# Chapter ᚛(Prologue): 0ur starting point *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ch-T2, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [Its glossary cards](ch02-cards.md) · [Its references](ch02-refs.md) · [Previous chapter](ch01.md) · [Next chapter](ch03.md) *A footnote such as [r-2.3] is reference entry 2.3 and links to it. Section 2 of 28; chapters are named by their Ogham letter, never by number.* ## The first steps ᚛ – [Feather mark](ch02-cards.md#g-og-feather): the beginning, Section 2 ## Chapter ᚛(Prologue): 0ur starting point > “No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought.” > — Alfred North Whitehead · *An Introduction to Mathematics*, 1911 [r-2.21](ch02-refs.md#r-2.21) Formula: [𒑊](ch02-cards.md#g-babyzero)·[𝋠](ch02-cards.md#g-mayazero)·[ο](ch02-cards.md#g-greekzero)·[nulla](ch02-cards.md#g-dionzero)·[०](ch02-cards.md#g-devzero)·[០](ch02-cards.md#g-khmerzero)·[N](ch02-cards.md#g-bedezero)·[٠](ch02-cards.md#g-arabzero)·[○](ch02-cards.md#g-gwalior)·[•](ch02-cards.md#g-bakhzero)·[0](ch02-cards.md#g-latinzero)·[〇](ch02-cards.md#g-chinazero)·[零](ch02-cards.md#g-jpzero) Key concepts: Numeral systems: thirteen ways people have written zero, from Babylon to Japan. It includes symbols you may not have seen before, such as: [𒑊](ch02-cards.md#g-babyzero), [𝋠](ch02-cards.md#g-mayazero), [०](ch02-cards.md#g-devzero), and [零](ch02-cards.md#g-jpzero). Zero here is history: a placeholder, then a digit, then a number you can do arithmetic with. A section about nothing holds only zeros. This chapter’s symbol, the first on the right-hand side of the equation, is the Pictish Ogham [feather mark](ch02-cards.md#g-og-feather), ᚛, a chevron that is not a letter and makes no sound. It tells the reader which way to go. [r-2.13](ch02-refs.md#r-2.13) I use it for the same purpose: to open a journey whose chapters are landmarks, leading from the simplest ideas to problems so difficult that their solutions carry million-dollar prizes. [r-28.1](ch28-refs.md#r-28.1) Similarly, in mathematics, 0 often marks the starting point from which we choose a direction, so it is a fitting initial subject for our compendium, before the “cool” numbers, physics, or the various frameworks. Before diving into the rich history and different meanings of 0, some quick housekeeping about our formula. Its first factor is 0: thirteen zeros multiplied together. That is enough to make the entire right-hand product zero, since every factor that follows contributes a finite number. In fact, every other chapter but one calculates to zero as well, acting as signposts all along the route, a small tribute to a number that owns nothing and, ironically, has more owners than any other ;) Field notes ### thirteen zeros Although we all “know” this number, its history could fill an entire book. Zero had to be invented, and *more than once*: in Babylon, the Maya world, and India. [r-2.9](ch02-refs.md#r-2.9) It is really three ideas wearing one name: a placeholder that keeps a column empty, a digit that can be written on its own, and a number that arithmetic can be done with. The thirteen signs represent all three uses, arranged in roughly historical order; the further back we go, the murkier the history becomes, and the references explain our choices. #### The displayed product There is no denominator: the chapter is a single product of thirteen zeros, grouped here by the job each sign did. Each sign is read as the ordinary number zero before the multiplication begins. Holding an empty place [𒑊](ch02-cards.md#g-babyzero) = 0; the oldest: the Babylonian placeholder, by ~300 BC. Scribes in Mesopotamia had counted in [base sixty](ch04-cards.md#g-sixty) for two thousand years, but they wrote 61 (one sixty and a one) and 3601 (one 60 × 60, an empty sixties column, and a one) the same way, and left the reader to guess. A tablet from Kish of about 700 BC marks the gap with three hooks; by ~300 BC [two slanted wedges](ch02-cards.md#g-babyzero) set in the empty column had become the standard sign, though the sources disagree on when the wedges first appeared. [r-2.23](ch02-refs.md#r-2.23) [r-2.22](ch02-refs.md#r-2.22) It held a place but never stood alone: nobody wrote the answer to 5 − 5 with it. [r-2.3](ch02-refs.md#r-2.3) [ο](ch02-cards.md#g-greekzero) = 0; Ptolemy’s Greek zero, around 150 AD, an astronomer’s inheritance from Babylon. Ptolemy used [a small circle, often written with a bar above it](ch02-cards.md#g-greekzero), in the *Almagest* to mark an empty place in his base-sixty fractions; the formula shows it as a plain Greek omicron. [r-2.3](ch02-refs.md#r-2.3) [〇](ch02-cards.md#g-chinazero) = 0; the Chinese circle, by the thirteenth century, still in use in Chinese and Japanese. Chinese mathematicians were using [a circle for the empty place](ch02-cards.md#g-chinazero) by then; Qin Jiushao’s *Mathematical Treatise in Nine Sections* (1247) is one example. [r-2.7](ch02-refs.md#r-2.7) A digit of its own [𝋠](ch02-cards.md#g-mayazero) = 0; the Maya shell, an invention of zero independent of the Old World. The Maya gave zero [a shell glyph](ch02-cards.md#g-mayazero) and a life as a digit in their calendar’s [base-twenty](ch04-cards.md#g-twenty) tradition. The oldest surviving Maya zero artifacts are fourth-century, but the positional [Long Count calendar](ch02-cards.md#g-longcount) is older still. Its two oldest surviving dates, 36 BC at Chiapa de Corzo and 32 BC at Tres Zapotes, are carved in Epi-Olmec rather than Maya style, which is why the calendar and its zero are thought to predate the Maya. The shell stands for that older calendrical tradition. [r-2.4](ch02-refs.md#r-2.4) [០](ch02-cards.md#g-khmerzero) = 0; the Khmer zero of 683 AD, with a story of being lost and found. Two stones from 683 AD, one in Cambodia and one in Sumatra, preserve some of the earliest dated zeros in the Hindu-Arabic [positional tradition](ch02-cards.md#g-placevalue). [r-2.6](ch02-refs.md#r-2.6) [The Cambodian one, from Sambor on the Mekong](ch02-cards.md#g-khmerzero), was published by George Cœdès in 1931, lost during the Khmer Rouge years, and found again in a shed in 2013. [r-2.14](ch02-refs.md#r-2.14) [○](ch02-cards.md#g-gwalior) = 0; India’s oldest undisputed zero in stone, 876 AD. [A small circle in an inscription on the wall of the Chaturbhuj temple at Gwalior](ch02-cards.md#g-gwalior), recording the size of a garden. [r-2.6](ch02-refs.md#r-2.6) [•](ch02-cards.md#g-bakhzero) = 0; the Bakhshali dot, placed with a deliberate question mark on its date. The Bakhshali manuscript, a birch-bark text dug up near Peshawar in 1881, uses [a plain dot for zero](ch02-cards.md#g-bakhzero). [r-2.2](ch02-refs.md#r-2.2) Oxford’s 2024 radiocarbon re-dating puts it somewhere between the late eighth and the early twelfth century. [r-2.1](ch02-refs.md#r-2.1) [r-2.10](ch02-refs.md#r-2.10) That span straddles [Gwalior’s date](ch02-cards.md#g-gwalior), so placing it after Gwalior is a best guess, not a settled order. A word for nothing [nulla](ch02-cards.md#g-dionzero) = 0; a zero supplied by Easter tables, 525 AD. Roman numerals had no zero and no empty column, but the people who computed the date of Easter needed a way to write nothing. Dionysius Exiguus wrote the Latin word *[nulla](ch02-cards.md#g-dionzero)*, nothing, in his Easter table where a Roman numeral could not go. [r-2.8](ch02-refs.md#r-2.8) [N](ch02-cards.md#g-bedezero) = 0; Bede’s N, my favorite of the thirteen. In Northumbria around 725, Bede or his colleagues wrote [N](ch02-cards.md#g-bedezero), the initial of [nulla](ch02-cards.md#g-dionzero) or of nihil, in tables of their own: a zero supplied by people whose inherited notation did not have one and who needed one anyway. [r-2.8](ch02-refs.md#r-2.8) [零](ch02-cards.md#g-jpzero) = 0; a word rather than a digit, closing the list of signs. Read rei in Japanese and líng in Chinese, the character once meant a drizzle or a small remainder and came to mean zero as well. The sources I found do not say when it first came to mean zero, so it stands last in the formula, after every sign that carries a date. [r-2.12](ch02-refs.md#r-2.12) A number with rules, and its travels [०](ch02-cards.md#g-devzero) = 0; Brahmagupta’s śūnya, 628 AD: the first zero with arithmetic rules. In India, zero acquired rules. In the *Brāhmasphuṭasiddhānta*, Brahmagupta treated [śūnya](ch02-cards.md#g-devzero), emptiness, as a number: a number minus itself is zero, a number times zero is zero, and, the rule he got wrong, zero divided by zero is zero. [r-2.5](ch02-refs.md#r-2.5) Everything in this book that divides is careful never to divide by zero; the zero under the bar on the left side is a [tropical zero](ch01-cards.md#g-tropical), read as [−∞](ch01-cards.md#g-neginf), not the ordinary 0 he divided by. [٠](ch02-cards.md#g-arabzero), [0](ch02-cards.md#g-latinzero) = 0, 0; the digit’s journey west, and the source of our words cipher and zero. From India the digit traveled west with the Arabic mathematicians, and its name traveled with it: [śūnya](ch02-cards.md#g-devzero) became [ṣifr](ch02-cards.md#g-arabzero), then Fibonacci’s Latin [zephirum](ch02-cards.md#g-latinzero) in the *Liber Abaci* of 1202, and eventually the words cipher and zero. [r-2.7](ch02-refs.md#r-2.7) The only product is 0: thirteen zeros multiplied together. **Fine print.** I ask the reader’s patience with the uncertain edges of the timeline; my intent is to illuminate how old this idea is, and how widely people found a use for it. [The Maya shell](ch02-cards.md#g-mayazero)’s place before Ptolemy reflects the older calendar tradition, not the age of the surviving glyphs. The age of the mathematics written on the Bakhshali leaves remains disputed, a question mark I am content to leave on it. What this chapter adds to the formula: [𒑊](ch02-cards.md#g-babyzero) · [𝋠](ch02-cards.md#g-mayazero) · [ο](ch02-cards.md#g-greekzero) · [nulla](ch02-cards.md#g-dionzero) · [०](ch02-cards.md#g-devzero) · [០](ch02-cards.md#g-khmerzero) · [N](ch02-cards.md#g-bedezero) · [٠](ch02-cards.md#g-arabzero) · [○](ch02-cards.md#g-gwalior) · [•](ch02-cards.md#g-bakhzero) · [0](ch02-cards.md#g-latinzero) · [〇](ch02-cards.md#g-chinazero) · [零](ch02-cards.md#g-jpzero) = 0 To ‘sum’marize a product: this chapter gathers thirteen ways of writing 0, from [Babylon’s wedges](ch02-cards.md#g-babyzero) to the word used in Chinese and Japanese, and multiplies them together—into nothing. [r-2.11](ch02-refs.md#r-2.11) Nothing certainly captivated plenty of imagination. The [feather mark](ch02-cards.md#g-og-feather) was our starting mark, acting as “ground zero.” The journey begins in our backyard. Let’s go! Watch and read - [A brief history of numerical systems - Alessandra King](https://www.youtube.com/watch?v=cZH0YnFpjwU) TED-Ed, five minutes: tally marks, Babylon’s base sixty, the Maya’s base twenty, and how the Indian positional system with zero became the one we all use. - [Problems with Zero - Numberphile](https://www.youtube.com/watch?v=BRRolKTlF6Q) Why 0/0 and 0⁰ are trouble: 0/0 is where Brahmagupta went wrong, and never dividing by zero is the rule this whole book is careful to obey. - [0](https://en.wikipedia.org/wiki/0) Wikipedia: the number, its history, and its names - [Brahmagupta](https://en.wikipedia.org/wiki/Brahmagupta) Wikipedia: the man who wrote the rules