mathematician named Brahmagupa stated the law of large numbers back in the

7th century AD, as I mentioned in the podcast. This got me curious, since I

hadn't heard much about this figure before. Who was Brahmagupta? What were

his mathematical contributions? I did a little web surfing, and found quite

a bit of colorful information.

Brahmagupta was born around 598 AD, in Ujjain, India. Eventually he

became head of the astronomical observatory there, and is most famous for

his book Brahmasphutasiddhanta (The Opening of the Universe), published in

628 AD, and its sequel Khandakhadyaka published in 667. His books were

published in elliptic verse, which seems kind of silly, but actually had

quite a good reason: when actual books are expensive or nonexistent, most

knowledge is transmitted orally. It's a lot easier to memorize and repeat

a verse rather than random prose.

One of his major contributions was to be the first known author to treat

zero as a true number, rather than a placeholder: he specified rules for

dealing with positive and negative numbers, including cases where they sum

to zero. Here's a piece of his translated verse I found on the web, where a

'fortune' refers to a positive number, and a 'debt' to a negative:

The product of zero multiplied by zero is zero.

The product or quotient of two fortunes is one fortune.

The product or quotient of two debts is one fortune.

The product or quotient of a debt and a fortune is a debt.

The product or quotient of a fortune and a debt is a debt.

Unfortunately, he spoiled this accomplishment a bit by going on to state

that 0 divided by 0 is 0, which does not really make sense in most contexts.

Once he was able to recognize zero as a number, this also enabled

a place-value multiplication algorithm, similar to the long

multiplication that we all learned in elementary school, or at least

should have if our teacher kept us away from calculators. He called this

method "gomutrika", which according to one source is translated as "like the

trajectory of a cow's urine".

Brahmagupta spent a lot of time on geometry and trigonometry, as would

be expected for an astronomer. He provided a table of sines, again

using poetic language: "twins" means 2, "Ursa Major" seven due to the

number of stars of Ursa Major, "Vedas" is 4 due to the 4 Vedas, and "dice"

represents the number 6, for example. He developed an interpolation

formula for computing the sine of a number when nearby larger and smaller

sines are known, which turned out to be a special case of a later formula

developed by Newton.

In the astronomy realm, an area of heated debate in Brahmagupta's day

was the question of whether the Earth could really be a spinning sphere. He

correctly (to some approximation) refuted the idea that everything would be

flung off by introducing the idea of gravity, a force attracting all objects

to the center of the earth. He also refuted the scriptural idea that the

Moon is farther from the earth than the sun, by looking at the moon's

patterns of waxing and waning and explaining it using the angles formed by

different positions of the sun, moon, and earth.

Of course, I'm just scratching the surface here. He also described

various geometric theorems, solutions for some Diophantine equations and the

first written description of the quadratic formula, and made many

astronomical contributions. Unfortunately, he often did not provide

proofs of his results, which is probably one reason why his name is

not as well known today as some Western mathematicians. His works

were brought into the Arab world and were a major influence to

al-Khwarazimi, remembered today as the father of algebra.

And this has been your math mutation for today.

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