Counting: number systems around the world

A small round clay tablet engraved with a square, its two diagonals and nail-shaped signs, with added annotations. — The Babylonian clay tablet YBC 7289: a square and its diagonals, with numbers written in base 60 that give an approximate value of the square root of 2 (annotated photograph). A. Urcia, Yale Peabody Museum; labels: Theodor Langhorne Franklin, CC0, source

Writing the number twenty is easy. Writing it as a Babylonian scribe did four thousand years ago is much less so.

The difficulty is that there is not just one way of counting. There are many, and several peoples invented very different ones, each on their own.

Counting means making bundles

When we count a great many objects, we cannot name them one by one. So we group them. We make bundles, then bundles of bundles.

The size of the bundle is called the base. In our system, the base is ten. This is no accident: we have ten fingers. Our ancestors counted on their hands, and our system still bears the trace.

But there is nothing compulsory about ten.

In Mesoamerica, people counted in bundles of twenty (the fingers and the toes).
In West Africa, the Yoruba language also uses base twenty, with a subtlety: the units from one to four are added to the ten, those from five to nine are subtracted from the next ten. Fifteen is said as “five less than twenty”, eighteen as “two less than twenty”. Sixty is said as “three twenties”.¹

Two further details are worth adding. The odd tens from fifty upwards are also formed by subtraction — fifty is “sixty minus ten” — exactly as in Danish. And the base twenty probably stems from the counting of cowries, the shells that served as money.²

Yet another number system born of accounting.

In Mesopotamia, people counted in bundles of sixty.

Three independent inventions

1. Additive numeration in Egypt

In ancient Egypt, a single symbol was drawn for 1, another for 10, another for 100, and so on up to a million. To write 300, the sign for 100 was repeated three times. This is an additive system: the total value corresponds to the sum of the symbols written.³

The method is simple but takes up a lot of space: writing 276 requires fifteen signs. To overcome this drawback, the Egyptians developed hieratic script, a second, more cursive system in which the number 9,999 is written with only four signs.

2. Counting rods in China

In China, small rods were laid out on a chequerboard to represent quantities. Each column was worth ten times more than the column to its right. An empty square indicated the absence of a value. To carry out operations, these rods were physically moved.⁴

3. The sexagesimal system in Mesopotamia

In Mesopotamia, scribes pressed wedge-shaped marks into fresh clay with a stylus. These accounting practices appeared alongside urbanisation, to record harvests, allocations and debts. Before numerals were formalised, clay tokens (calculi) were sealed inside hollow clay envelopes to secure transactions.

Their base sixty still structures our measurements today:

An hour is divided into 60 minutes.
A minute is divided into 60 seconds.
A full circle measures 360 degrees (6 × 60).

Babylonian units of mass followed the same progression: 60 shekels made a mina, and 60 minas a talent. This system, devised more than four millennia ago, still governs the display of today’s watches.

Why base sixty? It is often explained by its remarkable divisibility: 60 is the smallest number divisible by 1, 2, 3, 4 and 5, and it has 12 whole-number divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60). But historians find this reason rather too scholarly, and other explanations have been proposed. It makes it easy to divide quantities without leaving awkward remainders.⁵

The emergence of place value

In the number 55, the two digits look identical but do not express the same quantity. The 5 on the left represents fifty, the one on the right represents five units.

This is the principle of place value: the position occupied by a digit determines its value. This innovation makes it possible to write any number, however large, with a small set of symbols.

This system does, however, require marking the absence of a power. In two hundred and five (205), it is essential to indicate the absence of tens to avoid any confusion with twenty-five (25).

Mesopotamian scribes long made up for this lack with a simple space, before belatedly introducing (in the second half of the first millennium BCE) a separator made of two slanting wedges. However, this sign was used only in an internal position and never acquired the status of a mathematical operator.⁶

Interactive

One number, three ways of writing it

The Egyptians repeated a sign as many times as needed. The scribes of Mesopotamia and Chinese calculators gave a digit its value by its place.

The question

To write 9,999, do you need thirty-six signs, or just four?

Without JavaScript, here are the animation’s data.

The three systems
System	Principle	Source
Egypt	Additive, base 10: one sign for 1, for 10, for 100, for 1,000…, repeated as many times as needed	1
Mesopotamia	Positional, base 60: an empty place, long marked by a simple space	2
China (rods)	Positional, base 10: each column is worth ten times the next; an empty square for no value	3

A few numbers (calculated values)
Number	Egyptian signs	Places in base 60	Columns on the board
20	2	20	2 · 0
276	15	4 · 36	2 · 7 · 6
205	7	3 · 25	2 · empty · 5
3,601	10	1 · empty · 1	3 · 6 · empty · 1
9,999	36	2 · 46 · 39	9 · 9 · 9 · 9

The values in the second table are calculated. In the animation, the Egyptian signs are drawn in simplified form, and the base-60 places are written with our digits.

The takeaway In an additive system, the number of signs grows with the number. With place value, a few places are enough, but an empty place has to be marked.

Photo credits 6

The Rhind Mathematical Papyrus, detail: Egyptian calculation problems (British Museum, EA 10057). Ahmes (scribe), Public domain, source
Tablet YBC 7289, Old Babylonian period: a square, its diagonal and base-60 numbers giving an approximation of the square root of 2 (Yale Peabody Museum). A. Urcia, Yale Peabody Museum of Natural History, CC0, source
Lead counting rods, Western Han period (National Museum of China; Xi’an, 1982). Gary Todd, CC0, source
The Egyptian sign for ten (V20 in Gardiner’s list), drawn in the Abydos font. George Douros, Public domain, source
The Egyptian sign for a hundred (V1 in Gardiner’s list), drawn in the Abydos font. George Douros, Public domain, source
The Egyptian sign for a thousand (M12 in Gardiner’s list), drawn in the Abydos font. George Douros, Public domain, source

Sources 3

O’Connor, John J. and Robertson, Edmund F., “Babylonian Numerals” et “Egyptian Numerals”, MacTutor History of Mathematics, University of St Andrews, “Egyptian numerals” (MacTutor), 4th paragraph (“The Egyptians had a bases 10 system…”) and 6th paragraph (“As can easily be seen…”).
The Egyptians had a bases 10 system of hieroglyphs for numerals. By this we mean that they has separate symbols for one unit, one ten, one hundred, one thousand, one ten thousand, one hundred thousand, and one million. […] One just adds the individual symbols, but replacing ten copies of a symbol by a single symbol of the next higher value.
Open access open the source · checked on 24 October 2026
O’Connor, John J. and Robertson, Edmund F., “Babylonian Numerals”, example of tablet AO 17264.
Perhaps the scribe left a little more space than usual between the 6 and the 9
Open access open the source · checked on 27 October 2026
O’Connor, John J. and Robertson, Edmund F., “Chinese Numerals”, MacTutor History of Mathematics, University of St Andrews, January 2004, “Chinese numerals” article, part on the counting board.
A counting board consisted of a checker board with rows and columns. Numbers were represented by little rods made from bamboo or ivory. […] There was still no need for a zero on the counting board for a square was simply left blank.
Open access open the source · checked on 27 October 2026

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Zero, invented several times

No one travelled from one continent to another to pass on the recipe for zero. Peoples very far apart came up against the same problem and solved it, each on their own.

In Central America: More than 1,600 years ago, long before the arrival of Europeans, the Maya wrote a sign for zero. In their books, it took the form of a small shell. This sign served above all to count the days in a very precise calendar.
In South-East Asia: On a stone carved in Cambodia in the year 683, archaeologists found a small dot. It is the oldest written zero of the family from which our present-day digits come.⁷

The story of this stone

It was found in 1891. For forty years, no one paid it any attention.⁸

A French scholar finally deciphered it, and understood what he was holding. He published in 1931 — and it was not a neutral article: it opens with the thesis of a British scholar, G. R. Kaye, who saw positional zero as a Western invention, not an Indian one. The stone contradicted him.⁹

The rest of the inscription deserves attention. It is not a learned text: it is a bill of sale, mentioning slaves, five pairs of oxen and sacks of white rice. The oldest known zero appears on an invoice — which confirms what the beginning of this page says: numerals are born of accounting, not of philosophy.¹⁰

The story does not end there. Under the Khmer Rouge, more than ten thousand artefacts were destroyed in Cambodia. The stone disappeared.¹¹

A mathematician spent years searching for it — in libraries, then on the ground, in Thailand, Laos, Vietnam and Cambodia. He found it again in 2013, in a shed near Angkor, among thousands of fragments of broken statues.¹²

It is now on display in the National Museum of Cambodia.¹³

The world’s oldest zero was thus lost twice: once to oblivion, once to destruction. And found twice.

In India: Around the year 628, the mathematician Brahmagupta made a crucial discovery. For him, zero was no longer merely a sign to mark an empty square: it was a true number with which one could calculate.¹⁴

Material numeration: the Andean quipus

In the Andean world, the handling of numerical data relied on quipus (sets of knotted cotton or wool cords).

Among the Incas, the structure is strictly decimal. The height of a knot on a pendant cord determines its order of magnitude (units, tens, hundreds, thousands), while the absence of a knot represents zero.¹⁵

Quipus predate the Inca Empire: the oldest known specimens are attributed to the Wari culture (sixth–eleventh centuries). Although the numerical value of the knots has been deciphered, part of the coding (colours, twists of the threads) is still the subject of research.¹⁶

The spread to the West

The digits in everyday use in Europe come from the mathematics of South Asia:

Baghdad (early ninth century): The mathematician Al-Khwarizmi wrote a treatise on Indian calculation, whose Arabic original is lost, setting out methodically for the first time the Indian positional decimal system.
Western Europe (1202): Leonardo Fibonacci published the Liber Abaci. This work introduced Indo-Arabic calculation to Italian merchants and bankers.
Gradual adoption: Despite its efficiency, this system did not finally prevail over accounting in Roman numerals until some time in the sixteenth century.¹⁷

Comparative overview of number systems

Region	Period of appearance	Base	Structure	Representation of zero
Mesopotamia	Third millennium BC	60	Positional	Empty space, then internal double wedge (c. 300 BC)
Egypt	Third millennium BC	10	Additive	No dedicated symbol
China	Fifth century BC	10	Positional	Space left empty on the chequerboard
Mesoamerica	First century BC	20	Positional	Stylised glyph (shell)
South-East Asia	Seventh century (attested in 683)	10	Positional	Dot (bindu)
South Asia	Seventh century (attested in 628)	10	Positional and operational	Circle / dot with arithmetical rules
Andes (Incas)	Sixth – sixteenth century	10	Material positional	Absence of a knot on the cord

Note 1

For instance, to form numbers “eleven” to “fourteen”, numbers “one” to “four” are added to “ten” in that order (8a-d); but to form “fifteen” to “nineteen”, numbers “five” to “one” are subtracted from “twenty” in that reverse order (8e-i). […] Five less than 20 “15”

Adeniyi, K. and Jayeola, W. A., “Tonal Alternation in Yoruba Numerals”, JOLAN: Journal of the Linguistic Association of Nigeria, vol. 23, no. 1, 2020, p. 181-190