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What Do Mathematicians Do?

darshanhi — Tue, 23 Jan 2007 09:57:03 CST

As a break from the tradition of this newsletter, this article is meant to provoke discussion. Most research mathematicians are quite passionate about their subject. Yet they are aware that their enthusiasm is not shared (to put it mildly) by the public at large, and even, in many cases, by research scientists. Is this just a case of "love is blind", or is it possible that mathematicians are aware of something about mathematics that outsiders are not? I'd like to investigate this matter in this article. I hope that in doing so I will stimulate others, both mathematicians and non-mathematicians, to think about these questions, and maybe even contribute their thoughts to later issues of the newsletter.

I think it's very important to start by asking the right question. Typically, academic disciplines are defined by their subject matter. So, to ask what a geologist does is more or less the same thing as to ask what a geologist studies. Thus, for the Oxford English Dictionary, it is "the science which has for its object the investigation of the earth's crust, of the strata which enter into its composition, with their mutual relations, and of the successive changes to which their present condition and positions are due". Similarly, for the OED, biochemistry is "the science dealing with the substances present in living organisms and with their relation to each other and to the life of the organism". Moving away from science, we have the OED's definition of architecture, "the art or science of building or constructing edifices of any kind for human use"; economics, the study of "the development and regulation of the material resources of a community or nation"; and linguistics, "the study of languages". These examples were chosen at random. In every case I expect that the reader's definition would be very similar to the one given by the dictionary.

Well, if such definitions are so easy for other disciplines, why not for mathematics? Like most people, the OED assumes that mathematics too can be defined by its subject matter and tries "the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; and, in a wider sense, so as to include those branches of physical or other research which consist in the application of this abstract science to concrete data". A good effort, but one gets the strong impression that whoever wrote it was struggling! The "spatial and numerical relations" obviously cover geometry and arithmetic, but then algebra had to be added because it wasn't dealt with. However, that's nowhere near good enough. Important "divisions" like analysis, probability, set theory and operational research are completely ignored by this definition, so clearly it's very inadequate. Should we compensate by listing the titles of, say, all mathematics modules taught at NUS, in the hope that we'll cover the subject that way? That attempt is doomed too, because a glance at the list soon reveals courses on topics like filter banks, chaos and fractals, cryptography, game theory, etc., that weren't there ten or twenty years ago. If the subject is to be defined by a list constructed at a certain time, then after that time, no newcomers can ever join. However, it's clear that mathematics is continuing to grow, its tentacles finding their way into areas of investigation previously thought beyond its reach.

The first attempt I heard to define mathematics by what it studies was by T G Room (a geometer commemorated by Room squares in combinatorics). He reckoned that mathematics is the study of relationships between concepts. Although this is helpful to the non-mathematician, it is clearly inaccurate. There are many concepts, like punishment and retribution, love and fidelity, whose relationships have failed to attract mathematical interest. In order to nail down those concepts that might yield mathematical investigation, the topologist D H Gottlieb claimed that mathematics is the study of well-defined things. This notion has some appeal to mathematicians, for whom the expression "well-defined" is part of the lingo, and for whom it excludes the above philosophical concepts. Yet I fear that an attempt to explain it to a non-mathematician would result in a "well-defined thing" as being "one that is amenable to mathematical inquiry". In other words, mathematicians study what mathematicians study.

Mathematicians tend to be pretty stubborn (we like to say that we persevere), but there comes a stage when one has unsuccessfully battled against a tough question for so long that one realises that the difficulty was simply that it was the wrong question in the first place. I believe that's what's happened here.

We shouldn't ask what a mathematician studies, we should ask how.

Put it in another way; instead of the question, "What do mathematicians study?" we should ask, "What do mathematicians do?" Interestingly, when one examines the OED's attempt at a definition, one sees that, in contrast to the definitions of the other disciplines, there's a how answer only partly suppressed, "Investigates deductively the conclusions implicit in the elementary conceptions ...". Since mathematicians get their fingers into pies that often have names very different from geometry, arithmetic and algebra, it's more fruitful to clarify the process of doing mathematics. Then, when a new topic is proposed for the next revision of the mathematics curriculum, one has some hope of answering the question, "But is it mathematics?" My guess is that the correct answer would be, "Yes, when you look at it the right way." No geology lecture would be about zebras, or blood vessels, or language grammars, or DNA. By definition, it would fail to be a geology lecture. However, I've known mathematics lectures about these four topics.

Specifically, the lecture about zebras was interested in how they and other quadrupeds move. Different speeds of walking or running result in different sequences of hooves hitting the ground. Which sequences can occur, and what is the relation between the sequence and the speed?

Blood vessels can be studied for the way in which cells move along them; this is the dynamics of fluid motion where the walls are not rigid. And what governs the shape of the vessel itself? Can one predict when the forces will be so great as to lead to rupture?

Are there common rules of manipulation of words and phrases that apply across different languages? What does the similarity of such rules suggest about the cultural or genetic links between the speakers of such languages?

It's recently been discovered that in the process of replication, the enormously long DNA molecules get tied into knots, which partly dissolve and recombine as different knots. By inspecting the knots that appear, one can attempt deductions about the biochemical process that is leading from one knot to the next.

There's a pattern to what is happening in each of the above examples. The mathematician immediately ignores many specific features of the object in question. He or she is unlikely to care about whose body the blood vessel inhabits, or the age of the zebra. But pretty soon he or she may even forget that it's a blood vessel or zebra that's being studied, and may talk to a colleague about fluid in a tube or configurations of moving rods. The process of abstraction (OED: "of considering ... an attribute or quality independently of the substance to which it belongs") takes on a life of its own, so that before long two mathematicians may be discussing the problem in such a way that a third mathematician listening in would find it difficult to guess its physical origins. (The degree of difficulty is probably the distinction between pure and applied mathematics. Put like this, it's apparent that the distinction is more arbitrary and less clear-cut than generally recognised.)

After reading the above, the Japan-based mathematician A Kozlowski observed: "I think it a very important point that mathematics is probably the only subject whose content could change entirely and yet we would still recognize it as mathematics. We would probably recognize mathematics of beings from another universe, though we may have problems in distinguishing their physics from their philosophy, their history from their mythology etc."

I believe that the process of abstraction is a vital characteristic of mathematical thought, probably more distinctive than the method of deduction that the OED emphasises. Most scientists practise deduction, although not necessarily to the extent of mathematicians. However, other disciplines are comparatively restricted in the amount of abstraction that they allow themselves. While physiologists might be comfortable with the general properties of all blood vessels belonging to humans, or of all those with a certain condition, would they still be at ease with a level of abstraction that considered equally other fluids moving in inorganic tubes? Or would they feel that such generalizations were no longer in the realm of physiology?

In fact, the desire for abstraction seems to be an essential part of a mathematician's psyche. It's not just a matter of abstracting from the physical world to the mathematical; many mathematicians commence work only long after that process has been completed. Within mathematics, researchers are all the time striving to find just the right level of abstraction for a given setting, seeking the perfect balance between the twin goals of utility and generality.

Another feature of scientific method is of course induction, the attempt to generalize conclusions from a number of particular instances. Mathematicians practise this more often than is usually realised, however, in a special way. For a scientist such a conclusion has the status of a probationary law. If it stands the test of time, that is, accords with, and even predicts, subsequent observations, then it becomes more widely accepted. This tends to be a gradual process that can be partly or wholly reversed (medical science provides many examples of reversibility and controversy). For a mathematician, the result of induction is just a hunch. A strongly held hunch is honoured with the title of conjecture. For example, there was the Fermat Conjecture:

If x,y,z,n are whole numbers greater than 1 and xn + yn = zn, then n = 2.

Evidently, Fermat produced this statement by induction after examining many specific cases (with no help from an electronic computer). (In the event that n = 2, then for any whole number k we can always take x = 2k +1, y = 2k2+2k and z = y+1.) After succeeding generations of mathematicians had played with it, the formulation received the status of a conjecture. For further generations it was widely believed to be a correct assertion, yet no mathematician would admit to the list of proved statements anything that was logically dependent on it. That all changed in this decade, when Andrew Wiles' famous proof (a chain of deductions occupying hundreds of printed pages) survived the rigorous checks of his peers. The statement is now called a theorem, and will always remain so. "Elevation to the theorage" is an irreversible process. The progression from hunch to conjecture is an example of scientific induction, but the final, irreversible graduation from conjecture to theorem has no parallel outside mathematics.

We have now distinguished three modes of thinking that highlight the difference between mathematics and other disciplines:

Abstraction, Deduction, Induction.

They are listed above in decreasing order of importance to mathematical research, but I would guess in increasing order of importance for scientists generally. One might object that if deduction and induction are seen as opposites, then why doesn't an opposite of abstraction appear in the list? Well, since abstraction consists in seeing common properties and patterns among different situations, then once one has obtained conclusions about the general, abstract setup, the process of deduction is enough to get one back to conclusions about the original, more concrete setting. So our list of three seems to be enough.

Just what the process of abstraction involves is a big topic, and not central to this article. At its heart seems to be one of the greatest joys of mathematical research - pattern recognition. The patterns are not usually the visual ones of everyday experience. Recall that humans also get excited by more subtle threads of similarity, for instance in hearing in a Wagnerian opera a musical motif that occurs elsewhere in the Ring Cycle. Mathematicians have available for painting their patterns the whole canvas of human experience. For example, the kinship ties of the Warlpiri people of the Australian outback exhibit the same pattern as the symmetries of a square (known algebraically as the dihedral group of order 8).

The drive to find common themes from disaparate areas seems to be part of the mathematician's temperament. At its most banal, it's a source of painful puns (like the one I had to resist earlier, after including the words "pie" and "fruitful" in the same sentence). Used more creatively, it helps to explain what has been called the "unreasonable effectiveness" of mathematics. Consider the following title of a research paper, by R Ghrist, that reached me today.

Configuration spaces ... it begins. I think: Yes, it's about topology.

... and braid groups ... Okay, about algebra too.

on graphs ... Combinatorics as well. This is getting pretty interesting. But now for the knockout blow ...

in robotics.

The discovery of a unifying pattern can be like lightning flashing from one discipline to another. The difference is that it can illuminate both subjects forever. So there is a simple message for the nonmathematical researcher reading this article. When all seems cloudy, contact a mathematician (as broadminded as possible). Then stand by for flashes of lightning!

Prof A J Berrick
Department of Mathematics

MATHEMATICS FAIR

darshanhi — Sat, 20 Jan 2007 02:16:19 CST

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Pictures of the stand (University Fair 2001)

darshanhi — Sat, 20 Jan 2007 02:15:17 CST

Pictures of the stand (University Fair 2001)

What is simplification

darshanhi — Fri, 19 Jan 2007 09:08:53 CST

Simplification means reviewing, reducing and removing regulatory burdens for the public, private and voluntary sectors, either through merging regulations in to a more manageable form or removing inconsistency within or between regulations.
A burden can be administrative e.g. the time taken to fill in forms or the time taken to meet requests for information from regulators or government departments. Or a burden can be policy related, e.g. a compulsory cost of buying new equipment to comply with a regulation.
The aim of simplification is to reduce regulatory burdens wherever possible but without removing the necessary protections regulation provides, for the environment or workers for example. An important part of the simplification process is to gather practical suggestions from those with experience of being regulated - businesses, voluntary and public sector organisations and individuals - that can inform Government thinking on how to simplify regulation in a range of areas.

Fields of Indian mathematics

darshanhi — Sat, 13 Jan 2007 03:47:17 CST

There is no abstract available for this page revision.

Indian mathematics

darshanhi — Sat, 13 Jan 2007 03:46:39 CST

The chronology of Indian mathematics spans from the Indus Valley civilization (3300-1500 BCE) and Vedic civilization (1500-500 BCE) to modern India (21st century CE).
Indian mathematicians have made major contributions to the development of mathematics as we know it today. One of the biggest contributions of Indian mathematics is the modern arithmetic and decimal notation of numbers used universally throughout the world (known as the Hindu-Arabic numerals). John Playfair, the famous Scottish mathematician published a dissertation titled "Remarks on the astronomy of Brahmins" in 1790. His following quotation shows the appreciation of the then European Scientific community on the achievements of ancient Indian mathematicians and scientists.
"The Constructions and these tables imply a great knowledge of geometry,arithmetic and even of the theoretical part of astronomy.But what, without doubt is to be accounted,the greatest refinement in this system, is the hypothesis employed in calculating the equation of the centre for the Sun,Moon and the planets that of a circular orbit having a double eccentricity or having its centre in the middle between the earth and the point about which the angular motion is uniform.If to this we add the great extent of the geometrical knowledge required to combine this and the other principles of their astronomy together and to deduce from them the just conclusion;the possession of a calculus equivalent to trigonometry and lastly their approximation to the quadrature of the circle, we shall be astonished at the magnitude of that body of science which must have enlightened the inhabitants of India in some remote age and which whatever it may have communicated to the Western nations appears to have received another from them...."
Albert Einstein in the 20th century also comments on the importance of Indian arithmetic: "We owe a lot to the Indians, who taught us how to count, without which no worthwhile scientific discovery could have been made."
Said Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of antiquity, Archimedes and Apollonius."[1]
Other examples include zero, negative numbers, and the trigonometric functions of sine and cosine, which have all provided some of the biggest impetuses to advances in the field. Concepts from ancient and medieval India were carried to China and the Middle East, where they were studied extensively. From there they made their way to Europe and other parts of the world.

An overview of Indian mathematics

darshanhi — Sat, 13 Jan 2007 03:38:54 CST

It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. What is quite surprising is that there has been a reluctance to recognise this and one has to conclude that many famous historians of mathematics found what they expected to find, or perhaps even what they hoped to find, rather than to realise what was so clear in front of them. We shall examine the contributions of Indian mathematics in this article, but before looking at this contribution in more detail we should say clearly that the "huge debt" is the beautiful number system invented by the Indians on which much of mathematical development has rested. Laplace put this with great clarity:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

We shall look briefly at the Indian development of the place-value decimal system of numbers later in this article and in somewhat more detail in the separate article Indian numerals. First, however, we go back to the first evidence of mathematics developing in India. Histories of Indian mathematics used to begin by describing the geometry contained in the Sulbasutras but research into the history of Indian mathematics has shown that the essentials of this geometry were older being contained in the altar constructions described in the Vedic mythology text the Shatapatha Brahmana and the Taittiriya Samhita. Also it has been shown that the study of mathematical astronomy in India goes back to at least the third millennium BC and mathematics and geometry must have existed to support this study in these ancient times. The first mathematics which we shall describe in this article developed in the Indus valley. The earliest known urban Indian culture was first identified in 1921 at Harappa in the Punjab and then, one year later, at Mohenjo-daro, near the Indus River in the Sindh. Both these sites are now in Pakistan but this is still covered by our term "Indian mathematics" which, in this article, refers to mathematics developed in the Indian subcontinent. The Indus civilisation (or Harappan civilisation as it is sometimes known) was based in these two cities and also in over a hundred small towns and villages. It was a civilisation which began around 2500 BC and survived until 1700 BC or later. The people were literate and used a written script containing around 500 characters which some have claimed to have deciphered but, being far from clear that this is the case, much research remains to be done before a full appreciation of the mathematical achievements of this ancient civilisation can be fully assessed. We often think of Egyptians and Babylonians as being the height of civilisation and of mathematical skills around the period of the Indus civilisation, yet V G Childe in New Light on the Most Ancient East (1952) wrote:-

India confronts Egypt and Babylonia by the 3rd millennium with a thoroughly individual and independent civilisation of her own, technically the peer of the rest. And plainly it is deeply rooted in Indian soil. The Indus civilisation represents a very perfect adjustment of human life to a specific environment. And it has endured; it is already specifically Indian and forms the basis of modern Indian culture.

We do know that the Harappans had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they belong to two series both being decimal in nature with each decimal number multiplied and divided by two, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimetres) which has been called the "Indus inch". Of course ten units is then 13.2 inches which is quite believable as the measure of a "foot". A similar measure based on the length of a foot is present in other parts of Asia and beyond. Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches. It is certainly surprising the accuracy with which these scales are marked. Now 100 units of this measure is 36.7 inches which is the measure of a stride. Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction. It is unclear exactly what caused the decline in the Harappan civilisation. Historians have suggested four possible causes: a change in climatic patterns and a consequent agricultural crisis; a climatic disaster such flooding or severe drought; disease spread by epidemic; or the invasion of Indo-Aryans peoples from the north. The favourite theory used to be the last of the four, but recent opinions favour one of the first three. What is certainly true is that eventually the Indo-Aryans peoples from the north did spread over the region. This brings us to the earliest literary record of Indian culture, the Vedas which were composed in Vedic Sanskrit, between 1500 BC and 800 BC. At first these texts, consisting of hymns, spells, and ritual observations, were transmitted orally. Later the texts became written works for use of those practicing the Vedic religion. The next mathematics of importance on the Indian subcontinent was associated with these religious texts. It consisted of the Sulbasutras which were appendices to the Vedas giving rules for constructing altars. They contained quite an amount of geometrical knowledge, but the mathematics was being developed, not for its own sake, but purely for practical religious purposes. The mathematics contained in the these texts is studied in some detail in the separate article on the Sulbasutras. The main Sulbasutras were composed by Baudhayana (about 800 BC), Manava (about 750 BC), Apastamba (about 600 BC), and Katyayana (about 200 BC). These men were both priests and scholars but they were not mathematicians in the modern sense. Although we have no information on these men other than the texts they wrote, we have included them in our biographies of mathematicians. There is another scholar, who again was not a mathematician in the usual sense, who lived around this period. That was Panini who achieved remarkable results in his studies of Sanskrit grammar. Now one might reasonably ask what Sanskrit grammar has to do with mathematics. It certainly has something to do with modern theoretical computer science, for a mathematician or computer scientist working with formal language theory will recognise just how modern some of Panini's ideas are. Before the end of the period of the Sulbasutras, around the middle of the third century BC, the Brahmi numerals had begun to appear.

Here is one style of the Brahmi numerals..

These are the earliest numerals which, after a multitude of changes, eventually developed into the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 used today. The development of numerals and place-valued number systems are studied in the article Indian numerals. The Vedic religion with its sacrificial rites began to wane and other religions began to replace it. One of these was Jainism, a religion and philosophy which was founded in India around the 6th century BC. Although the period after the decline of the Vedic religion up to the time of Aryabhata I around 500 AD used to be considered as a dark period in Indian mathematics, recently it has been recognised as a time when many mathematical ideas were considered. In fact Aryabhata is now thought of as summarising the mathematical developments of the Jaina as well as beginning the next phase. The main topics of Jaina mathematics in around 150 BC were: the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations. More surprisingly the Jaina developed a theory of the infinite containing different levels of infinity, a primitive understanding of indices, and some notion of logarithms to base 2. One of the difficult problems facing historians of mathematics is deciding on the date of the Bakhshali manuscript. If this is a work which is indeed from 400 AD, or at any rate a copy of a work which was originally written at this time, then our understanding of the achievements of Jaina mathematics will be greatly enhanced. While there is so much uncertainty over the date, a topic discussed fully in our article on the Bakhshali manuscript, then we should avoid rewriting the history of the Jaina period in the light of the mathematics contained in this remarkable document.

You can see a separate article about Jaina mathematics.

If the Vedic religion gave rise to a study of mathematics for constructing sacrificial altars, then it was Jaina cosmology which led to ideas of the infinite in Jaina mathematics. Later mathematical advances were often driven by the study of astronomy. Well perhaps it would be more accurate to say that astrology formed the driving force since it was that "science" which required accurate information about the planets and other heavenly bodies and so encouraged the development of mathematics. Religion too played a major role in astronomical investigations in India for accurate calendars had to be prepared to allow religious observances to occur at the correct times. Mathematics then was still an applied science in India for many centuries with mathematicians developing methods to solve practical problems. Yavanesvara, in the second century AD, played an important role in popularising astrology when he translated a Greek astrology text dating from 120 BC. If he had made a literal translation it is doubtful whether it would have been of interest to more than a few academically minded people. He popularised the text, however, by resetting the whole work into Indian culture using Hindu images with the Indian caste system integrated into his text. By about 500 AD the classical era of Indian mathematics began with the work of Aryabhata. His work was both a summary of Jaina mathematics and the beginning of new era for astronomy and mathematics. His ideas of astronomy were truly remarkable. He replaced the two demons Rahu, the Dhruva Rahu which causes the phases of the Moon and the Parva Rahu which causes an eclipse by covering the Moon or Sun or their light, with a modern theory of eclipses. He introduced trigonometry in order to make his astronomical calculations, based on the Greek epicycle theory, and he solved with integer solutions indeterminate equations which arose in astronomical theories. Aryabhata headed a research centre for mathematics and astronomy at Kusumapura in the northeast of the Indian subcontinent. There a school studying his ideas grew up there but more than that, Aryabhata set the agenda for mathematical and astronomical research in India for many centuries to come. Another mathematical and astronomical centre was at Ujjain, also in the north of the Indian subcontinent, which grew up around the same time as Kusumapura. The most important of the mathematicians at this second centre was Varahamihira who also made important contributions to astronomy and trigonometry. The main ideas of Jaina mathematics, particularly those relating to its cosmology with its passion for large finite numbers and infinite numbers, continued to flourish with scholars such as Yativrsabha. He was a contemporary of Varahamihira and of the slightly older Aryabhata. We should also note that the two schools at Kusumapura and Ujjain were involved in the continuing developments of the numerals and of place-valued number systems. The next figure of major importance at the Ujjain school was Brahmagupta near the beginning of the seventh century AD and he would make one of the most major contributions to the development of the numbers systems with his remarkable contributions on negative numbers and zero. It is a sobering thought that eight hundred years later European mathematics would be struggling to cope without the use of negative numbers and of zero. These were certainly not Brahmagupta's only contributions to mathematics. Far from it for he made other major contributions in to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine tables. The way that the contributions of these mathematicians were prompted by a study of methods in spherical astronomy is described in [ 19 (2) (1968), 49-72.',25)">25]:-

The Hindu astronomers did not possess a general method for solving problems in spherical astronomy, unlike the Greeks who systematically followed the method of Ptolemy, based on the well-known theorem of Menelaus. But, by means of suitable constructions within the armillary sphere, they were able to reduce many of their problems to comparison of similar right-angled plane triangles. In addition to this device, they sometimes also used the theory of quadratic equations, or applied the method of successive approximations. ... Of the methods taught by Aryabhata and demonstrated by his scholiast Bhaskara I, some are based on comparison of similar right-angled plane triangles, and others are derived from inference. Brahmagupta is probably the earliest astronomer to have employed the theory of quadratic equations and the method of successive approximations to solving problems in spherical astronomy.

Before continuing to describe the developments through the classical period we should explain the mechanisms which allowed mathematics to flourish in India during these centuries. The educational system in India at this time did not allow talented people with ability to receive training in mathematics or astronomy. Rather the whole educational system was family based. There were a number of families who carried the traditions of astrology, astronomy and mathematics forward by educating each new generation of the family in the skills which had been developed. We should also note that astronomy and mathematics developed on their own, separate for the development of other areas of knowledge. Now a "mathematical family" would have a library which contained the writing of the previous generations. These writings would most likely be commentaries on earlier works such as the Aryabhatiya of Aryabhata. Many of the commentaries would be commentaries on commentaries on commentaries etc. Mathematicians often wrote commentaries on their own work. They would not be aiming to provide texts to be used in educating people outside the family, nor would they be looking for innovative ideas in astronomy. Again religion was the key, for astronomy was considered to be of divine origin and each family would remain faithful to the revelations of the subject as presented by their gods. To seek fundamental changes would be unthinkable for in asking others to accept such changes would be essentially asking them to change religious belief. Nor do these men appear to have made astronomical observations in any systematic way. Some of the texts do claim that the computed data presented in them is in better agreement with observation than that of their predecessors but, despite this, there does not seem to have been a major observational programme set up. Paramesvara in the late fourteenth century appears to be one of the first Indian mathematicians to make systematic observations over many years. Mathematics however was in a different position. It was only a tool used for making astronomical calculations. If one could produce innovative mathematical ideas then one could exhibit the truths of astronomy more easily. The mathematics therefore had to lead to the same answers as had been reached before but it was certainly good if it could achieve these more easily or with greater clarity. This meant that despite mathematics only being used as a computational tool for astronomy, the brilliant Indian scholars were encouraged by their culture to put their genius into advances in this topic. A contemporary of Brahmagupta who headed the research centre at Ujjain was Bhaskara I who led the Asmaka school. This school would have the study of the works of Aryabhata as their main concern and certainly Bhaskara was commentator on the mathematics of Aryabhata. More than 100 years after Bhaskara lived the astronomer Lalla, another commentator on Aryabhata. The ninth century saw mathematical progress with scholars such as Govindasvami, Mahavira, Prthudakasvami, Sankara, and Sridhara. Some of these such as Govindasvami and Sankara were commentators on the text of Bhaskara I while Mahavira was famed for his updating of Brahmagupta's book. This period saw developments in sine tables, solving equations, algebraic notation, quadratics, indeterminate equations, and improvements to the number systems. The agenda was still basically that set by Aryabhata and the topics being developed those in his work. The main mathematicians of the tenth century in India were Aryabhata II and Vijayanandi, both adding to the understanding of sine tables and trigonometry to support their astronomical calculations. In the eleventh century Sripati and Brahmadeva were major figures but perhaps the most outstanding of all was Bhaskara II in the twelfth century. He worked on algebra, number systems, and astronomy. He wrote beautiful texts illustrated with mathematical problems, some of which we present in his biography, and he provided the best summary of the mathematics and astronomy of the classical period. Bhaskara II may be considered the high point of Indian mathematics but at one time this was all that was known [ 53 (1-4) (1985), 204-208',26)">26]:-

For a long time Western scholars thought that Indians had not done any original work till the time of Bhaskara II. This is far from the truth. Nor has the growth of Indian mathematics stopped with Bhaskara II. Quite a few results of Indian mathematicians have been rediscovered by Europeans. For instance, the development of number theory, the theory of indeterminates infinite series expressions for sine, cosine and tangent, computational mathematics, etc.

Following Bhaskara II there was over 200 years before any other major contributions to mathematics were made on the Indian subcontinent. In fact for a long time it was thought that Bhaskara II represented the end of mathematical developments in the Indian subcontinent until modern times. However in the second half of the fourteenth century Mahendra Suri wrote the first Indian treatise on the astrolabe and Narayana wrote an important commentary on Bhaskara II, making important contributions to algebra and magic squares. The most remarkable contribution from this period, however, was by Madhava who invented Taylor series and rigorous mathematical analysis in some inspired contributions. Madhava was from Kerala and his work there inspired a school of followers such as Nilakantha and Jyesthadeva. Some of the remarkable discoveries of the Kerala mathematicians are described in [ 53 (1-4) (1985), 204-208',26)">26]. These include: a formula for the ecliptic; the Newton-Gauss interpolation formula; the formula for the sum of an infinite series; Lhuilier's formula for the circumradius of a cyclic quadrilateral. Of particular interest is the approximation to the value of π which was the first to be made using a series. Madhava's result which gave a series for π, translated into the language of modern mathematics, reads

π R = 4R - 4R/3 + 4R/5 - ...

This formula, as well as several others referred to above, were rediscovered by European mathematicians several centuries later. Madhava also gave other formulae for π, one of which leads to the approximation 3.14159265359. The first person in modern times to realise that the mathematicians of Kerala had anticipated some of the results of the Europeans on the calculus by nearly 300 years was Charles Whish in 1835. Whish's publication in the Transactions of the Royal Asiatic Society of Great Britain and Ireland was essentially unnoticed by historians of mathematics. Only 100 years later in the 1940s did historians of mathematics look in detail at the works of Kerala's mathematicians and find that the remarkable claims made by Whish were essentially true. See for example [ 13 (1945), 92-98.',15)">15]. Indeed the Kerala mathematicians had, as Whish wrote:-

... laid the foundation for a complete system of fluxions ...

and these works:-

... abound with fluxional forms and series to be found in no work of foreign countries.

There were other major advances in Kerala at around this time. Citrabhanu was a sixteenth century mathematicians from Kerala who gave integer solutions to twenty-one types of systems of two algebraic equations. These types are all the possible pairs of equations of the following seven forms:

x + y = a, x - y = b, xy = c, x2 + y2 = d, x2 - y2 = e, x3 + y3 = f, and x3 - y3 = g.

For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. See [ 25 (1) (1998), 1-21.',12)">12] for more details. Now we have presented the latter part of the history of Indian mathematics in an unlikely way. That there would be essentially no progress between the contributions of Bhaskara II and the innovations of Madhava, who was far more innovative than any other Indian mathematician producing a totally new perspective on mathematics, seems unlikely. Much more likely is that we are unaware of the contributions made over this 200 year period which must have provided the foundations on which Madhava built his theories. Our understanding of the contributions of Indian mathematicians has changed markedly over the last few decades. Much more work needs to be done to further our understanding of the contributions of mathematicians whose work has sadly been lost, or perhaps even worse, been ignored. Indeed work is now being undertaken and we should soon have a better understanding of this important part of the history of mathematics.

Sense of scale

darshanhi — Sat, 13 Jan 2007 03:19:06 CST

The facts below give a sense of how large one billion (one thousand million, 109) is in the context of passage of time.

About a billion minutes ago, the Roman Empire was flourishing. (One billion minutes is roughly 1,900 years.)
About a billion hours ago, modern human beings and their ancestors were living in the Stone Age (more precisely, the Middle Paleolithic). (One billion hours is roughly 114,000 years.)
About a billion days ago, Australopithecus, an ape-like creature related to an ancestor of modern humans, roamed the African savannas. (One billion days is roughly 2.7 million years.)
About a billion months ago, dinosaurs walked the earth during the late Cretaceous. (One billion months is roughly 82 million years.)
About a billion years ago, the first multicellular organisms appeared on Earth. (The universe is now thought to be about 13.7 billion years old.)

In terms of distance:

A billion centimeters is about the distance from Chicago, Illinois, USA to Tokyo, Japan.
A billion inches is 15,783 miles, more than halfway around the world and sufficient to reach any point on the globe from any other point.
A billion meters is almost three times the distance from the Earth to the Moon.
A billion kilometers is over six times the distance from the Earth to the Sun.

In terms of count:

A is a cube; B consists of 1000 cubes of type A. C consists of 1000 Bs; and D 1000 Cs. Thus there are 1 million As in C; and 1 billion As in D. Likewise, there are a billion cubic millimeters in a cubic meter.
In finance, the possession of one billion United States dollars allows one to be ranked among the world's wealthiest individuals

Selected 10-digit numbers (1000000001 - 9999999999)

darshanhi — Sat, 13 Jan 2007 03:16:08 CST

1023456789 - smallest pandigital number
1073676287 - Carol number
1073741824 = 230
1073807359 - Kynea number
1111111111 - repunit
1129760415 - Motzkin number
1134903170 - Fibonacci number
1162261467 = 319
1220703125 = 512
1234567890 - pandigital number with the digits in order
1311738121 - Pell number
1382958545 - Bell number
1406818759 - Wedderburn-Etherington number
1836311903 - Fibonacci number
1977326743 = 711
2147483647 - Mersenne prime
2147483648 = 231
2214502422 - primary pseudoperfect number
2222222222 - repdigit
2357947691 = 119
2971215073 - Fibonacci prime
3166815962 - Pell number
3192727797 - Motzkin number
3323236238 - Wedderburn-Etherington number
3333333333 - repdigit
3486784401 = 320
4294836223 - Carol number
4294967296 = 232
4295098367 - Kynea number
4444444444 - repdigit
4807526976 - Fibonacci number
5555555555 - repdigit
5784634181 - alternating factorial
6210001000 - the only self-descriptive number in base 10
6227020800 = 13!
6666666666 - repdigit
6983776800 - colossally abundant number
7645370045 - Pell number
7777777777 - repdigit
7778742049 - Fibonacci number
7862958391 - Wedderburn-Etherington number
8589869056 - perfect number
8589934592 = 233
8888888888 - repdigit
9043402501 - Motzkin number
9814072356 - largest square pandigital number, largest pandigital pure power
9876543210 - largest pandigital number without redundant digits
9999999999 - repdigit

1000000000 (number)

darshanhi — Sat, 13 Jan 2007 03:14:34 CST

1000000000 (number)

From Wikipedia, the free encyclopedia

Jump to: navigation, search

List of numbers - Integers
100000000 1000000000 10000000000

Cardinal	One billion (short scale)
Ordinal	One billionth (short scale)
Factorization	29 · 59
Binary	111011100110101100101000000000
Hexadecimal	3B9ACA00

One thousand million (1,000,000,000) is the natural number following 999,999,999 and preceding 1,000,000,001.
In scientific notation, it is written as 109.
In modern ('short scale') English language usage, it is usually called a billion (although in many other languages and 'long scale' usage, a billion means a million millions (or 1,000,000,000,000), instead of a thousand millions).
The term milliard can also be used to refer to 1,000,000,000, though this terminology is rarely used in the English language, but often in other languages.
Physical quantities can be expressed using the SI prefix giga.
See Orders of magnitude (numbers) for larger numbers; and long and short scales.

Selected 7-digit numbers (1000000 - 9999999)

darshanhi — Sat, 13 Jan 2007 03:08:02 CST

1046527 - Carol number
1048576 = 220 (power of two), 2116-gonal number, an 8740-gonal number and a 174764-gonal number, the number of bytes in a mebibyte, the number of kibibytes in a gibibyte, and so on.
1048976 - Leyland number
1050623 - Kynea number
1058576 - Leyland number
1084051 - Keith number
1089270 - harmonic divisor number
1111111 - repunit
1136689 - Pell number, Markov number
1234567 - Smarandche consecutive number (base 10 digits are in numerical order)
1278818 - Markov number
1342269 - Fibonacci number
1346269 - Markov number
1421280 - harmonic divisor number
1441440 - colossally abundant number
1441889 - Markov number
1539720 - harmonic divisor number
1563372 - Wedderburn-Etherington number
1594323 = 313
1596520 - Leyland number
1647086 - Leyland number
1679616 = 68
1686049 - Markov number
1771561 = 116 = 13312, also, Commander Spock's estimate for the tribble population in the Star Trek episode "The Trouble With Tribbles"
1941760 - Leyland number
1953125 = 59
2012174 - Leyland number
2012674 - Markov number
2097152 = 221, power of two
2097593 - Leyland number
2124679 - Wolstenholme prime
2178309 - Fibonacci number
2222222 - repdigit
2356779 - Motzkin number
2423525 - Markov number
2674440 - Catalan number
2744210 - Pell number
2796203 - Wagstaff prime
2922509 - Markov number
3263442 - product of the first five terms of Sylvester's sequence
3263443 - sixth term of Sylvester's sequence
3276509 - Markov number
3301819 - alternating factorial
3333333 - repdigit
3524578 - Fibonacci number, Markov number
3626149 - Wedderburn-Etherington number
3628800 = 10!
4037913 - sum of the first ten factorials
4190207 - Carol number
4194304 = 222, power of two
4194788 - Leyland number
4198399 - Kynea number
4208945 - Leyland number
4213597 - Bell number
4400489 - Markov number
4444444 - repdigit
4561111 - telephone number of the White House (202 area code)
4782969 = 314
4785713 - Leyland number
4826809 = 136
5555555 - repdigit
5702887 - Fibonacci number
5764801 = 78
6000000 - The standard estimate of persons of Jewish origin murdered by the Nazis and their collaborators in the Holocaust.
6536382 - Motzkin number
6625109 - Pell number, Markov number
6666666 - repdigit
7365000 - Pennsylvania 6-5000, old style phone number of the Pennsylvania Hotel and name of a popular song
7453378 - Markov number
7777777 - repdigit
7861953 - Leyland number
7913837 - Keith number
8000000 - Used to represent infinity in Japanese mythology
8388608 = 223, power of two
8389137 - Leyland number
8399329 - Markov number
8436379 - Wedderburn-Etherington number
8675309 - Phone number in the popular song from 1981 titled 867-5309/Jenny by Tommy Tutone. This is also a prime number.
8888888 - repdigit
8946176 - self-descriptive number in base 8
9227465 - Fibonacci number, Markov number
9369319 - Newman-Shanks-Williams prime
9647009 - Markov number
9694845 - Catalan number
9765625 = 510
9865625 - Leyland number
9999999 - repdigit

Retrieved from "http://en.wikipedia.org/wiki/Million"

The word million

darshanhi — Sat, 13 Jan 2007 03:06:28 CST

In standard English, it is pronounced with an l-sound followed by a y-glide. However, as other languages use a fully palatalized 'l' in this word (such as Italian spells by 'gl'), some English-speakers have picked up this pronunciation, which does not occur elsewhere in the English language but in words of this model.
This word is the most common of words ending in -lion, and the fact that in English it does not have a distinct pronunciation due to its double 'll' causes spelling confusion with 'vermilion', 'pavilion', etc.

Million

darshanhi — Sat, 13 Jan 2007 03:04:37 CST

Million

From Wikipedia, the free encyclopedia

Jump to: navigation, search

List of numbers - Integers
100000100000010000000

Cardinal	One million
Ordinal	One millionth
Factorization	26 · 56
Roman numeral
Unicode representation of Roman numeral
Binary	11110100001001000000
Hexadecimal	F4240

Look up million in
Wiktionary, the free dictionary.

Artistic depiction of the well-known, large, round number emblazoned in gold.

One million (1,000,000), or one thousand thousand, is the natural number following 999,999 and preceding 1,000,001.
In scientific notation, it is written as 106. Physical quantities can also be expressed using the SI prefix mega, when dealing with SI units. For example, 1 megawatt equals 1 000 000 watts.
The million is sometimes used in the English language as a metaphor for a very large number, as in "Never in a million years" and "You're one in a million", or a hyperbole, as in "I've walked a million miles". Hence, a millionaire is a rich person, no matter the actual currency or the exact quantity. Il Milione is the title of Marco Polo's narration of his travel to China. The name is supposed to come from Polo's nickname after his tales of riches and multitudes.
The word "million" is common to the short scale and long scale numbering systems (and also to the proposed Rowlett numbering system), unlike the larger numbers, which have different names in the two systems.
The name is derived from Italian, where milla was 1,000, and 1,000,000 became millione, "a large thousand".

Comments:

darshanhi — Fri, 12 Jan 2007 04:25:30 CST

To solve this problem, first we will need to look at the denominators 3 and 5 and see what multiples of these numbers are common.
Multiples of 3 are:
3 6 9 12 15 18 21
Multiples of 5 are:
5 10 15 20 25
so, in multiples of 3 and 5 the least common multiple is 15.
now, we will change 1/3 and 2/5 into their equivalent fractions with denominator 15:

Examples OF FRACTION ADDITION

darshanhi — Fri, 12 Jan 2007 04:25:04 CST

Example 1

Solve:
changing both fractions to a common denominator (see comments on the right)

FRACTIONS ADDITIONS

darshanhi — Fri, 12 Jan 2007 04:24:19 CST

Adding fractions is done differently than the usual numbers. Normally, while adding or subtracting fractions you will find two types of problems:

Type 1: where the fractions being added have the same denominator eg.

Type 2: where the fractions being added have different denominators eg.

As you know, fractions represent parts of the whole. So, when these parts are from the whole broken into same number of parts it is easy to add them.
For Type 1 problems, we just need to add the top parts (numerator) of the fractions and leave the denominator as such. So, to solve the problem in the above example the solution will be:

For Type 2 problems where the denominator is different, we can not add these fractions by simply adding the numerators. In order to solve these problems first we will need to make into fractions with the same denominator. See how this is done in examples below.

GEOMETRY

darshanhi — Fri, 12 Jan 2007 04:14:36 CST

IF YOU HAVE ANY FACT OF GEOMETRY,PLEASE E-MAIL US AT mathsofdarshandudhoria@gmail.com

MATHS FACTS

darshanhi — Fri, 12 Jan 2007 04:12:30 CST

IF YOU HAVE ANY FACT OF MATHEMATICS,PLEASE E-MAIL US AT mathsofdarshandudhoria@gmail.com