# Glossary cards — Chapter ᚛(Prologue): 0ur starting point *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#group-T2, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch02.md) *Numeral systems: Babylon, the Maya, Ptolemy, India, Dionysius and Bede, Cambodia, the Arab world, Gwalior, Fibonacci, China, and Japan. Each card: its symbol, type, one-line headline, the longer note, the reference entries it cites and the cards it links to.* ### the feather mark — `᚛` {#g-og-feather} *Ogham mark* Not a letter but punctuation: the ogham feather mark stands at the beginning of a text and can show which way it is to be read. U+169B OGHAM FEATHER MARK is classed as open punctuation rather than as one of the twenty-six ogham letters, and the Unicode Standard describes the mark and its partner as used at the beginning and the end of ogham text, particularly in manuscript ogham; where only the feather mark appears it can indicate the direction of the text. Because it is not a letter it has no name-meaning, no word-ogham, and no plant: the three layers the other cards in this section keep apart do not exist for it. Here the pair frames the equation as it frames an inscription. Cites: [2.13](ch02-refs.md#r-2.13), [2.15](ch02-refs.md#r-2.15), [2.17](ch02-refs.md#r-2.17) ### the Mayan zero — `𝋠` {#g-mayazero} *Integer or rational* A shell glyph, base twenty, an invention of zero independent of the Old World, in use by the fourth century. The Maya wrote numbers in base twenty with dots and bars, and used a stylized shell for zero. The Long Count calendar, which needs a positional zero to work, is attested before the Maya: the oldest dates are Epi-Olmec, Stela 2 at Chiapa de Corzo (36 BC) and Stela C at Tres Zapotes (32 BC), and the oldest surviving shell-glyph zeros are fourth-century. It is one of the independent inventions of zero. Type: integer, exactly 0. Cites: [2.4](ch02-refs.md#r-2.4), [2.9](ch02-refs.md#r-2.9) See also: [the Long Count calendar](ch02-cards.md#g-longcount), [the Mayan base](ch04-cards.md#g-twenty) ### the Babylonian placeholder — `𒑊` {#g-babyzero} *Integer or rational* Two slanted wedges marking an empty sexagesimal place, in use by ~300 BC; a zero that could hold a column but never stood alone. Babylonian scribes counted in base sixty and for centuries left empty places blank, so 61 and 3601 looked alike. A tablet from Kish of perhaps 700 BC marks the gap with three hooks; by ~300 BC, in the Seleucid period, two slanted wedges had become the standard placeholder. When the wedges came in is not settled: three sources give three dates, about 700 BC, about 400 BC, and 300 BC, and none of them argues its date, so the book gives the latest, by which all three have the sign in use. It is a zero, but only a placeholder: it never appears alone and was not used at the end of a number, so 1 and 60 still had to be told apart by context. Type: integer, exactly 0 in the place it holds. Cites: [2.22](ch02-refs.md#r-2.22), [2.23](ch02-refs.md#r-2.23), [2.3](ch02-refs.md#r-2.3), [2.9](ch02-refs.md#r-2.9) See also: [the Babylonian base](ch04-cards.md#g-sixty) ### Brahmagupta’s zero — `०` {#g-devzero} *Integer or rational* The first zero with rules: 628 AD, a − a = 0, a·0 = 0, and the Sanskrit name śūnya, emptiness. In 628 AD, in the Brāhmasphuṭasiddhānta, Brahmagupta gave zero the first written rules: a − a = 0, a + 0 = a, a × 0 = 0, and rules for negative numbers as debts. He also claimed 0/0 = 0, which is wrong, and wrote a/0 only as a fraction with zero below, never assigning it a value, which is as close to “undefined” as the seventh century got. The Sanskrit name śūnya, emptiness, is the ancestor of the names cipher and zero. The oldest Indian zero in stone that everyone agrees is genuine is the small circle in the Chaturbhuj temple inscription at Gwalior, 876 AD, which has its own entry; the Bakhshali manuscript’s dot is on birch bark that Oxford’s re-dating places in the eighth to eleventh centuries. Type: integer; this is the moment zero became a number rather than a gap. Cites: [2.1](ch02-refs.md#r-2.1), [2.5](ch02-refs.md#r-2.5), [2.6](ch02-refs.md#r-2.6) See also: [the Bakhshali dot](ch02-cards.md#g-bakhzero), [the Gwalior zero](ch02-cards.md#g-gwalior) ### the Khmer zero — `០` {#g-khmerzero} *Integer or rational* One of the oldest securely dated zeros in the Hindu-Arabic positional tradition, on a stone at Sambor, Cambodia, 683 AD. Inscription K-127, on a stone from Sambor on the Mekong in Cambodia, is dated to the Śaka year that corresponds to 683 AD, and the year’s three digits carry a dot for zero in the middle position. George Cœdès published it in 1931, and an Old Malay inscription at Kedukan Bukit in Sumatra carries a zero in the same Śaka year, so the two cannot be ranked against each other. The stone vanished during the Khmer Rouge years and was tracked down in 2013 by Amir Aczel, who told the story in Finding Zero. Type: integer, exactly 0. Cites: [2.14](ch02-refs.md#r-2.14), [2.6](ch02-refs.md#r-2.6) ### ṣifr — `٠` {#g-arabzero} *Integer or rational* The Arabic zero, whose name became both cipher and zero. The Indian digit reached the Arabic world by the ninth century; al-Khwarizmi’s treatise on Indian reckoning, around 825, spread it westward. The Arabic name ṣifr translates śūnya, emptiness, and became the Latin zephirum, the Italian zefiro, and the English cipher and zero, so both of those English names are fossils of the Sanskrit one. Type: integer, exactly 0. Cites: [2.7](ch02-refs.md#r-2.7) See also: [Brahmagupta’s zero](ch02-cards.md#g-devzero), [zephirum](ch02-cards.md#g-latinzero) ### the Chinese circle zero — `〇` {#g-chinazero} *Integer or rational* A circle for the empty place, in use by the thirteenth century, still used in Chinese and Japanese today. Chinese mathematicians used counting rods and a blank space for an empty place for centuries. Qin Jiushao’s Mathematical Treatise in Nine Sections of 1247 is the oldest surviving Chinese mathematical text to write zero as the circle 〇, which also survives in the Suzhou numerals, the last living form of the rod system; the book itself circulated in manuscript and was not printed until 1842. The origin of the circle is unknown; one hypothesis makes it a modified □, the text-space filler. 〇 is still used in Chinese and Japanese for zero today. Type: integer, exactly 0. Cites: [2.7](ch02-refs.md#r-2.7) ### rei — `零` {#g-jpzero} *Integer or rational* A Japanese and Chinese word for zero, written as a character rather than a digit. The character 零, read rei in Japanese and líng in Chinese, originally meant a drizzle of rain or a small leftover, and was pressed into service for zero as a word rather than a digit. It is the ordinary word for zero in Chinese; in Japanese the preferred spoken reading of 0 is the English loanword zero, with rei beside it. Type: integer, exactly 0, written as a word rather than a glyph. Cites: [2.12](ch02-refs.md#r-2.12) ### Bede’s nulla — `N` {#g-bedezero} *Integer or rational* The letter N for nulla or nihil, both meaning nothing, used by Bede around 725 AD in tables that Roman numerals could not otherwise fill. Bede, writing in Northumbria in 725 in De temporum ratione, built tables for calculating the date of Easter, and needed to record that a remainder, the epact, was nothing. Roman numerals have no zero, so he or his colleagues wrote N, the initial of nulla or of nihil. He was following Dionysius Exiguus, who had written the word nulla in his Easter table two centuries earlier, in 525; Bede’s N may be that word abbreviated to a letter. A zero invented on the spot by people whose number system did not have one. Type: integer, exactly 0. Cites: [2.8](ch02-refs.md#r-2.8) See also: [Dionysius Exiguus’s nulla](ch02-cards.md#g-dionzero) ### zephirum — `0` {#g-latinzero} *Integer or rational* Fibonacci’s Latin for the Arabic ṣifr, 1202, the zero the West finally kept. Leonardo of Pisa, Fibonacci, popularized the Indian digits in Latin Europe in the Liber Abaci of 1202, and Latinized ṣifr as zephirum. It was not widely used for a long time, but the digit won, and the word shortened through the Italian zefiro to zero. This is the 0 the rest of the book uses. Type: integer, exactly 0. Cites: [2.7](ch02-refs.md#r-2.7) See also: [ṣifr](ch02-cards.md#g-arabzero) ### Ptolemy’s zero — `ο` {#g-greekzero} *Integer or rational* A small circle with a bar, used by Ptolemy in the Almagest around 150 AD as the empty place in sexagesimal fractions; the Greek zero. Ptolemy, writing the Almagest in Alexandria around 150 AD, tabulated astronomy in Babylonian base-sixty fractions and marked an empty place with a small circle carrying a long overbar, the Greek zero sign (Unicode 𐆊). It was a placeholder in tables rather than a number to calculate with, and its origin is argued over: the popular story derives it from omicron, the first letter of οὐδέν, nothing, but Neugebauer rejected that, since omicron already meant 70 in Greek alphabetic numerals; it was free to serve as zero only because a sexagesimal place never runs past 59. In ninth-century Byzantine manuscripts it is still a full-size uncial omicron with a macron, and the form alters markedly only in some late medieval copies. Type: integer, exactly 0. Cites: [2.3](ch02-refs.md#r-2.3) See also: [the Babylonian base](ch04-cards.md#g-sixty) ### the Bakhshali dot — `•` {#g-bakhzero} *Integer or rational* A dot for zero in the Bakhshali manuscript, an Indian birch-bark text; Oxford’s 2024 re-dating places the sampled leaves in about 773–1032 AD and models the manuscript’s production at roughly 799–1102 AD. The Bakhshali manuscript, seventy birch-bark leaves of arithmetic found near Peshawar in 1881 and kept in the Bodleian Library, writes zero as a dot, hundreds of times. In 2017 the Bodleian radiocarbon-dated three leaves to 224–383, 680–779, and 885–993 AD and announced the oldest known written zero; Plofker, Wujastyk, Hayashi, Keller, and Montelle replied that a manuscript is dated by its latest leaf, so the text as we have it may be ninth-century or later. Oxford’s Radiocarbon Accelerator Unit then re-dated the manuscript and published the revised results on 14 October 2024: the earlier Folio 16 determination was inaccurate, Folio 17 was treated as an outlier, and Folios 15, 16, 23, 33, and 55 date to 773–986, 901–1032, 890–1014, 885–995, and 774–991 AD at 95.4% confidence, with a phase model putting production at roughly 799–1102 AD. Whatever its date, the dot is a written zero used in calculation and not only a placeholder, and so it belongs in the product. Type: integer, exactly 0. Cites: [2.1](ch02-refs.md#r-2.1), [2.10](ch02-refs.md#r-2.10), [2.2](ch02-refs.md#r-2.2) ### Dionysius Exiguus’s nulla — `nulla` {#g-dionzero} *Integer or rational* The Latin word for nothing, written in an Easter table in 525 AD where a Roman numeral could not go; Bede’s N may abbreviate it. Dionysius Exiguus, the monk who invented the Anno Domini count, drew up an Easter table in 525 AD in which some entries had to be nothing. Roman numerals cannot say so, so he wrote the word nulla. It is a zero as a word rather than a glyph, like 零, and it may be the ancestor of Bede’s N two centuries later. Type: integer, exactly 0. Cites: [2.8](ch02-refs.md#r-2.8) See also: [Bede’s nulla](ch02-cards.md#g-bedezero), [rei](ch02-cards.md#g-jpzero) ### the Gwalior zero — `○` {#g-gwalior} *Integer or rational* The oldest Indian zero in stone that everyone agrees is genuine: a small circle in an inscription of 876 AD at the Chaturbhuj temple in Gwalior. An inscription of 876 AD at the Chaturbhuj temple in the fort of Gwalior records a land grant of 270 by 187 hastas and a gift of 50 garlands, and writes the zeros of 270 and 50 as small circles. It is the oldest dated Indian zero in stone that nobody disputes: older Indian inscriptions have been proposed, among them inscriptions of AD 672, 683, and 684, but their dates or their authenticity are questioned. It is later than the Khmer stone of 683. Type: integer, exactly 0. Cites: [2.6](ch02-refs.md#r-2.6) See also: [the Khmer zero](ch02-cards.md#g-khmerzero) ### positional notation — `dₙ·bⁿ` {#g-placevalue} *Function or operator* The column a digit stands in tells you what it is worth, so a small fixed set of signs writes every number—and an empty column has to be written down. In positional notation the value of a digit is its face value times a power of the base, fixed by the column it occupies, so a small fixed alphabet of signs writes every number and arithmetic becomes a matter of columns. The Babylonian system, base 60, is credited as the first positional one, and it is exactly the system that made a zero necessary: with no sign for an empty column, 61 and 3601 look alike, which is why three hooks by about 700 BC, and later two slanted wedges, were pressed into service; they were never written at the end of a number, so 1 and 60 still looked alike. The Indians were the first to build a base-10 positional system, and the one on this page is its descendant. This is the invention that the chapter’s three separate zeros—placeholder, digit, number—are three stages of: a sign-value system like Roman numerals has no empty column to name, and so never needs the first of them. A notation of the framework, uncolored. Cites: [2.22](ch02-refs.md#r-2.22), [2.3](ch02-refs.md#r-2.3) See also: [the Babylonian placeholder](ch02-cards.md#g-babyzero) ### the Long Count calendar — `13.0.0.0.0` {#g-longcount} *Function or operator* A positional, mostly base-twenty count of days used for Mesoamerican dates, needing its own zero place to work at all. The Long Count numbers days from a fixed starting point using five or more place values, all base twenty except the second, which is base eighteen, so that the third place counts 360 days, roughly a year, rather than 400. Its two oldest surviving dates, Stela 2 at Chiapa de Corzo and Stela C at Tres Zapotes, are Epi-Olmec rather than Maya and predate the Common Era; the Maya inherited the system and came to write its zero place with a stylized shell glyph. A positional count needs a symbol for an empty place to be positional at all, which is why the calendar and the Mayan zero belong to the same story. Cites: [2.24](ch02-refs.md#r-2.24) See also: [the Mayan zero](ch02-cards.md#g-mayazero), [the Mayan base](ch04-cards.md#g-twenty)