Counting is older than writing
People were keeping count long before they wrote numbers. Notches on bones and sticks, pebbles, knots and finger patterns all served this purpose, and the evidence for prehistoric counting is mostly tally marks such as those on the Ishango bone, whose exact purpose scholars still debate. Written numerals appeared when administrators, in places like Mesopotamia and Egypt, needed durable records of grain, livestock, land and labour.
Those first systems differed in a way that matters more than the shapes of the signs. Broadly, ancient numerals fall into three families: additive, alphabetic and place-value. Many real systems mixed features, so the families are a convenient guide rather than strict boxes.
Additive notations: repeat the sign
An additive system has a sign for one, another for ten, another for a hundred, and so on. A number is written by repeating signs until they add up to the total. The order of the signs hardly matters, because each one carries its own value wherever it stands.
The Egyptian hieroglyphic numerals work this way, with separate pictures for powers of ten, and so do Roman numerals, which use letters of the alphabet for the same purpose. The early Athenian signs known as acrophonic numerals are another example. Their great strength is readability: a scribe or a viewer can understand a small number at a glance. Their weakness is length, since a number like 888 needs many signs, and multiplying or dividing with them on paper is awkward.
Alphabetic notations: letters as digits
The second family assigns a number to each letter of an alphabet. The Greek Milesian system used the first nine letters for 1 to 9, the next nine for 10 to 90, and a third group for 100 to 900, with a few archaic letters kept to fill the gaps. Hebrew and Arabic writers developed similar schemes. Arabic scholars used what are called abjad numerals, which follow the old order of the Semitic alphabet.
This design is compact, since three symbols can write any number up to 999. It also had a side effect that shaped later culture. Because every word could be read as a number, readers could add up the letters of a name or phrase, a practice behind Hebrew gematria and the Greek habit later called isopsephy. Whether such sums carry any meaning is a matter of belief; what is historically clear is that the notation made the practice possible.
Place-value notations: position carries meaning
In a place-value system, the same sign means different amounts depending on where it stands. The 5 in 52 and the 5 in 25 are not worth the same. This allows a handful of symbols to express any number, and it makes written arithmetic far easier.
Several cultures arrived at positional notation, in different ways:
- Babylonian scribes wrote numbers with wedge-shaped marks in base 60, a form of what is often called the Mesopotamian sexagesimal system. For a long time they had no sign for an empty place.
- Chinese calculators laid out counting rods on a board in decimal positions, leaving a blank space where a digit was missing.
- Maya scribes wrote numbers vertically with dots, bars and a shell-shaped sign for zero, in a base-20 system with a modification for calendar use. The details are covered in the history of Maya numerals.
- Indian mathematicians developed ten digits with a zero in a decimal place-value system, the ancestor of the numbers we use now.
Why the empty place mattered
A positional system raises a problem that additive ones avoid: how do you show that a position is empty? Without a marker, 305 and 35 can look alike, and the reader must guess from context. Babylonian scribes did this for centuries, and an empty-place marker sign appears only later in their tradition. Chinese rod calculation used blank space. The development of a true zero, first as an empty-place marker and then as a number in its own right, is its own story, and the history of zero in India and beyond is usually told as one of the great turning points in mathematics.
Notation and calculation
The notation a society used often shaped how it calculated. Romans and Greeks did much of their arithmetic on counting boards or with a device whose descendants we call an abacus, and then recorded only the result in numerals. Abacus history shows how long such tools lived alongside written figures. A place-value system, by contrast, lets a person do long addition and multiplication directly on the page, which is one reason it eventually spread so widely.
It would be wrong to say that cultures using additive or alphabetic numerals were less capable mathematically. Greek geometers achieved remarkable results with a clumsy notation, and the Babylonians solved problems that look advanced even today. The notation affected convenience and speed more than it limited what could be thought.
How the digits we use reached Europe
The decimal digits used worldwide today descend from Indian forms, passed to the Islamic world and from there to Europe over many centuries. Arabic-language writers described the Indian methods of calculation around the ninth century, and in 1202 the Italian mathematician Leonardo of Pisa wrote a book that taught them to merchants and scholars. Even so, Roman numerals stayed in common use in Europe for a long time, and they still appear on clock faces, building inscriptions and book chapters.
A closer look at each system
The articles in this cluster take each system in turn: the Egyptian signs for powers of ten, the Athenian acrophonic numerals, the Greek letter system of Miletus, Roman numerals and their subtractive forms, Babylonian wedges, Chinese counting rods, the shapes of Brahmi numerals and their descendants, and the book that helped bring Hindu-Arabic digits to Europe.
No single system was the “best” in every setting. Each solved the problems of its own users, whether tax scribes, merchants, astronomers or teachers, and the differences between them still tell us a good deal about how people thought about quantity.