Brahmagupta

Chapter 1 · Bhillamala and Ujjain

An astronomer in a tradition of astronomers

598 – c. 628 · Bhillamala · Ujjain

Brahmagupta was born in 598, the son of Jishnu, probably at Bhillamala in what is now Rajasthan — a city of the Gurjara kingdom. Almost nothing else about his personal life is recorded, which is normal for an Indian scholar of this period and worth saying rather than filling in.

What is known is the tradition he worked in. Indian astronomy by the seventh century was a mature, competitive discipline organised around texts called siddhantas — systematic treatises covering planetary positions, eclipses, the length of the year, and the mathematics needed to compute them. Astronomers were judged on whether their predictions matched the sky.

He later directed the observatory at Ujjain, which was the foremost centre of astronomical work in India and had been for centuries. That position matters for understanding him: he was not a philosopher speculating about the nature of number. He was a working computer of planetary positions who needed arithmetic that behaved reliably, and who was prepared to write down rules for quantities other people preferred to avoid.

It is worth holding on to that. The reason zero and negative quantities got their arithmetic from an astronomer rather than from a philosopher is that an astronomer had to subtract one position from another and needed an answer whichever way round they came.

Why this matters

The rules that made negatives respectable came out of practical computation, not out of an argument about what numbers really are.

A working astronomer in a competitive tradition. What would you ask him?

Ask Brahmagupta

• “What did an astronomer at Ujjain actually do all day?”
• “What is a siddhanta, and why did astronomers write them?”
• “How were rival astronomers judged against one another?”
• “Why would an astronomer need negative quantities at all?”
• “Why is so little recorded about your life?”

Chapter 2 · The Eighteenth Chapter

Fortunes, debts, and the arithmetic of nothing

628 · Bhillamala

In 628, at the age of thirty, Brahmagupta composed the Brahmasphutasiddhanta — the Correctly Established Doctrine of Brahma — in Sanskrit verse. Most of it is astronomy. Its eighteenth chapter is why he is known outside India.

In it he sets out, as plainly as anyone had ever set out the rules for ordinary numbers, what happens when you calculate with zero and with quantities less than zero. He does not call them positive and negative. He calls them fortunes and debts, and the rules read like statements about money: the sum of two debts is a debt; a fortune and an equal debt make nothing; a debt subtracted from nothing becomes a fortune; the product of two debts is a fortune, and the product of a debt and a fortune is a debt.

Zero he treats as a quantity in its own right rather than as a gap. A number subtracted from itself gives zero; zero added to anything leaves it unchanged; zero multiplied by anything is zero.

Two things are worth being precise about. He did not invent the numeral or the place-holder — an empty column had been marked in Indian arithmetic before him. And he did not prove any of this; his tradition stated rules rather than demonstrating them. What he did was decide that these awkward quantities deserved rules at all, and then write a complete and correct set of them.

Why this matters

This is the point at which negative quantities stop being something to be argued away and become something you can simply calculate with.

Fortunes and debts, set down in verse. What would you ask him?

Ask Brahmagupta

• “What are your rules for calculating with nothing?”
• “Why do you speak of fortunes and debts rather than signs?”
• “What does a debt multiplied by a debt come to, and why that?”
• “Is zero a number, or the absence of one?”
• “Did you prove these rules, or simply state them?”

Chapter 3 · The Rule That Was Wrong

Nothing divided by nothing

628 · Bhillamala

Having given rules for addition, subtraction and multiplication that are all correct, Brahmagupta came to division, and here he made a mistake that is still worth studying.

He stated that zero divided by zero is zero. It is not. There is no quantity that satisfies the requirement, because any number multiplied by zero gives zero, so nothing distinguishes one candidate answer from another. His account of a non-zero quantity divided by zero is also unsatisfactory: he leaves it as a fraction with zero underneath and does not resolve what that means.

It would be easy to be embarrassed on his behalf, and it would miss the point. He was extending arithmetic into territory nobody had mapped, and he got three operations out of four completely right on the first attempt. The fourth is genuinely hard — it took centuries more before division by zero was properly understood as having no value at all, rather than as having some value nobody had yet found.

Later Indian mathematicians noticed. Bhaskara II, five hundred years afterwards, argued that a quantity divided by zero is infinite, which is closer in spirit and still not right by modern standards. The error is a useful thing for a student to see: being first is not the same as being finished, and a person can be radically correct and locally wrong in the same chapter.

Why this matters

A clear, checkable case of a great mathematician being wrong — and of why division by zero is genuinely harder than the other three operations.

Three operations right, one wrong, all in the same chapter. What would you ask him?

Ask Brahmagupta

• “Why did you say that nothing divided by nothing is nothing?”
• “What makes division by zero harder than the other operations?”
• “Does getting one rule wrong undermine the other three?”
• “What did later Indian mathematicians say about your division rule?”
• “How would you have tested whether a rule of yours was right?”

Chapter 4 · Against Aryabhata

What disagreement looked like in Sanskrit verse

628 – 665 · Bhillamala · Ujjain

Brahmagupta was not a gentle colleague. A substantial part of the Brahmasphutasiddhanta consists of attacks on the positions of earlier astronomers, and Aryabhata — who had written a century before — comes in for the sharpest treatment. Brahmagupta names him and disputes him repeatedly, on the length of the day, on the causes of eclipses, on the rotation of the earth.

Some of the criticism is well founded and some of it is not. Aryabhata had proposed that the earth rotates on its axis, which is correct and which Brahmagupta rejected. On other points Brahmagupta's computations were the better ones. The mixture is characteristic of live scientific disagreement, and it is a useful corrective to the idea that older thinkers formed a single agreed body of knowledge.

His tone was strong even by the standards of the genre. Later commentators noted it, and some defended Aryabhata against him. The polemic was not decorative: these were rival systems for predicting where a planet would be, and the disagreements had consequences for calendars and for religious observance.

In 665 he produced a second and more practical work, the Khandakhadyaka, a handbook for computation which, somewhat awkwardly, adopts several of Aryabhata's parameters after all.

Why this matters

Scientific traditions argue with themselves, and a student who thinks of the past as a settled consensus will misread almost everything about it.

A century-old rival, attacked by name and in verse. What would you ask him?

Ask Brahmagupta

• “What did you think Aryabhata got wrong?”
• “He said the earth rotates. Why did you reject that?”
• “Why attack a man who had been dead a hundred years?”
• “Why did your later handbook adopt his numbers?”
• “What was actually at stake in getting a planet's position right?”

Chapter 5 · The Road to Baghdad

How a Sanskrit treatise reached Europe

c. 668 – c. 1500 · Ujjain · Baghdad · Europe

Brahmagupta died around 668; the date is not securely known. What happened to his book afterwards is better documented than his life.

Around 770, an embassy from India arrived at the court of the caliph al-Mansur in Baghdad, bringing astronomical works. Brahmagupta's treatise was among them, and it was translated into Arabic as the Sindhind. It became one of the foundations of Islamic astronomy, and the Indian methods it carried — including the numerals and the arithmetic of zero — passed into the work of scholars there, al-Khwarizmi among them. From Baghdad they travelled west, reaching Latin Europe through translations and, later, through Fibonacci's Liber Abaci.

The negative numbers took very much longer than the numerals. European mathematicians of the Renaissance were still calling negative solutions absurd, false or fictitious in the sixteenth century, nearly a thousand years after Brahmagupta had given them a complete and correct arithmetic. Cardano, in the 1540s, worked with them while describing them as fictitious.

That gap is the most striking thing about him. The mathematics was not lost, and it was not obscure; it was simply not accepted, because accepting it required treating a quantity less than nothing as a real answer rather than as a sign that a problem had been set up wrongly.

Why this matters

A concrete transmission route from seventh-century India to European classrooms — and a thousand-year illustration that having a correct idea is not the same as it being believed.

A book that travelled further than its author ever did. What would you ask him?

Ask Brahmagupta

• “How did your book come to be translated in Baghdad?”
• “Why did Europe take so long to accept negative numbers?”
• “What would you say to a mathematician who calls a negative answer absurd?”
• “Did you expect your rules to travel beyond India?”
• “Which of your results are you proudest of, and is it the famous one?”

What Brahmagupta changed

Brahmagupta gave zero and negative quantities a complete arithmetic in 628, treating both as numbers rather than as absences or impossibilities. Every schoolchild who works with a temperature below freezing or an overdrawn account is using rules first set down in the eighteenth chapter of his treatise. His formula for the area of a cyclic quadrilateral still carries his name, and the transmission of his book to Baghdad around 770 began the chain that carried Indian arithmetic into Islamic and then European mathematics.

A debate that continues

Very little about his life is documented beyond his father's name and his approximate dates. His rule that zero divided by zero is zero is straightforwardly wrong, and his sharp criticism of Aryabhata included rejecting the rotation of the earth, which Aryabhata had right.

Keep exploring — ask Brahmagupta

• “Is zero a number in the same sense that three is?”
• “Why is division by zero different from the other three operations?”
• “How can a correct idea sit unaccepted for a thousand years?”

Related lives

• Muhammad al-Khwarizmi — Who Named Algebra And The Algorithm
• Leonardo Fibonacci — Who Brought Europe The Nine Figures
• Euclid — Who Made Proof The Standard
• Ibn al-Haytham — Who Tested What the Ancients Wrote

Related themes

Negative numbers · Zero and place value · Transmission of mathematics