40 Fun Facts About Brahmagupta

Brahmagupta called numbers above zero fortunes; what money word did he use for numbers below zero?

Debts were his picture of numbers below zero, and the image made the rules easy to follow. Take a debt away from nothing, he wrote, and you are left with a fortune.

Brahmagupta's multiplication method gomutrika takes its name from the trail of urine left by which animal?

Cow urine gave gomutrika its name, yet the method itself is close to the long multiplication taught in schools today. It sets out shifted rows of partial products and adds them up.

Which of Brahmagupta's rules for calculating with zero is the one that modern mathematics rejects?

0 ÷ 0 = 0 was his answer, where mathematicians now call that division undefined. The other three rules shown here, set down in 628, are still taught today.

Brahmagupta wrote that it is the earth's nature to attract and keep things, describing which force?

Gravity was a pull in his account: heavy things fall to the earth, he wrote, by a law of nature. He set that hold beside the way water naturally flows.

Brahmagupta's great treatise has no mathematical symbols because the whole book is written in what?

Verse carries the entire work, so every rule and every number is spelled out in words. Even so, it holds the first clear description of a formula taught in every algebra class.

Brahmagupta's arithmetic reached Europe via al-Khwarizmi, whose garbled name gave English which word?

Algorithm comes from Algoritmi, the European spelling of al-Khwarizmi's name. His textbook on Indian reckoning carried Brahmagupta's methods of arithmetic west.

Brahmagupta said a person who could solve his equation x² − 92y² = 1 within what span was a mathematician?

A year was the allowance, which shows how hard he judged the puzzle to be. His own smallest solution was x = 1151 and y = 120.

Brahmagupta rebutted the old idea that which body lies farther from Earth than the Sun does?

Moon phases were his proof: if it sat beyond the Sun, he reasoned, the half facing us would always be lit. He compared it to a pot standing in sunlight, bright on one side and dark on the other.

Brahmagupta gave the square root of ten as his accurate value for which Greek-letter constant?

Pi came out as about 3.162 by this rule, less than one percent too high. For everyday sums he simply used 3.

Brahmagupta shares the name of an identity about sums of two squares with which Italian mathematician?

Fibonacci published the identity in Europe in 1225, six centuries after Brahmagupta had used it in India. It says that multiplying two sums of two squares always gives another sum of two squares.

Brahmagupta defined zero as the result you get from doing what simple thing to a number?

Subtracting it from itself was his definition, which turned zero from an empty space into a number you could calculate with. He added that putting zero on or taking it off leaves a number unchanged.

Brahmagupta gave the first clear description of the formula that solves which type of equation?

Quadratic equations get their famous formula from his treatise of 628, where it is written out entirely in words. He gave two equivalent versions of the rule.

Brahmagupta's formula gives the area of a four-sided figure whose corners all lie on which shape?

A circle through all four corners is the condition, and then the area follows from the four side lengths alone. Brahmagupta never stated the condition himself, though his rules make it clear.

Brahmagupta's arithmetic, retold by al-Khwarizmi, reached medieval Europe written in which language?

Latin translations of al-Khwarizmi's book introduced positional Indian numerals to the Western world. Brahmagupta's rules for arithmetic travelled with them.

Brahmagupta's astronomy was translated into Arabic in the 770s in which newly built capital?

Baghdad was then a brand-new city, built by the same caliph who ordered the translation. The texts had arrived with an embassy from Sindh, possibly carried in an astrologer's memory.

In which classical language of India did Brahmagupta compose both of his treatises on astronomy?

Sanskrit was the language of both books, and its titles can be a mouthful: Brahmasphutasiddhanta means the correctly established doctrine of Brahma. He composed it at the age of thirty.

Brahmagupta gave a rule for making whole-number right triangles, sets named after which Greek?

Pythagoras lends his name to the triples, such as 3, 4 and 5. Brahmagupta dressed his rule up as a story about two travellers leaving a mountain top for the same city.

Brahmagupta's square root method matches a formula named after Raphson and which English scientist?

Isaac Newton's name is on the formula, yet Brahmagupta's rule for square roots is equivalent and was written in the seventh century. Each round of the method closes in on the true root.

Brahmagupta's translated books helped carry which number system from India into the Islamic world?

Decimal place-value numbers were used throughout his texts, so the translation spread them at once. Europe received the same system centuries later.

For all his skill as an astronomer, what did Brahmagupta believe about the motion of the Earth itself?

It stands still, he held, keeping to the prevailing view that the sky does the turning. An earlier Indian astronomer had already argued that the planet rotates each day.

One whole chapter of Brahmagupta's Brahmasphutasiddhanta is devoted to doing what to earlier astronomers?

Attacking their theories fills the entire eleventh chapter, and digs at rivals run through the ten before it. The quarrels were over astronomical numbers and models, not the mathematics.

Brahmagupta's four-sided area formula becomes which Greek's triangle formula when one side shrinks to nothing?

Heron's triangle rule is a special case of the Indian formula, because a triangle is just a four-sided figure with one side of length zero.

Which mathematician wrongly named the equation Brahmagupta pioneered after the Englishman John Pell?

Euler credited the wrong man: the solution was Brouncker's, and Pell had only revised a book that discussed it. Brahmagupta had tackled the equation about ten centuries before either.

The method Brahmagupta used to break numbers into ever smaller pieces is known by what vivid name?

The pulverizer, kuttaka in the original, is essentially Euclid's procedure for finding common divisors. Brahmagupta recommended it whenever an equation had many unknowns.

Brahmagupta devised a rule in 665 to fill the gaps in a table of which trigonometric function?

Sine tables listed values only at fixed steps, so astronomers had to estimate the ones in between. His rule used second differences, a step beyond the straight-line estimates made before him.

In Brahmagupta's number code, where objects replace digits, which constellation's name means seven?

Ursa Major has seven bright stars, so its name stood for seven, just as twins meant two and dice meant six. The code was a common way to set down numerical data in Indian treatises.

Brahmagupta nicknamed astronomical calculation after which material that covered the boards used for sums?

Dust work, dhulikarman, was his phrase for the sums of astronomy. Indian students wrote out their working and drew diagrams on boards covered with the stuff.

Brahmagupta wrote his great treatise at Bhillamala, now the town of Bhinmal, in which modern Indian state?

Rajasthan's Bhinmal was then Bhillamala, capital of the kingdom of Gurjaradesa and a centre of learning. A later commentator called Brahmagupta simply the teacher from Bhillamala.

Brahmagupta is said to have headed the astronomical observatory in which ancient Indian city of learning?

Ujjain was the foremost mathematical centre of ancient India, where Varahamihira had worked before him. Brahmagupta moved there in later life and is presumed to have died in the city.

Which twelfth-century Indian called Brahmagupta a gem among mathematicians and extended his algebra?

Bhaskara II, working around 1150, set out a general method for the equations Brahmagupta could solve only in special cases. That was about five centuries after the older master's death.

Brahmagupta's handbook Khandakhadyaka answered which earlier astronomer, namesake of India's first satellite?

Aryabhata is mentioned by name in more than a hundred places in Brahmagupta's writing. The Khandakhadyaka was composed as a response to one of the older astronomer's works.

Brahmagupta's second book, the astronomy handbook Khandakhadyaka, has a title that translates as what?

Edible bite suits a book meant to be easy to digest: a practical manual for students, written when Brahmagupta was 67.

Brahmagupta made zero a true number, where which earlier people had used it only as a placeholder?

The Babylonians marked an empty position inside a larger number but went no further. Brahmagupta's treatise is the earliest known text to treat zero as a number in its own right.

Brahmagupta generalised an identity first found by which Greek algebraist, author of the Arithmetica?

Diophantus was content with one solution to an indeterminate equation, where Brahmagupta gave every whole-number solution. The two used some of the same examples, which has led historians to suspect a link between them.

The later Indian algebra text Bijaganita went beyond Brahmagupta by declaring that a number divided by zero equals what?

Infinity was the verdict, likened in the text to an unchanging god who gains or loses nothing whatever is added or removed. Modern mathematics leaves the division undefined instead.

Brahmagupta's handbook Khandakhadyaka uses a system in which each day begins at which moment?

Midnight starts the day in this handbook, as it does on modern clocks. The idea came from an earlier astronomer whose lost work can be partly rebuilt from Brahmagupta's discussion of it.

A Brahmagupta triangle has a whole-number area and three side lengths that follow which pattern?

Consecutive integers such as 3, 4, 5 or 13, 14, 15 make the sides. Brahmagupta listed the first eight triangles of this kind without saying how he found them.

The Arabic version of Brahmagupta's great treatise became known by which name?

Sindhind, a garbling of the Indian word siddhanta, became the name for a whole family of astronomical tables in the Islamic world. Even al-Khwarizmi produced his own version.

Which Jain mathematician reworked Brahmagupta's book and said a number divided by zero is unchanged?

Mahavira wrote the earliest Indian text devoted entirely to mathematics, leaving the astrology out. His rule for dividing by zero was wrong, though, as that division has no defined result.

Brahmagupta's theorem concerns a four-sided figure whose two diagonals are what to each other?

Perpendicular diagonals are the condition: a line from their crossing point, dropped square onto one side, then always cuts the opposite side in half.