Ancient Near East

Ancient Egyptian numerals were of base 10. They used hieroglyphs for the digits and were not positional. By 1770 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was also used to indicate the base level in drawings of tombs and pyramids, and distances were measured relative to the base line as being above or below this line. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated base 60 positional numeral system. The lack of a positional value (or zero) was indicated by a space between sexagesimal numerals. In a tablet unearthed at Kish (dating to as early as 700 BC), the scribe Bêl-bân-aplu used three hooks as a placeholder in the same Babylonian system. By 300 BC, a punctuation symbol (two slanted wedges) was co-opted to serve as this placeholder. The Babylonian placeholder was not a true zero because it was not used alone, nor was it used at the end of a number. Thus numbers like 2 and 120 (2×60), 3 and 180 (3×60), 4 and 240 (4×60) looked the same, because the larger numbers lacked a final sexagesimal placeholder. Only context could differentiate them

Pre-Columbian Americas

The Mesoamerican Long Count calendar developed in south-central Mexico and Central America required the use of zero as a placeholder within its vigesimal (base-20) positional numeral system. Many different glyphs, including the partial quatrefoil were used as a zero symbol for these Long Count dates, the earliest of which (on Stela 2 at Chiapa de Corzo, Chiapas) has a date of 36 BC. Since the eight earliest Long Count dates appear outside the Maya homeland, it is generally believed that the use of zero in the Americas predated the Maya and was possibly the invention of the Olmecs. Many of the earliest Long Count dates were found within the Olmec heartland, although the Olmec civilization ended by the 4th century BC, several centuries before the earliest known Long Count dates. Although zero became an integral part of Maya numerals, with a different, empty tortoise-like "shell shape" used for many depictions of the "zero" numeral, it is assumed not to have influenced Old World numeral systems. Quipu, a knotted cord device, used in the Inca Empire and its predecessor societies in the Andean region to record accounting and other digital data, is encoded in a base ten positional system. Zero is represented by the absence of a knot in the appropriate position.

Classical antiquity

The ancient Greeks had no symbol for zero (μηδέν), and did not use a digit placeholder for it.[13] According to mathematician Charles Seife, the ancient Greeks did begin to adopt the Babylonian placeholder zero for their work in astronomy after 500 BC, representing it with the lowercase Greek letter ό (όμικρον) or omicron.[14] However, after using the Babylonian placeholder zero for astronomical calculations they would typically convert the numbers back into Greek numerals.[14] Greeks seemed to have a philosophical opposition to using zero as a number.[14] Other scholars give the Greek partial adoption of the Babylonian zero a later date, with the scientist Andreas Nieder giving a date of after 400 BC and the mathematician Robert Kaplan dating it after the conquests of Alexander.[15][16] Greeks seemed unsure about the status of zero as a number. Some of them asked themselves, "How can not being be?", leading to philosophical and, by the medieval period, religious arguments about the nature and existence of zero and the vacuum. The paradoxes of Zeno of Elea depend in large part on the uncertain interpretation of zero.[17]By AD 150, Ptolemy, influenced by Hipparchus and the Babylonians, was using a symbol for zero ( — ° )[18][19] in his work on mathematical astronomy called the Syntaxis Mathematica, also known as the Almagest.[20] This Hellenistic zero was perhaps the earliest documented use of a numeral representing zero in the Old World.[21] Ptolemy used it many times in his Almagest (VI.8) for the magnitude of solar and lunar eclipses. It represented the value of both digits and minutes of immersion at first and last contact. Digits varied continuously from 0 to 12 to 0 as the Moon passed over the Sun (a triangular pulse), where twelve digits was the angular diameter of the Sun. Minutes of immersion was tabulated from 0′0″ to 31′20″ to 0′0″, where 0′0″ used the symbol as a placeholder in two positions of his sexagesimal positional numeral system,[b] while the combination meant a zero angle. Minutes of immersion was also a continuous function 1 / 12 31′20″ √d(24−d) (a triangular pulse with convex sides), where d was the digit function and 31′20″ was the sum of the radii of the Sun's and Moon's discs.[22] Ptolemy's symbol was a placeholder as well as a number used by two continuous mathematical functions, one within another, so it meant zero, not none. The earliest use of zero in the calculation of the Julian Easter occurred before AD 311, at the first entry in a table of epacts as preserved in an Ethiopic document for the years AD 311 to 369, using a Ge'ez word for "none" (English translation is "0" elsewhere) alongside Ge'ez numerals (based on Greek numerals), which was translated from an equivalent table published by the Church of Alexandria in Medieval Greek.[23] This use was repeated in AD 525 in an equivalent table, that was translated via the Latin nulla or "none" by Dionysius Exiguus, alongside Roman numerals.[24] When division produced zero as a remainder, nihil, meaning "nothing", was used. These medieval zeros were used by all future medieval calculators of Easter. The initial "N" was used as a zero symbol in a table of Roman numerals by Bede—or his colleagues—around AD 725.[25]

China

The Sūnzĭ Suànjīng, of unknown date but estimated to be dated from the 1st to 5th centuries AD, and Japanese records dated from the 18th century, describe how the c. 4th century BC Chinese counting rods system enabled one to perform decimal calculations. As noted in Xiahou Yang's Suanjing (425–468 AD) that states that to multiply or divide a number by 10, 100, 1000, or 10000, all one needs to do, with rods on the counting board, is to move them forwards, or back, by 1, 2, 3, or 4 places,[27] According to A History of Mathematics, the rods "gave the decimal representation of a number, with an empty space denoting zero".[26] The counting rod system is considered a positional notation system.[28] In AD 690, Empress Wǔ promulgated Zetian characters, one of which was "〇"; originally meaning 'star', it subsequently[when?] came to represent zero. Zero was not treated as a number at that time, but as a "vacant position".[29] Qín Jiǔsháo's 1247 Mathematical Treatise in Nine Sections is the oldest surviving Chinese mathematical text using a round symbol for zero.[30] Chinese authors had been familiar with the idea of negative numbers by the Han Dynasty (2nd century AD), as seen in The Nine Chapters on the Mathematical Art.[31]

India

Pingala (c. 3rd/2nd century BC[32]), a Sanskrit prosody scholar,[33] used binary numbers in the form of short and long syllables (the latter equal in length to two short syllables), a notation similar to Morse code.[34] Pingala used the Sanskrit word śūnya explicitly to refer to zero.[32] The concept of zero as a written digit in the decimal place value notation was developed in India.[35] A symbol for zero, a large dot likely to be the precursor of the still-current hollow symbol, is used throughout the Bakhshali manuscript, a practical manual on arithmetic for merchants.[36] In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from AD 224–383, AD 680–779, and AD 885–993, making it South Asia's oldest recorded use of the zero symbol. It is not known how the birch bark fragments from different centuries forming the manuscript came to be packaged together.[37][38][39] The Lokavibhāga, a Jain text on cosmology surviving in a medieval Sanskrit translation of the Prakrit original, which is internally dated to AD 458 (Saka era 380), uses a decimal place-value system, including a zero. In this text, śūnya ("void, empty") is also used to refer to zero.[40] The Aryabhatiya (c. 500), states sthānāt sthānaṁ daśaguṇaṁ syāt "from place to place each is ten times the preceding".[41][42][43] Rules governing the use of zero appeared in Brahmagupta's Brahmasputha Siddhanta (7th century), which states the sum of zero with itself as zero, and incorrectly division by zero as:[44][45] A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Epigraphy

A black dot is used as a decimal placeholder in the Bakhshali manuscript, portions of which date from AD 224–993.[46] There are numerous copper plate inscriptions, with the same small o in them, some of them possibly dated to the 6th century, but their date or authenticity may be open to doubt.[8] A stone tablet found in the ruins of a temple near Sambor on the Mekong, Kratié Province, Cambodia, includes the inscription of "605" in Khmer numerals (a set of numeral glyphs for the Hindu–Arabic numeral system). The number is the year of the inscription in the Saka era, corresponding to a date of AD 683.[47] The first known use of special glyphs for the decimal digits that includes the indubitable appearance of a symbol for the digit zero, a small circle, appears on a stone inscription found at the Chaturbhuj Temple, Gwalior, in India, dated 876.[48][49]