From counting marks to the language of infinity. Explore when notation appeared, how it changed, and the ideas that moved mathematics forward. Every numeral system in the collection is included, with approximate origins or dated evidence where known.
⅟n=∫∀MARKS BECOME A LANGUAGE
5,000 years, at a glance
40 numeral systems · 84 entries · 3 ways to explore
3000 BCE2000 BCE1000 BCE1 CE1000 CEPresent
Bar height shows entries in this collection, not the pace of discovery. Select a period to explore. Traditions with unknown origins appear after the dated entries.
Periods are shown on the same 3000 BCE–present scale. Solid bars show selected attested periods; arrows mark living notation. A lone dot means no reliable end date is assigned. The date filter includes periods that overlap your selection; unknown origins appear only with all history selected; undated endings are matched by their starting evidence only.
c. 3000 BCEEgypt
Numeral systemsHistorical
Egyptian hieroglyphic numerals
Separate signs for powers of ten make large quantities visible in writing.
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Hieroglyphic numerals and the more compact hieratic forms coexisted for centuries. Changes in script and practice were gradual; this entry does not assign one final year to all Egyptian numeral use.
The position of a cuneiform number determines its value in a base-60 system.
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The period shown is the selected Old Babylonian variant used in this collection, c. 1800–1600 BCE. Later Babylonian mathematics continued; 1600 BCE is not an extinction date for base 60.
A surviving collection of practical problems covers fractions, arithmetic, areas, and volumes.
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The British Museum dates this manuscript to about 1550 BCE. It records a scribal mathematical tradition, rather than the invention of arithmetic on that date.
Shang inscriptions preserve a decimal tradition with additive and multiplicative structure.
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This approximate date marks early written evidence. Modern Chinese numerals descend from evolving written forms; today’s characters are not identical to the oracle-bone signs.
Period of use An evolving tradition · still in use
1st millennium BCELevant and Mediterranean trading networks
Numeral systemsHistorical
Phoenician
Right-to-left additive notation with signs for units, ten, twenty, and one hundred.
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This broad period locates Phoenician and Punic numerical practice. Its first year is only a browsing anchor; the numeral repertoire has no precisely established invention date. Phoenician writing spread across Mediterranean networks and continued in Punic forms. Its numerical notation uses repeated signs, with dedicated marks for ten, twenty, and hundred alongside small tally groups.
Period of use Phoenician and Punic traditions · 1st millennium BCE onward
Phoenician and Punic traditions · 1st millennium BCE onward
c. 6th century BCEAttica, ancient Greece
Numeral systemsHistorical
Greek Attic
Number signs based on the initial letters of Greek number names.
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This approximate date locates the Attic acrophonic tradition within the first millennium BCE. Regional forms and earlier acrophonic practices vary. The Attic acrophonic system appears in ancient inscriptions, including financial records. Several of its signs derive from initial letters of Greek number words. Regional acrophonic systems also existed; the shapes and values shown here belong to the Attic numerical repertoire.
Greek letters represent units, tens, and hundreds.
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Alphabetic numerical inscriptions are attested at the tunnel of Eupalinus on Samos around this date; the convention’s precise invention is unknown. The alphabetic system, also called Milesian or Ionic, assigns numerical values to twenty-seven letters. Three additional signs retain places in the sequence beyond the standard twenty-four-letter Greek alphabet. Its forms evolved in manuscripts and printing, and Greek numerical letters are still used in particular contexts.
Additive cuneiform signs for units, ten, twenty, and one hundred.
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Achaemenid inscriptions attest the selected numeral tradition from around this period. The date is approximate. Old Persian script was developed for royal monumental inscriptions associated with Darius I. Its wedge-shaped signs are a distinct writing system, not simply Babylonian cuneiform with the same mathematical rules.
Period of use Achaemenid inscriptions · from c. 525 BCE
Letters represent values that are added or subtracted: XLII = 50 − 10 + 2.
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Roman numerals developed gradually from earlier Italic practices. This approximate origin does not date today’s standardized subtractive spellings. Roman numeral conventions developed over centuries and varied in inscriptions, manuscripts, and later printing. They remain visible on clocks, monuments, and numbered volumes. This converter uses the common modern subtractive convention, a standardized selection from that wider historical practice.
Rods on a counting board turn decimal place value into a physical calculation tool.
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Counting boards are described from the fourth century BCE. Abacus use later expanded, but no single year marks the disappearance of rods. The collection demonstrates a later Song-period orientation convention.
Dots represent one, bars five, and a shell-shaped sign zero. Vertical positions count powers of twenty.
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A San Bartolo mural preserves the calendar notation “7 Deer” in this date range. The timeline uses its earlier boundary as an evidence date, not the invention of the complete positional system or zero. Maya scribes recorded numbers in monuments and manuscripts using dots for units, bars for fives, and several forms of zero. Numbers were used in astronomical and calendar records. This display uses the familiar dot-and-bar forms rather than the elaborate head variants.
Period of use Pre-Columbian; living mathematical heritage
Early Indian numeral signs become ancestors of many later digit forms.
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Early Brahmi notation was not positional. Inscriptions attest these forms into the fourth century CE; the displayed span is an evidence window, not a precisely dated disappearance.
A foundational Chinese text brings together practical arithmetic, geometry, and methods for simultaneous equations.
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This is a browsing date within a debated compilation period, roughly 200 BCE–50 CE, for a text with multiple layers. Liu Hui’s influential commentary follows in 263 CE.
c. 1st century CE · selected formsGandhāra · northern Pakistan and eastern Afghanistan
Numeral systemsHistorical
Kharoshthi
Right-to-left notation with signs for one, two, three, four, ten, twenty, and one hundred.
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The selected manuscript forms belong to the first century CE. Kharoshthi writing is older; this date identifies the represented evidence, not the invention of all Kharoshthi numerals. Kharoshthi accompanied Gāndhārī manuscripts and inscriptions in the northwest of the Indian subcontinent. The numeral shapes here follow the first-century manuscript tradition documented by Unicode.
Period of use Manuscript forms · c. 1st century CE
Chinese counting-board methods extend powers of ten below one, making tenths and hundredths part of calculation.
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A fifth-century mathematical manual attributed to Xiahou Yang describes multiplying or dividing by powers of ten by shifting counting rods. These methods show an understanding of decimal fractions: the same place-value pattern can handle parts of a whole as well as whole numbers. Counting in base ten is much older than this; decimal fractions are the further step. The date marks textual evidence, not a known first invention or the appearance of today’s decimal point.
Brahmagupta sets out rules for zero and for positive and negative quantities.
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His Brahmasphutasiddhanta treats zero as part of arithmetic. This is a textual milestone, not the first appearance of a zero placeholder anywhere; his division rules also differ from modern ones.
725 CE · Bede’s accountNorthumbria, England; wider Latin manuscript tradition
Numeral systemsHistorical
Medieval European Finger Counting
Human hands become a compact number display: the left carries units and tens, the right hundreds and thousands.
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Bede describes finger reckoning in De temporum ratione. This dates a surviving account of an older practice, rather than its invention. Bede opened De temporum ratione in 725 with instructions for numerical hand positions before explaining calendars and the computation of Easter. His account preserves an older Mediterranean tradition. Medieval authors transmitted and modified these positions; Fibonacci’s later placement of hundreds and thousands differs from Bede’s.
Period of use 725 CE; ancient precedents and later medieval variants
Cursive number signs from Coptic-Arabic manuscripts.
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Unicode dates the development of these administrative numeral signs to the tenth century; 900 anchors the century, not an exact invention year. Epact numerals developed in the Coptic-Arabic community and appear in astronomical and administrative manuscripts. Their forms derive from cursive Coptic letters, but constitute a distinct number repertoire rather than the ordinary Coptic alphabetic numeral convention.
The documented period of use begins in the tenth century and extends through the seventeenth. This is an attested period, not a precise invention date. Rumi numerals, also called Fasi, belong to a manuscript tradition spanning North Africa and al-Andalus. They appear in accounting, astronomy, and manuscript numbering. Their historical shapes vary; this collection follows the variant represented in the Unicode Rumi repertoire.
Al-Uqlidisi gives an early surviving treatment of decimal fractions while adapting Hindu-Arabic arithmetic for pen and paper.
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His book shows calculations with tenths, including repeatedly increasing a number by one tenth. Decimal fractions let fractional results stay within base-ten arithmetic. He also modifies methods that had relied on moving and erasing digits on a dust board so they can be written with ink. Historians debate how fully he formulated the decimal idea, and Chinese precedents are earlier. The surviving copy dates to 1157; 0.1 is a modern illustration, not his original notation.
A single stem with attached marks can encode a number up to 9,999.
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The bar marks the principal medieval manuscript period, approximately the thirteenth through fifteenth centuries. Sixteenth-century instruments and later marginal revivals survive, so the endpoint is not a last-ever use.
Period of use Main manuscript period · c. 1200–1500
A practical arithmetic book helps spread Hindu-Arabic numerals among European readers.
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Fibonacci learned the system in Béjaïa and through Mediterranean travel. This is an important transmission milestone; he did not invent decimal digits or introduce them to every part of Europe at once.
Knotted cords carry numerical records for an empire.
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This date marks Inka administrative use, not the invention of all knotted-cord records. Earlier khipu and colonial-era continuations are documented; conquest did not instantly end the practice.
Period of use Earlier origins · colonial continuations
Infinite expansions for trigonometric functions open new ways to calculate.
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Madhava’s mathematical work is known largely through later Kerala authors. The date and attribution reflect that transmitted evidence, rather than a surviving manuscript written in exactly 1400.
Dots count units; flags represent twenty, feathers four hundred, and bags eight thousand.
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This is the manuscript tradition represented in the collection. Its starting century is a browsing anchor, not a known invention date. Nahua scribes used numerical signs alongside images of objects and tribute goods. The Codex Mendoza and Matrícula de Tributos preserve examples of flags, feather signs, and bags marking large quantities. Their composition is connected to the records in which they occur; it is not simply a replacement font.
Period of use 15th–16th-century manuscript tradition
Simon Stevin’s De Thiende promotes decimal fractions to simplify calculation for merchants, surveyors, and craftspeople.
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Tenths, hundredths, and thousandths let people calculate with fractional quantities using the same basic methods as whole numbers. Stevin explains practical decimal arithmetic and advocates dividing money, weights, and measures into powers of ten. His booklet helps spread decimal fractions in Europe; earlier Chinese and Arabic work means he did not invent them. He labels decimal places with small numbers rather than today’s decimal point. The modern illustration 0.25 means 25 hundredths, or one quarter.
Logarithm tables make long multiplication and division manageable through addition and subtraction.
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Napier’s original logarithms differ in normalization from modern natural and common logarithms. Tables become essential tools long before electronic calculators.
Descartes’ La Géométrie connects geometrical problems with algebraic equations.
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Fermat develops related methods independently. The modern coordinate-pair illustration summarizes the idea; it is not a facsimile of Descartes’ notation.
Leibniz publishes his differential calculus in Acta Eruditorum.
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Newton develops calculus independently in earlier unpublished work. The publication date marks public communication, rather than a single moment when all of calculus was invented.
18th–19th centuries · account booksChina · commercial traditions including Hong Kong and Macau
Numeral systemsHistorical
Suzhou
Decimal merchant numerals with alternating stroke orientations for one, two, and three.
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The cited Tong Taisheng account books document commercial numeral forms in the eighteenth and nineteenth centuries. This is an evidence window within an older tradition, not its origin. Suzhou numerals appear extensively in surviving Chinese merchant account books. Related to counting-rod notation, they became a compact companion to abacus bookkeeping and persisted in some later commercial settings.
Period of use Ming–Qing commerce and later survivals
1703 · Leibniz’s accountWorldwide · digital computing
Numeral systemsStill in use
Binary
Two digits, 0 and 1, count powers of two. Decimal 42 becomes 101010 in binary.
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Leibniz describes arithmetic using only 0 and 1 in this dated account. This is a documented mathematical milestone, not a claim that binary was invented in 1703 or that digital computing existed then. Leibniz described arithmetic with only 0 and 1 in a 1703 account. Binary place value later became central to digital computing, where each bit has two possible values. This entry presents the modern computing notation rather than assigning binary a single place or date of invention.
Period of use Modern computing; earlier mathematical roots
1824 · published La Trappe sign listLa Trappe Abbey, Normandy, France; Loccum comparison
Numeral systemsHistorical
Cistercian Numerical Gestures
A documented La Trappe rule answers the time by showing one finger for each hour, with parts of a finger for half and quarter hours.
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Louis Du Bois publishes a Cistercian/Trappist sign vocabulary including gestures for hours, half-hours, and quarter-hours. This dates the selected documentation, not the beginning of monastic signing. Louis Du Bois printed a vocabulary of La Trappe signs in 1824, attributing its renewal to the Trappist reform of de Rancé. Its entry Heure describes numerical replies about time. This dated Cistercian Strict Observance witness is used here, without assigning its exact gestures to the medieval order as a whole. Earlier Loccum lists, printed by Leibniz in 1717, independently document COUNT and HALF.
Period of use 1824 witness of Trappist signs; earlier monastic tradition
1824 witness of Trappist signs; earlier monastic tradition
Late 1820sCherokee Nation, southeastern North America
Numeral systemsHistorical
Sequoyah’s Cherokee numerals
Every amount from zero to one hundred in the Cherokee Nation chart of Sequoyah’s proposed numerals.
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Sequoyah proposed this distinct numerical notation in the late 1820s. 1828 is an approximate browsing anchor within that period, separate from his syllabary. The numeral proposal did not enter general use. The Cherokee Nation preserves its forms in an educational poster covering zero to one hundred, while Unicode notes that Sequoyah’s numerical signs are not encoded in the standard.
Period of use Proposed by Sequoyah in the late 1820s
Lobachevsky publishes a geometry with a different theory of parallels.
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Bolyai independently publishes his related work in 1832. These developments show that changing the parallel postulate can produce a coherent alternative geometry.
Right-to-left digit groups with attached subscripts for teens and decimal magnitudes.
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The script and its distinctive numeral repertoire developed in the early twentieth century. 1900 anchors that broad period rather than an exact invention year. Kikakui is an indigenous script for the Mende language. Its numerical repertoire combines nine unit signs with marks for decimal magnitudes and a separate mark for the teens. The numeral arrangement follows the script’s right-to-left reading direction.
Ten signs shared with the Bamum syllabary, used here as decimal digits.
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Bamum writing developed from 1896 to 1910. This entry places the final A-ka-u-ku repertoire near the end of that process; its modern zero convention must not be projected onto earlier signs. King Ibrahim Njoya developed Bamum writing through a succession of forms beginning in the late nineteenth century. This collection uses the final, A-ka-u-ku repertoire. Its last ten letters also serve as numerals.
Decimal digits from Osman Yusuf Kenadid’s Somali script.
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Osman Yusuf Kenadid devised Osmanya in 1920–1922. The timeline anchors its script and digit repertoire at the beginning of that interval. Osmanya is a writing system devised for Somali in the twentieth century by Osman Yusuf Kenadid. Its character repertoire includes ten decimal digits. The forms preserve the visual identity of this writing tradition while expressing familiar decimal mathematics.
The numeral signs were devised in the 1920s, later than the syllabary’s nineteenth-century origins. The decade is approximate. The Vai syllabary is an indigenous West African writing system. A set of ten numeral signs was devised in the 1920s, giving European decimal digits forms compatible with Vai writing. Script-specific numeral identity does not imply a different numerical base.
Twenty digits used in Medefaidrin, a language and script developed in Nigeria.
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The numeral repertoire belongs to the community’s early twentieth-century script tradition. The decade is approximate; 1930 is not a documented invention day or year. Medefaidrin belongs to the Oberi Okaime community in southeastern Nigeria. Its script developed alongside a spiritual language during the early twentieth century. The encoding proposal includes handwritten numerals supplied by Reverend Richmael Ekit in 2016.
Turing describes abstract machines and shows limits to what algorithms can decide.
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His work on computable numbers helps establish theoretical computer science. Church independently obtains related undecidability results through a different formalism.
Solomana Kanté devised N’Ko in 1949, including its right-to-left decimal numeral tradition. N’Ko writing was created by Solomana Kanté for Manding languages. Unlike Arabic numbers embedded in Arabic prose, N’Ko numerals follow the right-to-left direction of the surrounding script.
Students create a base-20 numeral system that fits Iñupiaq counting.
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The design groups units and fives within each digit. This is a modern Indigenous invention, with community and educational use, rather than an ancient notation.
Origin unknownChoctaw communities, southeastern North America and Oklahoma
tuchina
Numeral systemsStill in use
Choctaw
Build number words from one through one hundred using Choctaw Nation teaching forms.
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The collection uses contemporary Choctaw Nation teaching forms. These living number words have no documented invention date. The counting words follow Choctaw Nation of Oklahoma language lessons. Its October 2018 Biskinik teaching chart documents the teens and compounds through thirty-one; additional Nation lessons and translated reports document the multiplied tens and hundred expression.
Period of use Contemporary Choctaw Nation teaching orthography
Signs for units and tens combine with markers for one hundred and ten thousand.
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The numeral signs derive from Greek or Coptic forms. The linked sources explain their ancestry and use but do not establish a secure date for the numeral repertoire. The numeral signs form part of the Ethiopic writing tradition. Their barred, letter-like forms are related to Greek and Coptic numeral traditions. They survive alongside familiar decimal digits and have their own rules for composing larger quantities.
Period of use Historical tradition; continuing use
Hawaiian decimal counting through one hundred: named tens join extra units with kūmā. Traditional group names are explained separately.
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Traditional grouping words survive in a living language. The sources document vocabulary rather than a securely dated origin. The selected converter follows Kaliko Trapp’s contemporary number-building lesson. It uses the bare counting roots and the -kūmā- pattern. Hawaiian also has traditional terms for groups of objects, such as kāuna (four) and kaʻau (forty); these remain in the dictionary with their own context, like English dozen rather than a substitute for every ordinary number name.
Period of use Contemporary decimal counting; traditional object-group vocabulary retained
Contemporary decimal counting; traditional object-group vocabulary retained
Origin unknownWorldwide · digital computing
Numeral systemsStill in use
Hexadecimal
Digits 0–9 and letters A–F count powers of sixteen. Decimal 42 becomes 2A in hexadecimal.
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This entry uses the modern computing alphabet 0–9 and A–F. The linked references document its rules and use without establishing a single invention date. Hexadecimal makes binary data compact enough to read easily: one hexadecimal digit represents four bits. Modern computing uses the digits 0–9 followed by A–F for the values ten through fifteen. The notation is used internationally to display computer data.
Origin unknownMangareva, Gambier Islands, French Polynesia
tataua
Numeral systemsStill in use
Mangarevan
Spoken counting with groups of 10, 20, 40, and 80. The counting unit depends on the object.
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The traditional counting practice is documented in nineteenth- and twentieth-century accounts. Its invention predates that documentation by an unknown interval. Bender and Beller’s 2013 analysis draws especially on Te Rangi Hiroa’s ethnography. The specialized system counted valued goods alongside an ordinary decimal system; it was not a universal replacement for decimal counting.
Period of use Traditional object-specific counting, documented in the 19th–20th centuries
Traditional object-specific counting, documented in the 19th–20th centuries
Origin unknownAotearoa New Zealand
rima tekau mā ono
Numeral systemsStill in use
Māori
Contemporary Māori decimal number words, composed from thousands, hundreds, tens, and additional units.
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A living spoken tradition with historical variation. Modern number words do not establish a dated invention of Māori counting. Modern counting belongs to a living language. Earlier practices cannot simply be read back from today’s terminology: Te Aka records that tekau, now ten, originally denoted twenty, as related forms still do in other eastern Polynesian languages.
Period of use Contemporary decimal counting; historical variants explained separately
Origin unknownNavajo Nation, southwestern North America
dízdiin dóó bi’aan naaki
Numeral systemsStill in use
Diné / Navajo
Decimal number words, using forms documented by Diné College.
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The collection uses contemporary Diné College learning forms. Their documentation does not establish when Diné counting originated. The examples follow the number words and spelling in the reference guide published by Diné College’s Learning Center.
Period of use Contemporary Diné College learning forms
Origin unknownOksapmin area, central New Guinea, Papua New Guinea
14 · nose · center
Numeral systemsStill in use
Oksapmin body counting
A 27-position body count, from one thumb to the opposite little finger.
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Ethnographic studies from 1978 onward document body counting; 1978 is a research date, not the invention of the practice. Geoffrey Saxe’s field research documents both the counting practice and its changes through trade, currency and schooling. Traditional enumeration and later arithmetic strategies are related but historically distinct uses of the same body sequence.
Period of use Traditional practice; ethnographic studies from 1978 onward
Traditional practice; ethnographic studies from 1978 onward
Origin unknownOttoman administrative world
Numeral systemsHistorical
Ottoman Siyaq
Specialized number signs used in Ottoman accounting records.
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This specialized accounting repertoire belongs to the Ottoman period and derives from older Siyaq practices. The linked sources do not pin down a single invention date for the selected forms. Ottoman Siyaq belonged to specialized accounting and administrative practice. Its numeral repertoire has primary units, tens, hundreds, larger orders, alternate historical forms, and selected fractions.
Unit and tens signs combine with hundred and thousand multipliers.
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Sinhala Illakkam was used before 1815 and appears in the Kandyan Convention of that year. That late witness and the decline in common use do not date its invention. Sinhala Illakkam survives in royal documents and artifacts, including material from the Kandyan period. Sri Lankan research distinguished this non-positional set from the zero-bearing astrological numerals called Lith Illakkam.
Period of use Archaic numerals · in use before 1815
Origin unknownTamil-speaking South India and Sri Lanka
Numeral systemsStill in use
Traditional Tamil
Digits combine with signs for ten, one hundred, and one thousand.
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Traditional Tamil numeration evolved from earlier South Asian traditions. The linked technical account documents its non-positional rules without establishing an exact origin for this repertoire. Traditional Tamil numerals retain distinct signs for ten, hundred, and thousand. This additive-multiplicative notation differs from modern positional writing with Tamil digits and a zero.
Number words built around twenties, including subtraction: fifteen is twenty minus five.
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This spoken counting tradition has no securely dated invention. Contemporary teaching examples document its structure, not its beginning. Yorùbá numeration is a spoken language tradition, not an extinct set of written numeral glyphs. Its number names carry mathematical relationships, including multiples of twenty and subtraction from a nearby larger value. Tone marks and underdots belong to the spelling.
Evidence comes first. “c.” means approximately. A dated inscription, manuscript, or publication is evidence of use; it is rarely proof that nobody used the idea earlier. Illustrative symbols use modern forms unless a linked numeral specimen is shown.
Notation seldom has an expiry date. A historical period may describe one variant or a surviving set of documents. Traditions can overlap, evolve, decline, or return. A bar’s endpoint is explained in its entry.
Every system in this collection. All 40 supported numeral systems appear here alongside selected symbols and mathematical milestones. “Origin unknown” means the sources do not establish a reliable starting date; those traditions appear after the dated entries and are omitted from the histogram and narrower date ranges. Evidence dates and broad period boundaries do not claim an exact invention year. Many other people, regions, and ideas remain to be added. Era labels are browsing aids, not universal historical divisions.