In 628 CE, Brahmagupta Wrote Down Rules for Zero and Negative Numbers, a Quadratic Solution and an Area Formula for Four-Sided Figures

Brahmagupta’s Brahmasphutasiddhanta, completed in 628 CE, lays out how zero and negative numbers behave when they are added, subtracted and multiplied. The Bodleian Libraries at Oxford call it the first document to discuss zero as a number, and a Springer reference work on astronomers says its author was about 30 when he wrote it.

The rules sit inside a book of astronomy, written in verse, in a chapter on algebra that also holds a way to solve quadratic equations and a challenge problem for readers. This piece covers the mathematics. How the decimal system and the zero symbol grew over the centuries is in the article on India and zero.

A treatise finished at about 30, in Bhillamala

Both MacTutor, the history-of-mathematics archive at the University of St Andrews, and the Springer Biographical Encyclopedia of Astronomers give 598 as the year of Brahmagupta’s birth. On the place they differ: Ujjain is a possibility for MacTutor, Bhillamala the answer in Springer. Either way, he wrote the Brahmasphutasiddhanta at Bhillamala, now the town of Bhinmal in the southwest of Rajasthan and, in MacTutor’s description, the capital of lands ruled by the Gurjara dynasty. His father was Jisnugupta.

An older witness for 628 appears in the 1817 introduction by the English scholar Henry Colebrooke. Astronomers at Ujjain told William Hunter, who had stayed there with a British embassy, that Brahmagupta belonged to the Saka year 550, which Colebrooke converts to 628 CE. Their grounds, he adds, were not recorded.

Later he headed the astronomical observatory at Ujjain, which MacTutor calls the foremost mathematical centre of ancient India at the time. A second work, the Khandakhadyaka, followed in 665, when he was 67, MacTutor says. As for his death, MacTutor gives 670, while Springer says only “after 665”.

Twenty-five chapters, two of them mathematics

The Brahmasphutasiddhanta has 25 chapters in MacTutor’s count and 24 in Springer’s, which notes that some editions add a chapter of versified tables; MacTutor’s list includes those tables. Many historians think the first ten chapters were an earlier version, MacTutor says, and that the other fifteen are a major addition covering topics such as algebra, the gnomon (a vertical rod used to measure shadows), the sphere and instruments. Springer reports that Brahmagupta describes the work as an improved version of an earlier astronomical system attributed to Brahma.

Chapter 12, on arithmetic, and chapter 18, on algebra, hold the mathematics. English readers got both from Colebrooke, who translated them with the later commentary and published them in London in 1817. He argues in his introduction that Brahmagupta’s chapters are older than the Arabic works on algebra.

What the book says about zero

Zero, in MacTutor’s account of the treatise, is defined as the result of subtracting a number from itself. Brahmagupta’s rules for it come in the algebra chapter, in a section Colebrooke titles “Algorithm”, and cover addition, subtraction, multiplication and division.

For addition, Colebrooke’s translation reads: “The sum of cipher and negative is negative; of affirmative and nought is positive; of two ciphers is cipher.” Here “cipher” and “nought” mean zero. Subtraction reads: “Negative, taken from cipher, becomes positive; and affirmative, becomes negative.” In the language of money that MacTutor uses, positive numbers are fortunes and negative numbers are debts, and a debt subtracted from zero is a fortune.

Multiplication comes next: a negative times an affirmative is negative, two negatives make a positive, and zero times anything is zero. Pupils learn the same rules today; the treatise that states them is nearly 1,400 years old.

In MacTutor’s history of zero, Brahmagupta is the first person known to have tried to extend arithmetic to negative numbers and zero. Background comes from the Bodleian Libraries, which hold the Bakhshali manuscript: a dot used as a placeholder for an empty place appears throughout that birch-bark text. Carbon dating put its earliest material as early as the third or fourth century, though the manuscript’s age has been debated and an earlier study placed it between the eighth and twelfth centuries. In the Bodleian’s account, the step from placeholder to number in its own right was made in India, in 628, in Brahmagupta’s treatise. Aryabhata’s knowledge of zero, 129 years earlier in 499, is taken up in the piece on his mathematics.

A quadratic rule written in words

Chapter 18 also gives a rule for equations that contain a square, an unknown and a plain number. Colebrooke’s translation, in its section on the quadratic equation, says to multiply the plain number by four times the number in front of the square, and add the square of the number in front of the unknown. Take the square root of the result, subtract the number in front of the unknown, and divide by twice the number in front of the square.

Written in modern symbols as ax² + bx = c, the rule gives x equal to the square root of (4ac + b²), minus b, all divided by 2a. A reader can test it on x² + 2x = 3: the square root of 16 is 4, less 2 is 2, divided by 2 is 1, so x is 1. Only the positive answer appears in the verse.

The pulveriser and a challenge to solve in a year

Chapter 18 opens by calling the pulveriser (kuttaka) the key to solving questions. The method finds a whole number that leaves given remainders when divided by given numbers. Colebrooke’s text works an example: which number leaves 5 when divided by 6, 4 when divided by 5, 3 when divided by 4 and 2 when divided by 3? It gives 59 as the answer. Aryabhata’s shorter verses on the same method are covered in the article linked above.

Brahmagupta also set a problem for his readers. Take a number, square it, multiply by 92, add one, and make the result a perfect square. Colebrooke’s translation says that a person who solves it within a year is a mathematician. Its smallest answer is 120, which can be checked by hand: 92 times 120 squared, plus 1, is 1,324,801, which is 1,151 squared. MacTutor adds that he found a series of solutions to the related equation 8x² + 1 = y², among them (1, 3), (6, 17) and (35, 99).

Areas, circles and the case of the quadrilateral

The arithmetic chapter gives a formula for the “exact” area of a triangle or four-sided figure. Colebrooke translates it as: half the sum of the sides, set down four times and each lessened by a side, are multiplied together, and “the square-root of the product is the exact area.” In a triangle, one of the four lessened numbers is left as the half-sum itself, Colebrooke notes. His commentary works a triangle with sides 14, 13 and 15: half the sum is 21, the four numbers are 7, 8, 6 and 21, their product is 7,056, and its root is 84.

MacTutor describes the same rule as a formula for the area of a four-sided figure whose corners lie on a circle, with a matching formula for its diagonals. It adds that Brahmagupta does not say the formula holds only for such figures. Some historians treat that as an error, MacTutor says, while others read him as meaning only that case.

Circles get two values each for circumference and area in Colebrooke’s translation. Multiplying by three gives the “practical” value; the square root of ten times the square gives the “neat” one. Applied to the circumference, that means the diameter times the square root of ten, about 3.16 times the diameter.

Everyday problems fill the arithmetic chapter too. One, quoted by MacTutor, concerns 500 drammas lent at an unknown rate of interest. Four months’ interest on it is lent to another person at the same rate and comes to 78 drammas in ten months; the reader must find the rate.

A critic of Aryabhata, and the one place he is silent

Brahmagupta’s eleventh chapter criticises earlier astronomers, mostly Aryabhata. In W. E. Clark’s account, Brahmagupta refers to Aryabhata some sixty times, and most of the harshest criticism concerns points where Aryabhata departed from tradition. Clark adds that no stanza from Aryabhata’s mathematics section is quoted or criticised, a silence he suggests may mean Brahmagupta found nothing to object to.

On astronomy, Brahmagupta believed in a static Earth, MacTutor says, and gave the length of the year as 365 days 6 hours 5 minutes 19 seconds in the first work. In the Khandakhadyaka he moved to 365 days 6 hours 12 minutes 36 seconds, which MacTutor does not count as an improvement. Springer adds that in the first part of that later work he followed the Ardharatrika school, which it says Aryabhata I founded. On the question of the Earth’s rotation, see the earlier article on Aryabhata.

From Ujjain to Baghdad

In MacTutor’s overview of Indian mathematics, Brahmagupta is the next figure of major importance at the Ujjain school, with one of the major contributions to number systems through his work on negative numbers and zero. His astronomy reached Arabia in the latter half of the eighth century, Springer says.

M. S. Khan, in a paper for a 1987 International Astronomical Union colloquium, reports that al-Biruni recorded the Brahmasphutasiddhanta as translated on the orders of the Caliph al-Mansur of Baghdad, in the year 154 or 156 of the Islamic calendar, which falls between 770 and 773 CE. That Arabic version was called the Great Sindhind. Colebrooke, citing the preface to Arabic astronomical tables published in 920, reports that an Indian astronomer brought tables to al-Mansur’s court in the year 156 of the Islamic calendar (773 CE) and that Muhammad al-Fazari translated the book as the greater Sind-hind. The link between that name and the Sanskrit siddhanta is a conjecture, Colebrooke says.

According to MacTutor’s biography of al-Khwarizmi, the Indian text behind his own Sindhind zij (an astronomical table book) reached the Baghdad court around 770 as a gift from an Indian political mission. His work on the Hindu art of reckoning was the first in what is now Iraq to use zero as a placeholder, MacTutor’s history of zero adds.

Later Indian mathematicians kept working on Brahmagupta’s open question, division by zero. In his rules he wrote that zero divided by zero is zero, which MacTutor judges to be wrong, while calling the whole effort a brilliant attempt to extend arithmetic. About 830, Mahavira wrote the Ganita Sara Samgraha as an update of Brahmagupta’s book, and more than 500 years after Brahmagupta, Bhaskara was still working on the same problem.

MacTutor’s history of zero records that Fibonacci described the nine Indian digits together with the sign 0 in Liber Abaci around 1200, calling 0 a sign and the other nine symbols numbers.