Number systems: how the world writes numbers

Egyptian hieroglyphic numerals

Script:

Egyptian hieroglyphs

Where:

Egypt and Nubia

When:

Attested from around 3000 BCE, near the beginning of Egyptian writing itself, and in use for some three millennia

Structure:

Additive

Base:

10

Place value:

No

Zero:

No zero numeral; a word is attested

On zero. No zero numeral and no need for one. The word nfr marks a zero reference level in architectural drawings and a zero balance in accounts — a word standing in for the idea, not a digit.

Today. None. Read by Egyptologists; written by nobody.

How it composes

• One distinct sign per power of ten, from 1 up to 1,000,000.
• Each sign is repeated as many times as needed and the values are added: 243 is two coils of rope, four hobbles and three strokes.
• There is no place value, so order carries no meaning — scribes grouped repeated signs into neat blocks for legibility.
• Hieroglyphic text runs left to right or right to left (and sometimes in columns); the numerals follow whichever direction the surrounding text takes.
• Hieratic, the cursive script used for everyday documents, compressed these repeated signs into single abbreviated forms — a genuinely different notation built on the same decimal structure.

What is not settled

• Sign shapes vary considerably by period and by scribe; the forms encoded in Unicode follow one standardised palaeographic convention among several.
• The reading of the 1,000,000 sign is debated — it may have meant a specific very large quantity or simply "many", and it is rare in practice.
• Hieratic and demotic numeral forms differ substantially from the hieroglyphic ones shown here and are not modelled.

Display note. Lead with the description — "2 coils of rope, 4 hobbles, 3 strokes" — and treat the hieroglyphs as an enhancement. Most readers have no font covering U+13000–1342F and will see empty boxes. Never make the glyphs the only channel, and give any diagram real alt text describing the signs and counts.

Babylonian sexagesimal numerals

Script:

Sumero-Akkadian cuneiform

Where:

Mesopotamia — southern Iraq and the wider region

When:

Developed from Sumerian precursors around 2000 BCE and used for scribal mathematics and astronomy into the Seleucid period, roughly the last centuries BCE

Structure:

Positional

Base:

60

Place value:

Yes

Zero:

Placeholder only, not a number

On zero. No zero for most of the system's life, and never a zero that was a number. A two-wedge placeholder appears from roughly the sixth century BCE, used between digits; a value could still not end in it unambiguously.

Today. Not written today, but its base survives: 60 minutes in an hour, 60 seconds in a minute, and 360 degrees in a circle all descend from sexagesimal fractions — by way of Hellenistic Greek astronomy, not directly.

How it composes

• Two wedge signs only: a vertical wedge for 1 and a corner wedge for 10.
• Within one place, the wedges are grouped additively to write any value from 1 to 59.
• Places are positional and read left to right in descending powers of 60: the group on the left is worth sixty times the group on its right.
• Places are separated on the tablet by a space rather than any punctuation.
• There is no sexagesimal point. Whether a numeral meant 2, 120 or 1/30 was settled by context — a working astronomer knew the scale of the quantity being recorded.

What is not settled

• The absence of a place marker in the early system makes some tablets genuinely ambiguous, and modern readings of specific values are sometimes contested.
• How consistently the late placeholder sign was used, and whether any scribe treated it as a number, is debated.
• Why base 60 was chosen is unresolved. Its unusually rich divisibility (2, 3, 4, 5, 6, 10, 12, 15, 20, 30) is the standard explanation; a merger of two earlier counting traditions is another. Neither is established.

Display note. Lead with the wedge description — "2 corner wedges + 4 vertical wedges" — because the cuneiform block (U+12000 onward) renders in very few installed fonts. Show place separation with visible spacing and label which place is which; a reader cannot infer it from the glyphs.

Attic (acrophonic) Greek numerals

Script:

Greek capitals, plus compound signs found only as numerals

Where:

Attica and other Greek city-states, each with local variants

When:

Attested from roughly the 7th century BCE; largely displaced by the Ionic system during the Hellenistic period, though it survives in inscriptions for longer

Structure:

Additive

Base:

10

Place value:

No

Zero:

No zero

On zero. No zero, and no place for one — signs are repeated and summed.

Today. None. Attic numerals are read by specialists, not written by anyone.

How it composes

• Each sign is the first letter of the Greek word for its value — hence 'acrophonic'.
• Signs are repeated and added, largest first: ΔΔΠΙΙΙ is 10 + 10 + 5 + 1 + 1 + 1 = 28.
• The five-signs (50, 500, 5,000) are compounds: the sign for 10, 100 or 1,000 written inside the Π for 5.
• There is no subtraction. Four is ΙΙΙΙ.
• Athenian inscriptions also used dedicated signs for currency and weights — drachmas, talents — built on the same acrophonic principle.

What is not settled

• Local variants were the norm rather than the exception — different city-states used different signs, and the 'Attic' set is the Athenian one.
• Attic and Ionic numerals overlapped in use for a long period; the changeover was gradual and uneven, not an event.

Display note. Left to right. The compound signs for 50, 500 and 5,000 live in Unicode's Ancient Greek Numbers block (U+10140 onward) but are blank in most installed fonts, so describe them rather than emitting the codepoints.

Ionic Greek numerals

Script:

Greek alphabet, including three letters retained only as numerals

Where:

Ionia and Miletus first; then the Greek-speaking world and the Byzantine Empire

When:

In use from roughly the 5th–4th century BCE, displacing Attic numerals over the following centuries; still used for ordinal numbering in Greek today

Structure:

Ciphered additive

Base:

10

Place value:

No

Zero:

No zero

On zero. No zero. A ciphered-additive system has nothing for zero to do: each letter already states its own magnitude, so there is no column that could be empty.

Today. Greek still uses these letters as ordinals — chapter numbers, monarch and patriarch regnal numbers, list items — much as English uses Roman numerals.

How it composes

• Twenty-seven letters cover 1–9, 10–90 and 100–900, one symbol per value.
• Numbers are written in descending order and added: κδʹ is kappa (20) + delta (4) = 24.
• A keraia (ʹ), a mark like a prime, follows the numeral to show the letters are a number and not a word.
• Thousands reuse the unit letters with a mark below and to the left: ͵α is 1,000.
• Because each letter carries its own value, order is convention rather than meaning — δκ would still be 24, but nobody wrote it that way.

What is not settled

• The sign for 6 appears variously as digamma (ϝ), stigma (ϛ) or the two-letter στ, differing by period and by scribe. Numberwise shows stigma, the commonest printed form.
• Koppa for 90 has several shapes across periods (ϙ, ϟ), none of them wrong.
• Whether the Ionic system was adapted from a Phoenician-influenced model or devised independently in Miletus is not settled.

Roman numerals

Script:

Latin alphabet

Where:

Rome and its empire; then Europe, and now internationally as a stylistic convention

When:

c. 5th century BCE onwards; displaced for calculation by Hindu-Arabic numerals in Europe between roughly the 13th and 16th centuries

Structure:

Additive, with subtractive pairs

Base:

10

Place value:

No

Zero:

No zero numeral; a word is attested

On zero. No zero symbol, and no need for one: a non-positional system has no empty column to mark. The Latin word nulla (abbreviated N) appears in medieval Easter tables where a zero value was required.

Today. Still in everyday use for ordinal and decorative purposes — clock faces, book preliminaries, monarch and pope regnal numbers, film copyright dates, Super Bowl editions, and outline numbering. Nobody calculates with them.

How it composes

• Symbols are written in descending order and added: XVII is 10 + 5 + 1 + 1 = 17.
• A symbol may repeat up to three times in the canonical modern form: XXX is 30, and 40 is written XL rather than XXXX.
• Only I, X and C are subtracted, and only from the next two symbols up — giving exactly six subtractive pairs: IV, IX, XL, XC, CD, CM.
• That restriction is why 99 is XCIX (90 + 9) and not IC: I may only precede V or X.
• V, L and D are never repeated and never subtracted; each already represents a half-step, so DD would mean the same as M.

What is not settled

• Subtractive notation was never applied consistently in antiquity. IIII and XXXX are common in Roman inscriptions, and the strict six-pair form taught today was regularised in the medieval and early modern periods.
• The origin of the symbols is contested. The tally-mark account (I, V as the notch cut for five, X as a crossed notch) is well argued but not proven, and the derivation-from-Greek-letters account has its own supporters.
• The vinculum — an overline multiplying a numeral by a thousand — is attested but was never standardised, and competing conventions exist for millions. Numberwise does not generate it.

Hebrew numerals

Script:

Hebrew alphabet (right to left)

Where:

Wherever Hebrew is written; today chiefly Israel and Jewish communities worldwide

When:

Attested from roughly the Hellenistic period, likely on the Ionic Greek model; continuously in use since

Structure:

Ciphered additive

Base:

10

Place value:

No

Zero:

No zero

On zero. No zero. Nothing in the system needs one, and modern Hebrew writes 0 with the Hindu-Arabic digit.

Today. Ordinary numbers in modern Hebrew are written with Hindu-Arabic digits. The letter numerals remain standard for Hebrew calendar dates, Torah and Talmud references, and ordinal numbering — the way Roman numerals survive in English.

How it composes

• Twenty-two letters cover 1–9, 10–90 and 100–400.
• Letters are written in descending order and added: כד is kaf (20) + dalet (4) = 24.
• Above 400, tav (400) is repeated: 500 is תק, 800 is תת.
• 15 and 16 break the pattern. Written by the ordinary rule they would be יה and יו, which spell forms of the divine name, so tradition writes טו (9+6) and טז (9+7) instead.
• A geresh (׳) marks a single-letter numeral; a gershayim (״) sits before the last letter of a longer one, signalling that the letters are a number rather than a word.
• Thousands are usually written separately, often with their own geresh, rather than by extending the letter values.

What is not settled

• The 15/16 substitution is universal in practice but its earliest attestation is not precisely dated.
• Final letter forms (ך ם ן ף ץ) are sometimes assigned 500–900 in gematria; this is a gematria convention rather than standard numeral notation, and Numberwise does not use it for writing numbers.
• Whether the system was borrowed from Ionic Greek or developed in parallel is argued both ways.

Chinese numerals

Script:

Chinese characters (Han)

Where:

China, Taiwan, Hong Kong, Macau and Singapore; the same characters are read with different pronunciations in Japanese, Korean and Vietnamese

When:

Number characters appear on Shang oracle bones from around the 13th century BCE and have been in continuous use since

Structure:

Multiplicative-additive

Base:

10

Place value:

No

Zero:

Has a zero, used as a number

On zero. 〇 (and the fuller 零) is a genuine zero, used both as a digit and as a placeholder inside larger numbers. It entered Chinese numeral writing comparatively late — counting rods marked an empty place with a blank space first.

Today. Fully current. Hindu-Arabic digits are used for most arithmetic and prices, while the characters remain standard in prose, dates, formal documents and vertical text. The financial forms are legally required on Chinese cheques and many contracts.

How it composes

• Digits 一 to 九 combine with the named powers 十, 百, 千, 萬.
• A digit written before a power multiplies it, and the terms are added: 四十五 is four tens plus five.
• Ten to nineteen are conventionally written without the leading 一 — 十一 rather than 一十一 — when standing alone.
• Large numbers group by ten thousands (萬), not by thousands, so 100,000 is 十萬, ten wàn.
• The financial forms 壹貳參肆伍陸柒捌玖拾 replace the ordinary digits on cheques and contracts, because the simple characters can be altered with a single added stroke.
• 〇 or 零 marks a skipped power inside a larger number: 一百零五 is 105.

What is not settled

• Whether 〇 was borrowed from Indian notation, developed from the blank space left by counting rods, or arrived by both routes is not settled.
• Regional and register variation is real: 兩 rather than 二 before some measure words, and differing conventions for reading years digit by digit.
• Financial character sets differ slightly between mainland simplified usage and traditional usage in Taiwan and Hong Kong.

Chinese rod numerals

Script:

Arrangements of physical counting rods, and their written imitation

Where:

China, and adopted in Japan, Korea and Vietnam

When:

In use from at least the Warring States period (roughly the 5th–3rd century BCE) until the abacus displaced them, largely by the Ming dynasty

Structure:

Positional

Base:

10

Place value:

Yes

Zero:

Placeholder only, not a number

On zero. An empty place was originally left blank on the counting board. A written 〇 for the empty place appears later, once rod arrangements began to be recorded on paper rather than laid out physically.

Today. None, beyond the study of historical mathematics.

How it composes

• A fully decimal positional system, worked on a ruled counting board.
• Each place holds one to nine rods; a rod crossing the others counts as five, so 7 is one crossing rod plus two.
• Alternate places alternate orientation — vertical for units, horizontal for tens, vertical for hundreds — which is what keeps neighbouring places legible.
• An empty place is simply left blank on the board.
• Red and black rods (later, differently marked ones) distinguished positive from negative quantities — a working notation for negative numbers well before Europe accepted them.

What is not settled

• Dating the system's origin depends on textual references that predate surviving physical evidence.
• How early the blank place was consistently treated as a zero, rather than a gap needing interpretation, is debated.

Display note. Unicode encodes rod numerals at U+1D360 onward, with almost no font coverage, and inline text cannot convey the alternating orientation that makes the system readable. Present as a diagram with descriptive alt text.

Maya numerals

Script:

Maya hieroglyphic writing

Where:

The Maya region — southern Mexico, Guatemala, Belize, and western Honduras and El Salvador

When:

Attested in the Maya Long Count from the first centuries BCE and used through the Classic period (roughly 250–900 CE) and beyond; the underlying vigesimal counting is shared across Mesoamerica and older than the Maya evidence for it

Structure:

Positional

Base:

20

Place value:

Yes

Zero:

Has a zero, used as a number

On zero. A genuine positional zero, written as a shell, developed independently in Mesoamerica. Whether Maya mathematicians also treated zero as a number to calculate with — as Brahmagupta later did in India — is not clear from the surviving record, which is overwhelmingly calendrical.

Today. Not used for ordinary arithmetic today, though the numerals appear in Maya cultural and educational contexts and several Mayan languages still count vigesimally.

How it composes

• Each place holds a value from 0 to 19, written as up to three bars (five each) with up to four dots (one each) above them.
• Places stack vertically, smallest at the bottom; each place up is worth twenty times the one below.
• A shell fills a place that would otherwise be empty — the system's zero.
• Ordinary vigesimal places run 1, 20, 400, 8,000.
• The Long Count calendar uses the same marks with a modified third place worth 18 × 20 = 360 rather than 400, so that one cycle approximates a solar year. Calendar notation and vigesimal arithmetic are not the same system.

What is not settled

• Almost all surviving Maya numerals are calendrical. How the system was used for everyday commercial arithmetic is inferred rather than directly attested.
• Whether the positional zero originated with the Maya or with an earlier Mesoamerican tradition — Olmec or Epi-Olmec — is unresolved; the earliest Long Count dates are not Maya.
• The bar-and-dot numerals also appear in head-variant and full-figure forms, where each number is written as a deity's portrait. Those are the same values in an entirely different visual register.

Display note. Describe the marks — "2 bars + 3 dots over 1 dot" — as the primary channel; Unicode's Mayan Numerals block has almost no font coverage. Vertical stacking carries the place value, so any diagram must show it and any alt text must state which group is the upper place.

Base 2 — binary

A switch is reliably on or off and unreliably anything between. Two states is what electronics can hold without error, so the base is a physical constraint rather than a mathematical preference.

Where you meet it: Every digital device; file sizes; the powers of two that recur through computing.

On this site: 2, 4, 8, 16, 32, 64

Base 8 — octal

One octal digit is exactly three bits, which suited machines with word sizes divisible by three.

Where you meet it: Unix file permissions — the 755 in chmod 755 is three octal digits, one per permission group.

On this site: 8, 64

Base 10 — decimal

Ten fingers. There is no mathematical case for it: 12 divides more evenly and 2 suits machines better. Anatomy settled the question long before anyone thought to compare.

Where you meet it: Ordinary written numbers everywhere; the metric system; percentages.

On this site: 10, 100

Base 12 — duodecimal

12 divides by 2, 3, 4 and 6 where 10 divides only by 2 and 5, which matters enormously when goods are split by hand rather than by calculator. One account of its origin counts the three finger joints on each of four fingers with the thumb, giving twelve per hand.

Where you meet it: Dozens; inches in a foot; months; hours on a clock face; old British currency.

On this site: 12, 24, 60

Base 16 — hexadecimal

One hex digit is exactly four bits, so a byte is always two characters and the mapping to binary needs no arithmetic. Decimal has no such alignment, which is why nobody writes byte values in decimal.

Where you meet it: Memory addresses; colour codes such as #FF0000; MAC addresses; hashes.

On this site: 16, 32, 64

Base 20 — vigesimal

Fingers and toes. Counting the whole body rather than the hands gives twenty, and the pattern appears independently in Mesoamerica, in Celtic languages and in West Africa.

Where you meet it: Maya numerals; French quatre-vingts for eighty; the English 'score'; Danish number words.

On this site: 20, 80

Base 60 — sexagesimal

60 divides by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, so most fractions of it come out whole. Whether that is why the Babylonians chose it is not established, but it is why it survived.

Where you meet it: Minutes and seconds, of time and of angle; the 360-degree circle.

On this site: 12, 24, 30, 60