Greatest Discovery in Mathematics.

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.

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.

Archimedes never nailed it!!

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