Bakhshali Manuscript: Ancient Indian Mathematics and Zero

The Bakhshali manuscript is a fragmentary Indian mathematical text written on birch bark, found in 1881 near the village of Bakhshali in what is now north-western Pakistan. It is the oldest surviving manuscript of Indian mathematics and uses a dot to mark zero in decimal numbers. Radiocarbon tests in 2017 dated some of its leaves as early as AD 224–383, a claim that made headlines around the world and remains strongly debated.

Discovery

According to the earliest reports, a farmer found the manuscript in 1881 while digging among ruins at Bakhshali, near Mardan, in the north-western frontier region of British India. Much of it had already crumbled. The surviving pieces reached the Indologist Augustus Hoernle, who published a first description in 1883 and a fuller study later in the decade.

In 1902 the manuscript was given to the Bodleian Library in Oxford, where it is still kept. Between 1927 and 1933 the British scholar G. R. Kaye published a full edition. The standard modern study is the edition and translation by the Japanese historian of mathematics Takao Hayashi, published in 1995.

Mathematics in early India

Indian mathematics grew out of practical and religious needs. The Shulba Sutras, manuals for building Vedic fire altars composed in the 1st millennium BC, contain geometric rules for constructing shapes of equal area. Astronomy, used for calendars and rituals, demanded accurate calculation, and merchants needed arithmetic for trade, interest and partnership.

By the middle of the 1st millennium AD, Indian scholars had developed a decimal place-value system with nine digits and a sign for zero. Astronomers of the era of the Gupta Empire, such as Aryabhata, wrote their works in verse with number words, while working texts like the Bakhshali manuscript show how numbers were actually written and calculated with on the page.

What the manuscript contains

About 70 leaves survive, many broken, written in the Sharada script used in the north-west of India. The language is a form of Sanskrit with features of local Prakrit speech. The opening and ending are lost, so the original title and author are unknown.

The text follows the classic Indian pattern: a rule (sutra) in verse, then one or more worked examples and a solution. Topics include fractions, square roots, arithmetic progressions, problems of income and expenditure, profit and loss, and linear equations with several unknowns. There are also quadratic problems and questions of the kind modern mathematicians call indeterminate equations.

Many problems are framed as stories about everyday life. Typical examples involve merchants sharing profits, travellers moving at different speeds who must meet, or the amounts paid to workers and the prices of goods. A verification step usually follows the solution, in which the answer is checked by substituting it back into the problem, a habit that shows the text was used for teaching.

Notation and the dot for zero

The manuscript writes numbers in the decimal place-value system, with nine digits and a solid dot standing for an empty place. For example, a numeral with the dot in the middle shows the absence of tens, just as 0 does in 105. This dot is the feature that has drawn most public attention.

The same dot also served another purpose: in equations it marks an unknown quantity, like a blank to be filled in. The manuscript lays out problems in boxes, with fractions written one number above another without a dividing line, and marks negative quantities with a sign that looks like a modern plus written after the number.

The square root method

One celebrated rule gives a way of approximating the square root of a number that is not a perfect square. Starting from the nearest perfect square, it adds a first correction and then subtracts a second, smaller correction. For many numbers the result is accurate to several decimal places, and it is equivalent to applying a step of what is now called Newton’s method twice.

Worked examples in the manuscript apply this rule to particular numbers, showing a practical concern with precision rather than abstract theory.

The dating controversy

Dating the manuscript has always been difficult. Kaye placed it as late as the 12th century, while Hayashi argued from the content that the text was composed around the 7th century, with this copy perhaps a few centuries later.

In 2017 the Bodleian Library announced radiocarbon results for three samples. They gave three widely separated ranges: AD 224–383, AD 680–779 and AD 885–993. The library publicised the earliest date as evidence for the oldest recorded zero symbol.

Several leading historians of Indian mathematics, including Kim Plofker and Takao Hayashi, published a critique. They argued that a single manuscript in one hand and style is very unlikely to have been written over seven centuries, that birch bark could have been stored for a long time before use, and that the content fits best with the later dates. The question is unresolved, and many scholars treat the 3rd-century date with caution.

The debate also showed the limits of radiocarbon dating for manuscripts. The method measures when the birch tree’s bark stopped absorbing carbon, not when ink was put on it. Further tests on more leaves, together with study of the script and the mathematical content, may narrow the range in future.

Who used it

The manuscript itself says nothing about its owner, but its find-spot in the Peshawar valley, its script and its language point to a scholarly or commercial setting in the north-west of the Indian subcontinent. The many worked examples, step-by-step checks and repeated layouts suggest it was a commentary or teaching copy rather than a polished treatise. Its anonymous author, or authors, drew on an older body of rules that circulated widely in India, and some of its problems have parallels in later Sanskrit mathematical works.

Myths and facts

• Myth: The Bakhshali manuscript proves zero was invented in the 3rd century AD. Fact: The radiocarbon date applies to one leaf of bark, not necessarily to when it was written, and is disputed.
• Myth: The dot is zero as a number. Fact: In the manuscript it is mainly a placeholder; the arithmetic of zero as a number is set out by Brahmagupta in AD 628.
• Myth: The manuscript is the oldest evidence of the decimal system. Fact: Place-value numerals appear in Indian texts and inscriptions from roughly the same centuries; the manuscript is an important witness, not a unique one.

Timeline

• AD 224–383: Radiocarbon range for the oldest tested leaf.
• AD 628: Brahmagupta sets out rules for calculating with zero.
• AD 680–779 and 885–993: Radiocarbon ranges for two other tested leaves.
• 1881: The manuscript is found at Bakhshali near Mardan.
• 1883: Augustus Hoernle publishes the first description of the manuscript.
• 1902: The manuscript enters the Bodleian Library, Oxford.
• 1995: Takao Hayashi publishes the standard edition and translation.
• 2017: The Bodleian Library announces radiocarbon dates, prompting scholarly debate.

Why it matters

Whatever its exact age, the Bakhshali manuscript is a rare surviving copy of a working Indian mathematical text from the first millennium AD. It shows decimal place-value numbers with a sign for zero in everyday use, alongside sophisticated algebra, and so belongs to the same tradition as Aryabhata’s Aryabhatiya. That tradition later passed through the Islamic world to Europe and gave the world its modern numerals.