Wednesday, December 28, 2011

104: Fortunes, Debts, Ursa Major, and Cow Urine

    In my research for the last episode, I found that an Indian
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.

  • An Online Biography of Brahmagupta
  • Brahmagupta on Wikipedia
  • Another Article
  • Brahmagupta at Wolfram Science World
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