The zero in the decimal arithmetic used today developed through mathematical work in the Indian subcontinent. Indian scholars developed decimal place-value notation, from which the numerals familiar today descended. In 628 CE the astronomer and mathematician Brahmagupta set down surviving rules for treating zero as a number. He described addition, subtraction and multiplication involving zero. His rules for division by zero differed from the rules used in mathematics today.
Centuries before Brahmagupta, other mathematicians marked empty positions in written numbers. Babylonian scribes needed such a sign in calculations based on 60, and Mesoamerican mathematicians independently developed a zero glyph for another positional system. Any claim to have invented zero depends on what is meant: a space left empty in a number, a symbol that records that empty place, or a number governed by arithmetic rules. The evidence gives different histories for each.
The written evidence survives across centuries and on several kinds of material. Clay tablets preserve some Babylonian notation; Mesoamerican monuments carry dates cut into stone. Indian mathematical works survive in manuscripts that were sometimes copied long after their contents were written. These records reveal particular uses of zero, but cannot identify the first person to think about an absent quantity. No surviving document records a single moment of discovery.
Why writing 101 is different from calculating with zero
A positional numeral gives each digit a value according to where it stands. In 101, the first 1 represents a hundred and the second represents one unit. The zero indicates that there are no tens. Without something to preserve that position, the difference between 11 and 101 could disappear on the page. A scribe could use that placeholder without ever adding or multiplying zero as a number.
A merchant who starts with five coins and spends them all has none left. Arithmetic describes the result: 5 minus 5 equals 0, while multiplying 0 by 5 still gives 0. Those calculations require more than marking an unused position between digits. The University of St Andrews history of zero separates the early history of placeholders from the later development of explicit operations with zero, a distinction that matters when comparing Babylonian and Indian evidence.
People did not need a numeral for zero to know that a container was empty or a debt paid off. They could keep accounts, describe a missing quantity, study geometry and survey land with notations that lacked a zero sign. Astronomers also achieved detailed calculations without modern decimal numerals. Historians can investigate when people began writing a stable zero sign and formulating rules for it. They cannot date the first human awareness of an empty quantity.
Babylonian mathematicians had an earlier placeholder
In Mesopotamia, scribes worked with a sexagesimal, or base-sixty, system. Its place values followed powers of 60 instead of the powers of ten familiar today. The division of an hour into 60 minutes and a minute into 60 seconds preserves something of that tradition. Cuneiform groups of wedge-shaped marks could stand for different quantities according to their position. The same principle lets our digit 2 mean two, twenty or two hundred according to its position.
Some early tablets relied on spacing between cuneiform groups, leaving readers to work out whether a position was vacant. Later Babylonian mathematical and astronomical scribes, particularly in the first millennium BCE, used a sign made from two slanting wedges inside numbers. Like the zero in 101, the wedges marked an empty place between other positions. The sign reduced ambiguity between otherwise similar numerals, although tablets differ in date and practice. Babylonian tablets nevertheless show different conventions at different times.
The two-wedge sign generally did not fill the final place in the manner of the zeros in 20 or 200, so readers could still need context to determine a number’s scale. Babylonian scribes also did not ordinarily calculate with the placeholder as an independent quantity. The MacTutor account of Babylonian notation explains both the achievement and its limit. The notation helped readers follow complicated astronomical calculations. It did not provide explicit rules for adding or multiplying zero as a separate quantity.
The claim that the Sumerians invented modern zero thousands of years before Indian mathematicians confuses several developments. Sumerian and later Mesopotamian practices contributed to the long development of positional notation, and the Babylonian empty-place sign belongs to that history. Positional values, empty-place markers and arithmetic rules developed in different historical settings. The evidence does not support one invention date for all three.
Greek astronomers used a zero sign, but Greek mathematics was not uniform
Greek philosophy does not support the common story that mathematicians refused zero because they feared the void. Greek philosophers disagreed about emptiness, infinity, being and non-being, and those arguments did not dictate every mathematician’s notation. In astronomy, a practical need for empty positions could arise whether or not a particular philosophical account allowed a void. The surviving calculations show that Greek scholars sometimes used precisely such a convention.
Greek astronomers borrowed and adapted Babylonian sexagesimal methods. Tables associated with Ptolemy, working in Roman Egypt in the second century CE, could use a small circular mark for an absent position or zero value. Surviving manuscripts differ, and the mark is sometimes associated with the letter omicron. Not every surviving circular mark can therefore be identified confidently as the equivalent of our numeral 0. The history of mathematical zero at St Andrews places this Greek astronomical notation beside its Babylonian antecedents and explains what the evidence can support.
Ptolemy calculated angles and planetary positions with numerical tools that ordinary Greek alphabetic numerals did not need. Geometers approached ratios, magnitudes and figures through other conventions, rather than decimal positional writing. Each set of methods was suited to different mathematical problems. Greek and Babylonian astronomy were already mathematically developed before the Indian arithmetic rules for zero are documented, although neither tradition relied on the modern zero digit.
Roman numerals used a different system of notation. I, V, X, L, C, D and M represent quantities through combinations of signs, with X standing for ten wherever it appears. A separate sign for an empty tens place would not help distinguish ten from one hundred, because C already means one hundred. Romans used counting boards and other aids in accounting, and their engineers built roads and aqueducts without decimal positional numerals. Their written symbols did not require an empty-place marker.
The lack of a Roman zero sign cannot establish what Roman thinkers believed about nothingness. An administrator could subtract an entire debt or recognize an empty account without writing a zero digit. Roman notation served many purposes, even if it was less convenient for the written algorithms later developed with decimal places. The structure of Roman numerals removed the need for a positional placeholder, without implying a general philosophical ban on the concept.
Mesoamerica developed zero independently
Mesoamerican scribes also developed their own written zero. Maya numerals used dots for units, bars for sets of five and a shell-like glyph for an empty position. Their numerical writing generally followed powers of twenty. The Long Count calendar modified part of that sequence to accommodate the calendar’s periods. When a date contained no units of a particular period, a zero sign could record the vacancy plainly and unambiguously.
The Smithsonian National Museum of the American Indian illustrates the Maya dot, bar and shell notation. The Mathematical Association of America’s treatment of Maya number notation explains how ordinary base-20 positions differ from the calendar’s special 360-day position. The examples document how Mesoamerican mathematicians handled vacant numerical positions within their own system. There is no basis for supposing that this zero was imported from Indian or Mesopotamian arithmetic.
Early Long Count dates appear in inscriptions associated with several Mesoamerican cultures. The earliest carved zero cannot be dated from the recorded Long Count date alone: a carved text can commemorate an earlier time. Specific claims about first-century BCE examples must be checked against the surviving glyph and its archaeological context. Even with those uncertainties, positional notation using zero is securely attested in the Late Preclassic and Classic Mesoamerican record.
The Maya zero glyph had a practical role in written dates and numerical intervals. No surviving Maya treatise lays out abstract arithmetic rules for zero comparable to Brahmagupta’s. Mesoamerican mathematicians worked with distinct intellectual aims and materials. Their work belongs to a separate mathematical tradition with its own purposes, rather than an earlier version of the system that eventually reached Europe.
The Indian decimal system and the evidence of the Bakhshali manuscript
The decimal notation behind today’s digits developed along a different route, through Indian mathematical and astronomical practice over many centuries. A position could represent units, tens, hundreds or another power of ten, making an empty-place symbol particularly useful. The written numerals themselves changed with local scripts and writing materials. The familiar hollow circle developed through this history of changing written forms; earlier marks could look quite different.
The Bakhshali manuscript preserves an early example of that Indian notation. Found in 1881 near Bakhshali, in what is now Pakistan, it consists of mathematical problems written on birch bark and includes a dot used for zero in calculations. Oxford’s Bodleian Libraries preserve the manuscript. Its worked problems show a sign used in calculations, providing firmer evidence than philosophical discussions of emptiness alone.
In 2017 the Bodleian Libraries announced radiocarbon results for three pieces of the birch bark. The results placed the samples in different centuries; the earliest belonged to the third or fourth century CE. Reports then called the text the earliest recorded example of the sign that led to modern zero. The announcement left an important distinction unresolved. The test can estimate when the bark grew, but it cannot by itself establish when writing was added to it.
Historians Kim Plofker, Agathe Keller, Takao Hayashi, Clemency Montelle and Dominik Wujastyk questioned that interpretation. Their 2017 peer-reviewed response examined the dates alongside the manuscript’s language, script and mathematical content. The widely separated radiocarbon dates meant that the oldest piece of bark could not establish the age of all the writing. They also disputed a clean separation between placeholder and number in this manuscript, observing that some calculations give the zero mark more work to do than simply filling a visual gap.
The Bakhshali manuscript remains essential evidence, although scholars dispute exactly when its mathematical text was composed. Dates such as 224 CE or 300 CE cannot be presented as secure dates for its writing. Nor can the first surviving dot be identified as the first time zero entered someone’s thinking. This is why researchers compare the handwriting and mathematical methods with the radiocarbon dates of the bark.
What Aryabhata contributed in 499 CE
Aryabhata was born in 476 CE and completed the Aryabhatiya in 499. Indian astronomers of his period worked with positional values, immense astronomical periods and procedures for extracting roots. Aryabhata expressed numbers through an intricate arrangement of Sanskrit letters and vowels, allowing large quantities to fit into compact verses. His treatise covers arithmetic, geometry, planetary movement and eclipses, showing the range of computations Indian astronomers carried out.
The surviving Aryabhatiya does not display an unmistakable zero digit of the kind represented by our 0, nor does it present the explicit zero-arithmetic rules found in Brahmagupta’s later work. Its methods nevertheless show detailed knowledge of positional values. The University of St Andrews biography discusses the argument that Aryabhata probably knew place-value numerals and a zero sign while using his own alphabetic notation. The argument remains an inference; the surviving text contains neither the symbol nor Brahmagupta’s later rules. His alphabetic system makes the distinction between what the calculations may require and what the manuscript explicitly says particularly important.
Consider Aryabhata’s result for a circle with a diameter of 20,000 units. His rule produces a circumference of 62,832, implying a value for pi of about 3.1416. That close approximation does not establish that Aryabhata treated zero as an independent arithmetic number. Other ancient mathematical traditions also produced accurate numerical results with different scripts and techniques. His documented achievement is his mathematics and astronomy; the record does not justify making him zero’s sole inventor.
The Sanskrit term kha can refer to space or an empty position and is often cited in explaining Aryabhata’s notation. The vocabulary tells us how vacant positions were described. It does not, by itself, establish an arithmetic rule for zero. Aryabhata belongs in the history of Indian positional mathematics. The particular arithmetic later published by Brahmagupta cannot simply be attributed to him more than a century in advance.
Brahmagupta made zero an explicit subject of arithmetic
Born around 598 CE, Brahmagupta worked within Indian astronomical traditions that joined observation to elaborate computation. His Brāhmasphuṭasiddhānta, completed in 628, treats planetary calculations alongside arithmetic and algebra. The text provides more direct evidence than the shapes of written numerals: they can read specific rules for operating on zero and on quantities described as positive or negative.
Chapter 18 explains positive quantities through the language of fortunes and negative quantities through debts. Possessing five units counts as positive; owing five is negative. An equal fortune and debt cancel to produce zero. Brahmagupta also states that adding zero leaves a quantity unchanged, subtracting a positive quantity from zero yields a negative one, and any number multiplied by zero produces zero. These instructions specify how zero behaves in arithmetic, rather than where a scribe should place a sign.
These rules survive within the Sanskrit mathematical corpus and can be consulted in Henry Thomas Colebrooke’s 1817 English translation of Brahmagupta and Bhāskara, which includes the relevant algebraic chapter. The MacTutor discussion of Brahmagupta sets the passages in their historical context. Colebrooke’s early translation does not resolve every question of Sanskrit interpretation, but it reproduces the central rules attributed to Brahmagupta.
Brahmagupta’s account of division was less successful. He assigned zero to zero divided by zero, an expression treated as indeterminate in ordinary modern arithmetic. For a nonzero number divided by zero, he left what was effectively a fraction with zero as denominator, without establishing a valid answer. His mistaken division rules coexist with correct explicit rules for other operations. They remind us that the modern number system emerged through successive corrections rather than appearing complete in one treatise.
Brahmagupta had earlier mathematical traditions to draw on. His book is firm documentary evidence that sophisticated rules for zero existed in his time, but it cannot prove that nobody had previously used similar procedures. Calculations involving an empty quantity may have been practiced before any surviving author described them. The first surviving written explanation establishes a date by which the method existed, not necessarily when someone first used it.
The Gwalior inscription and the history of the written symbol
Meanwhile, the written sign changed independently of arithmetic theory. The Bakhshali zero was a dot, unlike the hollow circle used worldwide today. Indian numeral forms varied with scripts, regions and writing surfaces. Inscriptions supplement the manuscript evidence because numbers cut into stone sometimes have datable contexts, such as the construction or dedication of a building. A surviving circular sign can therefore tell us about notation even when it says nothing about who first calculated with zero.
At the Chaturbhuj temple in Gwalior, central India, an inscription dated to the ninth century CE contains a familiar early example of the circular zero. The Bodleian’s account of the Bakhshali manuscript points to this dated inscription as a comparison with the bark manuscript. The inscription establishes that the recognizable circular form was in use at Gwalior at that date. It does not establish that the invention occurred there, or that every earlier Indian zero looked identical.
A sign in a ninth-century inscription might represent a custom already well established, just as an old mathematical work could survive only in a younger manuscript. Composition, copying and the survival of physical evidence therefore have to be dated separately. The first surviving empty-place sign, the development of the modern circular glyph, and the earliest formal zero arithmetic are connected questions. No single object settles all three.
Did Indian philosophy make the invention possible?
Indian mathematical texts use words for emptiness, including śūnya. Related language appears in philosophical and religious discussions, though different schools did not all mean the same thing by emptiness. Scholars and popular writers have suggested that familiarity with these ideas helped Indian mathematicians conceive zero as a number. This influence is possible, but no surviving evidence directly derives the arithmetic rules from those philosophical ideas.
Nor does any surviving passage show Brahmagupta deriving his mathematics from Buddhist arguments, or show Greek and Roman religious beliefs preventing their scholars from writing zero. Positional notation already gave mathematicians a practical reason to mark vacant places. Repeated operations then raise the question of how zero behaves in subtraction, addition or multiplication. Cultural language could influence the form of an explanation without being the sole cause of the mathematical technique.
To reconstruct this history, the most dependable evidence comes from written calculations, symbols and the rules scholars taught. The familiar contrast between Indian acceptance of emptiness and supposed Western fear of it ignores differences among individual scholars. They also neglect the independent Mesoamerican system. Each mathematical tradition developed its methods through the work of people answering particular problems, and those methods could be passed on or revised over generations.
How Indian zero reached the Islamic world and Europe
Indian decimal numerals and their zero sign eventually spread through scholarly, commercial and political contacts. Arabic-speaking mathematicians were studying Indian computational and astronomical work by the eighth and ninth centuries. In Baghdad, scholars translated, adapted and developed mathematical texts. Transmission involved generations of scholars and their institutions, rather than one traveler carrying a book from one place to another.
Muḥammad ibn Mūsā al-Khwārizmī wrote about calculation with Indian numerals in the early ninth century. His original Arabic treatise on numerals is only partly recoverable, although its methods circulated in Latin. The St Andrews biography of al-Khwārizmī distinguishes that work from his separate and famous treatise on algebra. The Latinized form of his name gave rise to the English algorithm, while algebra derives from al-jabr, a word in the title of the other treatise.
Europeans learned the digits in part through works written in Arabic, which explains the name ‘Arabic numerals’. Their earlier origins lie in Indian mathematics. Arabic-speaking scholars also examined, taught, modified and extended the methods, rather than transmitting them without change. Numeral shapes themselves differed across the Islamic world. Their later European forms were the result of continued adaptation, not identical copies of one ancient set of signs.
Leonardo of Pisa, better known as Fibonacci, completed the Liber Abaci in 1202. For readers who worked with commercial accounts, he explained how Hindu-Arabic positional arithmetic could handle weights, prices, exchange rates and other calculations more efficiently than written Roman numerals. European scholars had encountered these numerals before 1202, and Fibonacci was teaching an existing system. The St Andrews history of zero describes the slower process by which merchants and teachers came to accept and use the system.
Decimal positional notation eventually became ordinary through repeated practical use and instruction. Commercial schools, printed arithmetic manuals and developing bookkeeping practices helped spread its methods in later centuries. Modern science and computing rely on forms of mathematics for which zero is indispensable. That development involved centuries of further mathematical work; Brahmagupta’s rules alone did not determine the forms of later science or computing.
Which civilization should receive credit for zero?
For the zero treated as a number within the decimal system used today, the clearest surviving written evidence points to India. Brahmagupta can be named, his treatise dated to 628 CE, and his rules examined. Aryabhata and the Bakhshali manuscript reveal the earlier mathematical setting in which such work developed. The documents show a continuing Indian mathematical tradition; they do not record one sudden invention of the complete system.
Babylonian scribes had already marked vacant places in sexagesimal calculations. Mesoamerican mathematicians independently gave zero a written role in their calendar and positional numerals. Greek astronomical tables also employed zero-like marks, whereas Roman numerals had no positional vacancy to mark. The differences in these practices explain why several civilizations deserve a place in the history without receiving identical claims about inventing the same arithmetic.
No surviving record identifies the first person to think about an absent quantity. We can trace more specific developments: the writing of an empty place and the statement of rules for calculating with a number called zero. Through Indian mathematical practice and its transmission in the Hindu-Arabic system, zero eventually became part of ordinary calculation across much of the world.
