. The main method was to write a fraction with numerator one, called a unit fraction, by placing a sign shaped like a mouth above the number for the denominator. So the sign over a 3 meant one-third. There were special signs for one-half and for two-thirds, and the parts of the Eye of Horus were used to write some fractions of grain measures. Other fractions were expressed as sums of different unit fractions. Mathematical papyri such as the Rhind papyrus, copied by a scribe around the sixteenth century BCE, contain tables and problems that show how this was done.
Numbers and the calendar
Egyptian records tie numbers closely to time and administration. The civil calendar had twelve months of thirty days, followed by five extra days, a structure that matters in the story of calendar systems. Time at night was marked by groups of stars, the decan stars, each rising at intervals of about ten days, a convenient fit with a decimal counting habit. The numerals described here were the tools for the accounts, tax records and inventories that kept this world running.
Why it mattered
The hieroglyphic numerals show one of the oldest ways of writing large numbers. The idea of counting by repeating signs probably grew out of tally marks, and a single stroke was close to the stroke used in prehistoric counting. What the Egyptians added was a distinct sign for each power of ten, which made counting in large quantities practical. They never took the next step to place value, but their arithmetic, built on doubling and halving, was workable for the tasks of the scribal state. Compared with the Mesopotamian sexagesimal system, which did use position, their notation was simpler to read and slower to compute with.