Mathematics, this 11 letters word brings trouble to most of people in the world we live today. If you ask 10 people if they hate mathematics there is high chance that more than 6 of them will respond with a big capital YES. Now if we all go back in time to 1500 years ago all of us will hate mathematics except few geniuses (am saying 0.0000001 of current around 7bn world population around 700 individuals in the whole world). I want to explain this and one wonderful discovery that changed the whole situation to mathematics which now an average person can easily learn.
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| We can do maths now |
What mathematical discovery more than 1500 years ago: Is one
of the greatest, if not the greatest, single
discovery in the field of mathematics? Which involved three subtle ideas
that eluded the greatest minds of antiquity, even geniuses such as Archimedes? Was
fiercely resisted in Europe for hundreds of years after its discovery? Even
today, in historical treatments of mathematics, is often dismissed with scant
mention, or else is ascribed to the wrong source? The answer is our modern system of
positional decimal notation with zero, together with the basic arithmetic
computational schemes, which were discovered in India about 500 CE.
As the 19th century mathematician
Pierre-Simon Laplace explained:” It is India that gave us the ingenious
method of expressing all numbers by means of ten symbols, each symbol receiving
a value of position as well as an absolute value; a profound and important idea
which appears so simple to us now that we ignore its true merit. But its very
simplicity and the great ease which it has lent to all computations put our arithmetic
in the first rank of useful inventions; and we shall appreciate the grandeur of
this achievement the more when we
remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”
As Laplace noted, the scheme is anything but “trivial,”
since it eluded the best minds of the ancient world, even extraordinary
geniuses such as Archimedes. Archimedes saw far beyond the mathematics of his
time, even anticipating numerous key ideas of modern calculus and numerical
analysis. He was also very skilled in applying mathematical principles to engineering
and astronomy. Nonetheless he used the traditional Greek-Roman numeral system
for performing calculations. It is worth noting that Archimedes’ computation of
π was a tour de force of numerical
interval analysis performed without either positional notation or trigonometry.
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| Archimedes never nailed it!! |
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| This is not true old man! |
Perhaps one reason this discovery gets so little attention
today is that it is very hard for us to appreciate the enormous difficulty of
using Roman numerals, counting tables and abacuses. As Tobias Dantzig (father
of George Dantzig, the inventor of linear programming) wrote, Computations
which a child can now perform required then the services of a specialist, and
what is now only a matter of a few minutes [by hand] meant in the twelfth
century days of elaborate work.
Michel de Montaigne, Mayor of Bordeaux and one of the, most learned men of his day confessed
in 1588 (prior to the widespread adoption of decimal arithmetic in Europe) that
in spite of his great education and erudition, “I cannot yet cast account either
with penne or Counters.” That is, he could not do basic arithmetic.
In a similar vein, at about the same time a wealthy German
merchant, consulting a scholar regarding which European university offered the
best education for his son, was told the following: If you only want him to be able
to cope with addition and subtraction, then any French or German university
will do. But if you are intent on your son going on to multiplication and
division—assuming that he has sufficient gifts—then you will have to send him
to Italy.
The best source currently available on the history of our
modern number system is by French scholar Georges Ifrah . He chronicles in encyclopedic
detail the rise of modern numeration from its roots in primitive hand counting
and tally schemes, to the Babylonian, Egyptian, Greek, Roman, Mayan, Indian and
Chinese systems, and finally to the eventual discovery of full positional
decimal arithmetic with zero in India, and its belated, kicking-and-screaming
adoption in the West. Ifrah describes the significance of this discovery in
these terms: Now that we can stand back
from the story, the birth of our modern number-system seems a colossal event in
the history of humanity, as momentous as the mastery of fire, the development
of agriculture, or the invention of writing, of the wheel, or of the steam
engine.
Indeed, the development of this system hinged on three key
abstract (and certainly non-intuitive) principles:
(a) The idea of attaching to each basic figure graphical
signs which were removed from all intuitive associations, and did not visually
evoke the units they represented;
(b) The idea of adopting the principle according to which
the basic figures have a value which depends on the position they occupy in the
representation of a number; and
(c) The idea of a
fully operational zero, filling the empty spaces of missing units and at the
same time having the meaning of a null number.
It is astonishing how many years passed before this system
finally gained full acceptance in the rest of the world. There are indications
that Indian numerals reached southern Europe perhaps as early as 500 CE, but
with Europe mired in the Dark Ages, few paid any attention. Similarly, there is
mention in Sui Dynasty (581–618 CE) records of Chinese translations of the
Brahman Arithmetical Classic, although sadly none of these copies seem to have
survived. The Indian system (also known as the Indo-Arabic system) was introduced
to Europeans by Gerbert of Aurillac in the tenth century. He traveled to Spain
to learn about the system first-hand from Arab scholars, prior to being named
Pope Sylvester II in 999 CE. However, the system subsequently encountered stiff
resistance, in part from accountants who did not want their craft rendered
obsolete, to clerics who were aghast to hear that the Pope had traveled to
Islamic lands to study the method. It was widely rumored that he was a
sorcerer, and that he had sold his soul to Lucifer during his travels. This
accusation persisted until 1648, when papal authorities reopened Sylvester’s
tomb to make sure that his body had not been infested by satanic forces.
The Indo-Arabic system was reintroduced to Europe by
Leonardo of Pisa, also known as Fibonacci, in his 1202 CE book Liber Abaci.
However, usage of the system remained limited for many years, in part because
the scheme continued to be considered “diabolical,” due in part to the mistaken
impression that it originated in the Arab world (in spite of Fibonacci’s clear
descriptions of the “nine Indian figures” plus zero). Decimal arithmetic began
to be widely used by scientists beginning in the 1400s, and was employed, for
instance, by Copernicus, Galileo, Kepler and Newton, but it was not universally
used in European commerce until after the French Revolution in 1793 . In
limited defense of the Roman system, it is harder to alter Roman entries in an
account book or the sum payable in a cheque, but this does not excuse the
continuing practice of using Roman numerals and counting tables for arithmetic.
The Arabic world, by comparison, was much more accepting of the Indian
system—in fact, as mentioned briefly above; the West owes its knowledge of the
scheme to Arab scholars.
So who exactly discovered the Indian system? Sadly, there is
no record of the individual who first discovered the scheme, who, if known,
would surely rank among the greatest mathematicians of all time. The very
earliest document clearly exhibiting familiarity with the decimal system is the
Indian astronomical work Lokavibhaga (“Parts of the Universe”). Here, for
example, the number 13,107,200,000 is written as panchabhyah khalu shunyebhyah
param dve sapta chambaram ekam trini cha rupam cha (“five voids, then
two and seven, the sky, one and three and the form”), i.e.0 0 0 0 0 2 7 0 1 3
1, which, when written in reverse order, is 13,107,200,000. One section of this
same work gives detailed astronomical observations that confirm to modern
scholars that this was written on the date it claimed to be written: 25 August
458 CE (Julian calendar). As Ifrah points out, this information not only
allows us to date the document with precision, but I also proves its
authenticity. Methods for computation were not explicitly mentioned in this work,
although the frequent appearance of large numbers suggests that advanced
arithmetic schemes were being used. Fifty-two years later, in 510 CE,
the Indian mathematician Aryabhata explicitly described schemes for
various arithmetic operations, even including square roots and cube roots,
which schemes likely were known earlier than this date. Aryabhata’s actual
algorithm for computing square roots, as described in greater detail in a
628 CE manuscript by a faithful disciple named Bhaskara I, . Additionally,
Aryabhata gave a decimal value of π = 3.1416 . From these and other
sources there can be no doubt that our modern system of
arithmetic—differing only in variations on the symbols used for the
digits and minor details of computational schemes—originated in India at
least by 510 CE and quite possibly by 458 CE. The Greatest Mathematical Discovery? David
H. Bailey ∗ Jonathan M. Borwein †
May 12, 2010.
Adapted from http://www.davidhbailey.com/dhbpapers/decimal.pdf written by David H. Bailey and Jonathan M. Borweiny
