Land pollutionFrom Wikipedia, the free encyclopedia

Land pollution is the demolition of Earth's land surfaces often caused by human activities and their misuse of land resources. It occurs when waste is not disposed properly. Health hazard disposal of urban and industrial wastes, exploitation of minerals, and improper use of soil by inadequate agricultural practices are a few factors. Urbanization and industrialization are major causes of land pollution. The Industrial Revolution set a series of events into motion which destroyed natural habitats and polluted the environment, causing diseases in both humans and other species of animals.Contents[hide]

1 Increased mechanization 2 Pesticides and herbicides 3 Mining 4 Increased waste disposal 5 Causes of soil pollution 6 See also 7 References

[

edit]Increased mechanization

The major increase in the concentration of population in cities, along with the internal combustion engine, led to the increased number of roads and all the infrastructure that goes with them. As the demand for food has grown exponentially with the increase of the human population, there is an increase in field size and mechanization. The increase in field size makes it economically viable for the farmer but results in loss of person and shelter for wildlife, as hedgerows and copses disappear. When crops are harvested, the naked soil is left open to wind after it has been compacted by heavy machinery. Another consequence of more intensive agriculture is the move to monoculture. This is unnatural, will deplete the soil of nutrients, allows diseases and pests to spread and, as it happens, it quickly exhausts all the natural resources in an area, causing the introduction of chemical fertilizers and foreign substances to the soil that poisons it. The chemical fertilizers in the soil cause its infertility.The farmer put artificial mannure to the plants to grow faster but it indirectly infertility occurs to soil.

[

edit]Pesticides and herbicides

A pesticide is a substance or mixture of substances used to kill a pest. A pesticide may be a chemical substance, biological agent (such as a virus or bacteria), antimicrobial, disinfectant or device used against any pest. Pests include insects, plant pathogens, weeds, mollusks, birds, mammals, fish, nematodes (roundworms) and microbes that compete with humans for food, destroy property, spread or are a vector for disease or cause a nuisance. Although there are benefits to the use of pesticides, there are also drawbacks, such as potential toxicity to humans and other organisms. Herbicides are used to kill weeds, especially on pavements and railways. They are similar to auxins and most are biodegrale by soil bacteria. However, one group derived from trinitrotoluene (2:4 D and 2:4:5 T) have the impurity dioxin, which is very toxic and causes fatality even in low concentrations. Another herbicide is Paraquat. It is highly toxic but it rapidly degrades in soil due to the action of bacteria and does not kill soil fauna. Insecticides are used to rid farms of pests which damage crops. The insects damage not only standing crops but also stored ones and in the tropics it is reckoned that one third of the total production is lost during food storage. As with fungicides, the first insecticides used in the nineteenth century were inorganic e.g.Paris Green and other compounds of arsenic. Nicotine has also been used since the late eighteenth century. There are now two main groups of synthetic insecticides Organochlorines include DDT, Aldrin, Dieldrin and BHC. They are cheap to produce, potent and persistent. DDT was used on a massive scale from the 1930s, with a peak of 72,000 tonnes used 1970. Then usage fell as the harmful environmental effects were realized. It was found worldwide in fish and birds and was even discovered in the snow in the Antarctic. It is only slightly soluble in water but is very soluble in the bloodstream. It affects the nervous and endocrine systems and causes the eggshells of birds to lack calcium causing them to be easily breakable. It is thought to be responsible for the decline of the numbers of birds of prey like ospreys and peregrine falcons in the 1950s - they are now recovering. As well as increased concentration via the food chain, it is known to enter via permeable membranes, so fish get it through their gills. As it has low water solubility, it tends to stay at the water surface, so organisms that live there are most affected. DDT found in fish that formed part of the human food chain caused concern, but the levels found in the liver, kidney and brain tissues was less than 1 ppm and in fat was 10 ppm which was below the level likely to cause harm. However, DDT was banned in Britain and America to stop the further build up of it in the food chain. The USA exploited this ban and sold DDT to developing countries, who could not afford the expensive replacement chemicals and who did not have such stringent regulations governing the use of pesticides.

Organophosphates, e.g. parathion, methyl parathion and about 40 other insecticides are available nationally. Parathion is highly toxic, methyl-parathion is less so and Malathion is generally considered safe as it has low toxicity and is rapidly broken down in the mammalian liver. This group works by preventing normal nerve transmission as cholinesterase is prevented from breaking down the transmitter substance acetylcholine, resulting in uncontrolled muscle movements.

[

edit]MiningModern mining projects leave behind disrupted communities, damaged landscapes, and polluted water.

Mining also affects ground and surface waters, the aquatic life, vegetation, soils, animals, and the human health.

Acid mine drainage can cause damage to streams which in return can kill aquatic life.

The vast variety of toxic chemicals released by mining activities can harm animals and aquatic life as well as their habitat.

Mining gas and petroleum also pollutes the land. Petroleum extraction and manufacturing contaminates the soil with bitumen, gasoline, kerosene and mining brine solutions. Opencast mining, which is a process where the surface of the earth is dug open to bring out the underground mineral deposits, destroys the topsoil and contaminates the area with toxic metals and chemicals.

[

edit]Increased waste disposal

In Scotland in 1993, 14 million tons of waste was produced. 100,000 tons was toxic waste and 260,000 tons was controlled waste from other parts of Britain and abroad. 45% of the special waste was in liquid form and 18% was asbestos - radioactive waste was not included. Of the controlled waste, 48% came from the demolition of buildings, 22% from industry, 17% from households and 13% from business - only 3% were recycled. 90% of controlled waste was buried in landfill sites and produced 2 million tons of methane gas. 1.5% was burned in incinerators and 1.5% were exported to be disposed of or recycled. There are 900 disposal sites in Scotland. There are very few vacant or derelict land sites in the north east of Scotland, as there are few traditional heavy industries or coal/mineral extraction sites. However some areas are contaminated by aromatic hydrocarbons (500 cubic meters). The Urban Waste Water Treatment Directive allows sewage sludge to be sprayed onto land and the volume is expected to double to 185,000 tons of dry solids in 2005. This has good agricultural properties

due to the high nitrogen and phosphate content. In 1990/1991, 13% wet weight was sprayed onto 0.13% of the land; however, this is expected to rise 15 fold by 2005. There is a need to control this so that pathogenic microorganisms do not get into water courses and to ensure that there is no accumulation of heavy metals in the top soil.

[

edit]Causes of soil pollution

Soil is polluted by many ways: 1. When pollutants get mixed with air, this causes acid rain. Acid rain degrades the top soil. 2. Garbage dumping, specially plastics, degrade the soil fertility as they are non biodegradable. 3. Chemical fertilizers and pesticides,when over used pollute the soil and also penetrate into ground water and make it non potable.

A Maths Starter of The Day

Maths Crossword 11 2

Across2. This type of number is the most important to factor. 3. Twelve inches.

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4. Letters representing numbers.4

6. Three squared. 7. Nothing with a belt. 9. An angle less than right. 11. A regular polygon with interior and exterior angles equal.

5

6

7

8

9

10

Down1. An all round perimeter. 2. An American government polygon. 5. Powerful numbers found in fishy places. 8. This sum is mixed up alott. 10. A selfish average.

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Indian mathematicsFrom Wikipedia, the free encyclopediaads not by this site

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Indian mathematics emerged in the Indian subcontinent[1] from 1200 BC

until the end of the 18th

century. In the classical period of Indian mathematics (400 AD to 1200 AD), important contributions

were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today[3] was first recorded in Indian mathematics. [4] Indian mathematicians made early contributions to the study of the concept of zero as a number,[5]negative numbers,[6] arithmetic, and algebra.[7] In addition, trigonometry[8] was further advanced in India, and, in particular, the modern definitions of sine andcosine were developed there.[9] These mathematical concepts were transmitted to the Middle East, China, and Europe[7] and led to further developments that now form the foundations of many areas of mathematics. Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved. [1][10] All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE. [11][12] A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th century CE. Their remarkable work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series). [13] However, they did not formulate a systematic theory ofdifferentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.[14][15][16][17]Contents[hide]

1 Fields of Indian mathematics 2 Prehistory 3 Vedic period

o o

3.1 Samhitas and Brahmanas 3.2 ulba Stras

4 Jain Mathematics (400 BCE 200 CE) 5 Oral tradition

o

5.1 Styles of memorization

o

5.2 The Stra genre

6 The written tradition: prose commentary 7 Numerals and the decimal numeral system 8 Bakhshali Manuscript 9 Classical Period (400 1200)

o o o

9.1 Fifth and sixth centuries 9.2 Seventh and eighth centuries 9.3 Ninth to twelfth centuries

10 Kerala mathematics (13001600) 11 Charges of Eurocentrism 12 See also 13 External links 14 Notes 15 Source books in Sanskrit 16 References 17 External links

[edit]Fields

of Indian mathematics

Some of the areas of mathematics studied in ancient and medieval India include the following:

Arithmetic: Decimal system, Negative numbers (see Brahmagupta), Zero (see Hindu numeral system), Binary numeral system, the modern positional notation numeral system, Floating point numbers (see Kerala school of astronomy and mathematics), Number theory, Infinity (see Yajur Veda), Transfinite numbers

Geometry: Square roots (see Bakhshali approximation), Cube roots (see Mahavira), Pythagorean triples (see Sulba Sutras; Baudhayana and Apastambastate the Pythagorean theorem without proof), Transformation (see Panini), Pascal's triangle (see Pingala)

Algebra: Quadratic equations (see Sulba Sutras, Aryabhata, and Brahmagupta), Cubic equations and Quartic equations (biquadratic equations) (see Mahavira and Bhskara II)

Mathematical logic: Formal grammars, formal language theory, the Panini Backus form (see Panini), Recursion (see Panini)

General mathematics: Fibonacci numbers (see Pingala), Earliest forms of Morse code (see Pingala), infinite series, Logarithms, indices[disambiguationneeded]

(see Jaina mathematics),Algorithms, Algorism (see Aryabhata and

Brahmagupta)

[edit]Prehistory

Trigonometry: Trigonometric functions (see Surya Siddhanta and Aryabhata), Trigonometric series (see Madhava and Kerala school)

Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilization have uncovered evidence of the use of "practical mathematics". The people of the IVC manufactured bricks whose dimensions were in the proportion 4:2:1, considered favorable for the stability of a brick structure. They used a standardized system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass produced weights in regular geometricalshapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.[18] The inhabitants of Indus civilization also tried to standardize measurement of length to a high degree of accuracy. They designed a rulerthe Mohenjo-daro rulerwhose unit of length (approximately 1.32 inches or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. [19][20]

[edit]Vedic

periodand Brahmanaswere being included in the texts. [2] For

See also: Vedanga and Vedas[edit]Samhitas

The religious texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurvedasahit (1200900 BCE), numbers as high as example, the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the avamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:[2] "Hail to ata ("hundred," million," ), hail to sahasra ("thousand," ), hail to ayuta ("ten thousand," ), hail toarbuda ("ten , literally ),

hail to niyuta ("hundred thousand,"

), hail to prayuta ("million,"

), hail to nyarbuda ("hundred million,"

), hail to samudra ("billion,"

"ocean"), hail to madhya ("ten billion," "end"), hail to parrdha ("one trillion,"

, literally "middle"), hail to anta("hundred billion,"

,lit.,

lit., "beyond parts"), hail to the dawn (uas), hail to the

twilight (vyui), hail to the one which is going to rise (udeyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to the heaven (svarga), hail to the world (loka), hail to all."[2] The Satapatha Brahmana (ca. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.[21]

[edit]ulba

Stras

Main article: ulba StrasThe ulba Stras (literally, "Aphorisms of the Chords" in Vedic Sanskrit) (c. 700400 BCE) list rules for the construction of sacrificial fire altars. [22] Most mathematical problems considered in the ulba

Stras spring from "a single theological requirement,"[23] that of constructing fire altars which havedifferent shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. [23] According to (Hayashi 2005, p. 363), the ulba Stras contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians." The diagonal rope (akay-rajju) of an oblong (rectangle) produces both which the flank ( prvamni) and the horizontal (tiryamn) produce separately."[24] Since the statement is a stra, it is necessarily compressed and what the ropes produce is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student. [24] They contain lists of Pythagorean triples,[25] which are particular cases of Diophantine equations.[26] They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."[27] Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: , , and[28]

,

,

as well as a statement of the Pythagorean theorem for the sides

of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square."[28] It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." [28] Baudhayana gives a formula for the square root of two,[29]

The formula is accurate up to five decimal places, the true value being[30]

This formula is similar in structure to the

formula found on a Mesopotamian tablet[31] from the Old Babylonian period (19001600 BCE):[29]

which expresses

in the sexagesimal system, and which too is

accurate up to 5 decimal places (after rounding). According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE[32] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[33] indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE. "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."[34] Dani goes on to say: "As the main objective of the Sulvasutras was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the Sulvasutras. The occurrence of the triples in the Sulvasutras is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily."[34] In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by Manava (fl. 750650 BC) and

the Apastamba Sulba Sutra, composed by Apastamba (c. 600 BC), contained results similar to the Baudhayana Sulba Sutra. Vyakarana An important landmark of the Vedic period was the work of Sanskrit grammarian, Pini (c. 520460 BCE). His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the BackusNaur form (used in the description programming languages).

[edit]Jain

Mathematics (400 BCE 200 CE)

Although Jainism as a religion and philosophy predates its most famous exponent, Mahavira (6th century BCE), who was a contemporary of Gautama Buddha, most Jaina texts on mathematical topics were composed after the 6th century BCE. Jaina mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period." A significant historical contribution of Jaina mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and infinities, led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, they went on to define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jaina mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations (beejganita samikaran). Jaina mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. More than a millennium later, their appellation became the English word "zero" after a tortuous journey of translations and transliterations from India to Europe . (See Zero: Etymology.)

In addition to Surya Prajnapti, important Jaina works on mathematics included the Vaishali Ganit (c. 3rd century BCE); the Sthananga

Sutra (fl. 300 BCE 200 CE); the Anoyogdwar Sutra (fl. 200 BCE 100 CE); and the Satkhandagama (c. 2nd century CE). Important Jaina mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the Bhadrabahavi-Samhita and a commentary on the Surya Prajinapti; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called Tiloyapannati; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jaina philosophy and metaphysics, composed a mathematical work called Tattwarthadhigama-Sutra Bhashya. Pingala Among other scholars of this period who contributed to mathematics, the most notable is Pingala (pigal) (fl. 300200 BCE), a musical theorist who authored the Chhandas Shastra (chanda-stra, also Chhandas Sutra chhanda-stra), a Sanskrit treatise on prosody. There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both the Pascal triangleand Binomial coefficients, although he did not have knowledge of the Binomial theorem itself.[35][36] Pingala's work also contains the basic ideas of Fibonacci numbers (called maatraameru). Although the Chandah sutra hasn't survived in its entirety, a 10th century commentary on it by Halyudha has. Halyudha, who refers to the Pascal triangle as Meru-prastra (literally "the staircase to Mount Meru"), has this to say: "Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting 1 in the first square. Put 1 in each of the two squares of the second line. In the third line put 1 in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put 1 in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way.

Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ..." [35] The text also indicates that Pingala was aware of the combinatorial identity:[36]

Katyayana Though not a Jaina mathematician, Katyayana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the Katyayana Sulba Sutra, which presented much geometry, including the general Pythagorean theorem and a computation of the square root of 2 correct to five decimal places.

[edit]Oral

tradition

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (paita "learned man"),[37] who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vykaraa), exegesis (mms) and logic (nyya)."[37] Memorization of "what is heard" (ruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorization and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia." [38]

[edit]Styles

of memorization

Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity.[39] For example, memorization of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the ja-

pha(literally "mesh recitation") in which every two adjacent wordsin the text were first recited in their original order, then repeated

in the reverse order, and finally repeated again in the original order.[40] The recitation thus proceeded as: word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ... In another form of recitation, dhvaja-pha[40] (literally "flag recitation") a sequence of N words were recited (and memorized) by pairing the first two and last two words and then proceeding as: word1word2, wordN 1wordN; word2word3, wordN 3wordN 2; ..; wordN 1wordN, word1word2; The most complex form of recitation, ghana-pha (literally "dense recitation"), according to (Filliozat 2004, p. 139), took the form: word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ... That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the gveda (ca. 1500 BCE), as a single text, without any variant readings.[40] Similar methods were used for memorizing mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. 500 BCE).

[edit]The

Stra genre

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Vedas, which took the form of works called Vedgas, or, "Ancillaries of the Veda" (7th4th century BCE).[41] The need to conserve the sound of sacred text by use of ik (phonetics) and chhandas (metrics); to conserve its meaning by use of vykaraa (grammar) and nirukta (etymology); and to correctly perform the rites at the correct time by the use of kalpa (ritual) and jyotia (astronomy), gave rise to the six disciplines of the Vedgas.[41] Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the Vedgas immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral

literature. They were expressed in a highly compressed mnemonic form, the stra (literally, "thread"): The knowers of the stra know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable.[41] Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language," [41] using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables.[41] The stras create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called Guru-

shishya parampara, 'uninterrupted succession from teacher (guru) tothe student (isya),' and it was not open to the general public" and perhaps even kept secret.[42] The brevity achieved in a stra is demonstrated in the following example from the Baudhyana ulba

Stra (700 BCE).

The design of the domestic fire altar in the ulba Stra

The domestic fire-altar in the Vedic period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent

rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely.[43] The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhyana ulba Stra, this procedure is described in the following words: "II.64. After dividing the quadri-lateral in seven, one divides the transverse [cord] in three. II.65. In another layer one places the [bricks] Northpointing."[43] According to (Filliozat 2004, p. 144), the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit,rajju, f.), two pegs (Sanskrit, anku, m.), and clay to make the bricks (Sanskrit, iak, f.). Concision is achieved in the stra, by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the East-West direction, but that too is implied by the explicit mention of "North-pointing" in the second stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory.[43]

[edit]The

written tradition: prose commentary

With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation.

"India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally."[44] The earliest mathematical prose commentary was that on the work, ryabhaya (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the ryabhaya was composed of 33 stras (in verse form) consisting of mathematical statements or rules, but without any proofs. [45] However, according to (Hayashi 2003, p. 123), "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (upapatti). Bhaskara I's commentary on the ryabhaya, had the following structure:[45]

Rule ('stra') in verse by ryabhaa Commentary by Bhskara I, consisting of:

Elucidation of rule (derivations were still rare then, but became more common later) Example (uddeaka) usually in verse. Setting (nysa/sthpan) of the numerical data. Working (karana) of the solution. Verification (pratyayakaraa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favored by then. [45]

Typically, for any mathematical topic, students in ancient India first memorized the stras, which, as explained earlier, were "deliberately inadequate"[44] in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (i.e. boards covered with dust).

The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta (fl. 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit:dhulikarman).[46]

[edit]Numerals

and the decimal numeral system

It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe.[47] The Syrian bishop Severus Sebokhtwrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers.[47] However, how, when, and where the first decimal place value system was invented is not so clear. [48] The earliest extant script used in India was the Kharoh script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brhm script, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system.[49] The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE.[50] A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate.[50] Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.[50] There are older textual sources, although the extant manuscript copies of these texts are from much later dates. [51] Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. [51] Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when

in hundreds, one hundred."[51] Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."[51] A third decimal representation was employed in a verse composition technique, later labeled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books. [52] Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. [52] According to Plofker 2009, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "tooth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). [52] So, Veda/tooth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left.[52] The earliest reference employing object numbers is a ca. 269 CE Sanskrit text, Yavanajtaka(literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology. [53] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India. [52] It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. [54] According to Plofker 2009, These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-

place-value system; no documentary evidence survives to confirm either conclusion."[54]

[edit]Bakhshali

Manuscript

The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit"[12] in the rad script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE.[55] The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously datedas early as the "early centuries of the Christian era"[56] and as late as between the 9th and 12th century CE.[57] The 7th century CE is now considered a plausible date,[58] albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work."[59] The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples.[55] The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi) and algebra (simultaneous linear equations and quadratic equations), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." [55] Many of its problems are the so-called equalization problems that lead to systems of linear equations. One example from Fragment III-5-3v is the following: "One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each

animal and the total value for the animals possessed by each merchant."[60] The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.[60]

[edit]Classical

Period (400 1200)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, and Bhaskara II give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotistra) and consisted of three sub-disciplines: mathematical sciences (gaitaor tantra), horoscope astrology (hor or jtaka) and divination (sahit).[46] This tripartite division is seen in Varhamihira's 6th century compilation

Pancasiddhantika[61] (literally panca, "five,"siddhnta, "conclusion ofdeliberation", dated 575 CE)of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries. [46]

[edit]Fifth

and sixth centuries

Surya Siddhanta Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry.[citation needed] Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.[62] This ancient text uses the following as trigonometric functions for the first time:[citation needed]

Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya).

It also contains the earliest uses of:[citation needed]

Tangent. Secant.

Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East. Chhedi calendar This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals). Aryabhata I Aryabhata (476550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:

Quadratic equations Trigonometry The value of , correct to 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include: Trigonometry: (See also : Aryabhata's sine table)

Introduced the trigonometric functions. Defined the sine (jya) as the modern relationship between half an angle and half a chord. Defined the cosine (kojya). Defined the versine (utkrama-jya). Defined the inverse sine (otkram jya).

Gave methods of calculating their approximate numerical values. Contains the earliest tables of sine, cosine and versine values, in 3.75 intervals from 0 to 90, to 4 decimal places of accuracy. Contains the trigonometric formula sin(n + 1)x sin nx = sin nx sin(n 1)x (1/225)sin nx.

Spherical trigonometry.

Arithmetic:

Continued fractions.

Algebra:

Solutions of simultaneous quadratic equations. Whole number solutions of linear equations by a method equivalent to the modern method.

General solution of the indeterminate linear equation .

Mathematical astronomy:

Accurate calculations for astronomical constants, such as the:

Solar eclipse. Lunar eclipse. The formula for the sum of the cubes, which was an important step in the development of integral calculus.[63]

Varahamihira Varahamihira (505587) produced the Pancha Siddhanta (The Five

Astronomical Canons). He made important contributions totrigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:

[edit]Seventh

and eighth centuries

Brahmagupta's theorem states thatAF = FD.

In the 7th century, two separate fields, arithmetic (which included mensuration) and algebra, began to emerge in Indian mathematics. The two fields would later be called p-gaita (literally "mathematics of algorithms") and bja-gaita (lit. "mathematics of seeds," with "seeds"like the seeds of plantsrepresenting unknowns with the potential to generate, in this case, the solutions of equations).[64] Brahmagupta, in his astronomical work Brhma Sphua

Siddhnta (628 CE), included two chapters (12 and 18) devoted tothese fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).[65] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:[65] Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula), as well as a complete description of rational triangles(i.e. triangles with rational sides and rational areas).

Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by

where s, the semiperimeter, given by Brahmagupta's Theorem on rational triangles: A triangle with rational sides and rational area is of the form:

for some rational numbers

and

.[66]

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers[65] and is considered the first systematic treatment of the subject. The rules (which included and exception: ) were all correct, with one .[65] Later in the chapter, he gave the

first explicit (although still not completely general) solution of the quadratic equation:

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.This is equivalent to:

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,[68]

where

is a nonsquare integer. He did this by

discovering the following identity:[68] Brahmagupta's Identity: which was a generalization of an earlier identity of Diophantus:[68] Brahmagupta used his identity to prove the following lemma:[68] Lemma (Brahmagupta): If solution of and, of is a solution , then: is a solution of He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem: Theorem (Brahmagupta): If the equation solution for any one of equation: then Pell's has an integer is a

also has an integer solution. [69] Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:[68] Example (Brahmagupta): Find integers such that:

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician."[68] The solution he provided was:

Bhaskara I Bhaskara I (c. 600680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryab

hatiya-bhashya and Laghubhaskariya. He produced:Solutions of indeterminate equations.

A rational approximation of the sine function.

A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.

[edit]Ninth

to twelfth

centuriesVirasena Virasena (9th century) was a Jain mathematician in the court of Rashtrakuta King Amoghava rsha of Manyakheta, Karnataka. He wrote

the Dhavala, a commentary on Jain mathematics, which:

Deals with the concept of ardhaccheda, the number of times a number could be halved; effectively logarithms to base 2, and lists various rules involving this operation.[70][71]

First uses logarithms to base 3 (trakacheda) and base 4 (caturthacheda).

Virasena also gave:

The derivation of the volume of a frustum by a sort of infinite procedure.

It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 AD.[71] Mahavira Mahavira Acharya (c. 800 870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was

patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titledGanit Saar

Sangraha on numericalmathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:

Zero Squares Cubes square roots, cube roots, and the series extending beyond these

Plane geometry Solid geometry Problems relating to the casting of shadows

Formulae derived to calculate the area of an ellipse and quadrilatera l inside a circle.

Mahavira also:

Asserted that the square root of a negative number did not exist

Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules

for area and perimeter of an ellipse.

Solved cubic equations. Solved quartic equations. Solved some quintic equations and higherorder polynomials.

Gave the general solutions of the higher order polynomial equations:

Solved indeterminate quadratic equations.

Solved indeterminate cubic equations.

Solved indeterminate higher order equations.

Shridhara Shridhara (c. 870930), who lived in Bengal, wrote the books titled Nav Shatika, Tri

Shatika and Pati Ganita. Hegave:

A good rule for finding the volume of a sphere.

The formula for solving quadratic equations.

THE 3 MISTAKES OF MY LIFE

This is my story, a guy who thought that the JEE was much different than any other exam, that it stresses upon understanding rather than memorizing, that it is nothing like the CBSE. Is it really so?

Here is my story. I was always interested in philosophy and cosmology/astronomy. I read books upon books and encyclopedias on "space, astronomy, the universe etc." I wouldn't say, I was a nerd, but yes, these happened to be my side interests.

The interests were reawakened when an astronomy programme was started in our school by an Indian astronomy organization called S.P.A.C.E. (www.space-india.org). It was hell lot of fun and I won't lie that I was certainly amongst its top students. The thing directly appealed to my natural instincts. In fact i was bothered by the fact that quite some students found the programme just like any other "academic programme". But anyway, the turnaround came when we were taken on an outstation trip to some village.

Astronomical observations are done in non-urban areas for the simple reason that no city means no mess in the sky. In short, a clear sky. Now the as the night fell (it was an overnight trip and I was in seventh grade), we were called up on the terrace where telescopes were set up. We were provided with our own planispheres and latitude finders and whole lots of other astronomical gear by the SPACE guys themselves.

A turning point now came in my life, the moment I looked up on that night. And the moment I looked up, I didn't put my head down for like the next five minutes and was moving around the whole of terrace with my neck pointing straight up, until it started hurting.

Well, when you look up at night, from your balcony or terrace, what you see is basically "a black dark background, with a few twinkling dots on it (the stars)". What we saw, was the exact opposite. "A WHITE BACKROUND, AND ON TOP OF IT , A FEW BLACK DOTS." The sky was literally, FLOODED with stars. You just cannot possibly imagine what I am talking about. There was just no place where you couldn't see a star. And the intensity and brightness of each of these was easily tens of times more than the brightest star you can spot from your balcony or terrace.

I'll post a picture, which will give you some idea, but which is actually nowhere even near what I am talking about.

This is a real photograph. It is the night sky as seen from the maldives. The streak that you see in the middle is what most of the rest of our galaxy is. Our sun is just one out of a 400 billion stars in this galaxy, and our sun lies at the very edge. But anyway, after that fateful night, my life or my perceptions were never the same again. But in due time, the programme was stopped in the school, and I had no choice to keep myself busy with other things.

Now came class 11th. I am in this confusion to decide what stream to opt for. In fact I didn't want to opt for any of them and was planning a totally different career path, i.e. music, which is as close to me as the above.

But, then somehow I took up science. It was perhaps making those huge accounts and banking tables in commerce that made me go in for science. I hated anything to do with banks and finances and stuff. Science was still in some sorts "my thing". I joined a coaching institute, for the reason that everybody was doing that. I was a sheep, but so was everybody else.

I still remember my first day at the coaching institute. i had scored a mere 80% in tenth. I belonged to a totally different kind of crowd. And all my friends were musicians and commercees. We had great plans of forming the most deadly heavy metal band in the world. And here I was sitting amongst the toppers, the 90%agers, and so on. I believed that I wouldn't get even a single thing in that class. The first class was physics.

The guy comes in, starts off with vectors. In a few classes, he reaches Newtons laws. I don't know what exactly appealed to me, but something did! It was only physics. The math teacher was an ass*ole who was just busy eating paan. The chemistry guy was slightly psycho. And chemistry itself didn't make any sense to me, ever.

But soon I realized, physics is the thing I have always been looking for sub-consciously!! Old memories of the astronomy trip, and my hours of reading the books on cosmology, were awakened. I then searched the internet like mad. I was soon spending hours on youtube watching videos of Einsteins

special relativity, string theory, Quantum physics, and the list is endless. THIS WAS MY FIRST MISTAKE. All of this happened around the mid-year of my class eleventh. And even though I was failing in every subject, I loved physics classes. Also let me add here, that I got an exact 50% in my first physics Unit Test. Apart from this, I NEVER, EVER, PASSED A PHYSICS EXAM IN SCHOOL, APART FROM THE BOARDS. While I watched videos upon videos on youtube and devoured books like stephen hawking, michio kaku and many others, my friends were busy ratofying equations of physics, were busy chewing their institute packages, were busy talking about mark distributions in the next U.T. or were busy discussing how they plan on an M.B.A. after a B.Tech.

Thus, they passed, got applauses, and I failed. Though, being very honest, failing exams was like a highly cool thing, and so I maintained the "cool" image by failing. I then looked up courses on physics and found the IIT M.Sc. integrated course out there. I then abused CBSE and made up my mind to take JEE with full force, "so that I could get the physics course from the highly prestigious institute." Usually it is the opposite of what people do. They take JEE, get selected and then fill sites like goiit.com with their garbage of "I am getting so and so rank, should I go for this or that or this or that???".

These are the most "DESERVING MINDS OF THE COUNTRY WHO DON'T HAVE THE SLIGHTEST INKLING OF WHAT THEY WANNA DO. AND LET ME REPEAT, THEY ARE THE MOST INTELLIGENT FELLAS OUT THERE."

But anyway, I thought that JEE is certainly going to be much better than CBSE and "that it is an exam which REALLY tests you. That it really tests your APTITUDE. I didnt bother to ask the question, "an aptitude for what?"

And took it to mean the kind of aptitude that I possessed, i.e. a scientific and philosophical one. THIS WAS MY SECOND MISTAKE. So finally the D-day is there, and I give the exam. I open up the physics section, and all I remember is that I just ticked and ticked and ticked. It was really very easy. I hadn't done the entire syllabus, so I only attempted the questions of the chapters I had done. I got almost all of them right. I turned to chemistry and math, and I was blank. I was just left turning pages and looking at the other kids working out the problems furiously.

I calculated the marks after the exam. I had scored almost all my marks in physics. I had easily cleared the cutoff (in physics), I had something like 60% marks in physics and I scored 14 marks in math and chem. combined(!!).

I then thought that maybe I will be able to clear this thing next year. So I took a drop.

THIS WAS MY THIRD MISTAKE I spent most of my drop year "getting into the depths of the subjects." If I came upon the euler formula, it intrigued me. It intrigued me as to how sin and cos functions which are geometric ratios of a triangle can creep into the equations of almost all kinds from electricity and mag. to S.H.M. I couldnt spot a triangle anywhere in electrical circuits, and couldn't understand the presence of the sin and cos functions. I simply pursued my curiosity and read wikipedia on the history of mathematics all day long. And all this while I read about the euler function, many of my competitors who now most probably are sitting in the IIT's, were furiously working out the packages of bansals and brilliant and what not.

I went into complete philosophical depths to satisfy my curiosity and worked out everything, inside out, in all the three subjects. But at the end of this, I had only done the standard books, and barely had any practice. I never gave so much of importance to practicing thousands of questions, as I gave to getting a deep insight into science, especially physics. I was continuously committing huge mistakes without realizing them.

I then finally gave the JEE paper this year. The damn thing had 82 questions. I started off, chem. was quite normal. I then turned to math. I looked up the questions, and I knew I could solve quite a many IF GIVEN ENOUGH TIME. I saw many familiar and direct questions. But as I never bothered to keep formulas on my fingertips and always relied upon deriving them first hand, in every question, I was in a huge mess.

Same thing happened in physics. There were just too many questions. I read them, the integer type, and I knew I can answer all of them. But I would have to derive all the equations for most of them. In the end, I missed the JEE, by a good margin of 24 marks. I then analyzed the whole thing, and realized that I had never done what I was supposed to do. i.e. SHUT MY HEAD AND PRACTICE QUESTIONS LIKE A FCKING DOG. This is what the JEE wants. It is not an exam any different from the CBSE boards. All of them are the same. They want the same stuff. They ask you typical stuff, year upon year. Every exam has this typical prep. for it. You just have to do that right.

JEE is based on mindless practice of all kinds of questions. So much practice, that everything is on your fingertips. It does not test how well you have delved into the subject itself. It just tests how much have you fcked your ass off with the subject, simply mindlessly practicing as much as you can.

I would sound like a frustrated loser, being a cry baby after a lost race. But it was never a race for me. It was always my wanting to be a physicist and doing the things I loved to their fullest. It so turned out , that the JEE doesn't give a damn about your insights, about any true understandings you may have, any natural temperaments you may have.

All exams, out there are based on the same lines. Many would now like to argue that there is no other way to test people. I think there is. There are many.

But in the end, I was just another guy, not bothering about which branch will I get, or where will I be placed or how much will I earn, but just wanting to get a reputed physics course. And I failed, because I committed the 3 mistakes.

------------------------------

Mom, from the time I was really young, I realized I had someone...you, who always cared, who always protected me, who was always there for me no matter what. You taught me right from wrong, and pushed me to do the right thing, even when it was hard to do. You took care of me when I was sick, and your love helped make me well. You had rules, and I learned that when I obeyed them, my life was simpler, better, richer. You were and are the guiding light of my life. My heart is filled with love for you, my teacher, my friend, my mother.

Major charactersGovind: Govind Patel is an ordinary guy with whom anybody can relate. He has very few desires but he is obsessed with the desires he covets. His main ambition is to become a businessman as he thinks that being a Gujarati, business is in his blood. His best friends are Omi and Ish (Ishaan). Govind is anagnostic. His father has abandoned him and his mother, who runs a business of selling home-made food items. To support her financially, he takesmathematics tuitions. He continues these tuitions even after starting the cricket shop business. He is the narrator of this story and the one who makes the "Three Mistakes". During the course of the story he falls in love with Vidya, Ishan's younger sister for whom he is a private tutor. Govind is the one who looks after the financial part of the business as he has good business sense and mathematical skills. Ishaan: He is a big cricket freak and also a patriot at heart. Ishan has been the best cricketer in his locality and school. He suggests the name of their business as "Team India Cricket Shop". He

helps Govind's business by organising daily cricket coaching camps. He has a family which makes life situation by keeping quiet. He has a younger sister, Vidya, about whom he is quite protective. When he discovers that a boy called Ali is a very talented batsman, he decides to go any length to give Ali proper training. Ishan usually looks after day-to-day shop activities as he has genuine interest in any cricket-related thing. Omi: He is the son of the Hindu priest of the local temple. His family enjoys great respect among the people. Through Omi's parents and maternal uncle (who own few shops as a part of the temple trust property), they readily get a place to start their business. He is a rather dumb kind of boy and has not many dreams, but likes to concentrate on having a healthy body. However, he resents growing up and being a saint like his father. He is a religious person and actively takes part in his maternal uncle's (Bittoo Mama) religious politics. He is however confused about his religious views which are mainly influenced by Bittoo Mama. Vidya: She is Ishan's younger sister. She is a rebel at heart and dreams to break free from the constraints of a typical middle-class family and society, to go to Mumbai, do a course in PR and become independent. She however despises maths which is required for her medical entrance exams. Hence, Ish asks Govind to take her mathematics tuition. However, in between their tuition they fall in love, have intimate sexual relations, which is unaccepted by anyone. Only Omi figures out the relationship Govind and Vidya share and also reminds Govind about the consequences of Ish getting to know about it. Ali: One of the students in Ish's coaching classes and a great batsman because of a rare nature's gift. However, he doesn't play too much cricket as he gets tired really fast and enjoys playing marbles. He is a Muslim boy and respects Ish like a Guru. He too, like Ish is patriotic at heart. He denies the offer of Australian scholarship and wants to play in the Indian side. Bittoo Mama: He is the maternal uncle of Omi. Mama runs the trust of the temple and agrees to rent the place to the three friends for the Cricket Shop business. He is an active member of aHindu political Party. He follows the preachings of Parekh-ji, a political-cum-spiritual leader and has complete faith in him. He has locked his horns with Ali's father who belongs to the Secular Party. He has a son Dhiraj. Mama is not fond of Govind as he is an agnostic. He is the main antagonist of the story. Govind's mom: Gujarati woman. She wants her son to pursue a degree in Engineering. She runs a home-made food business. She cares about Govind a lot and supports him well through his hardships and struggles. Ali's Father: He is a devout Muslim and works for a secular party. He is a very kind-hearted person who wants Ali to take his education seriously despite his mediocre financial condition. He treats the three friends very well when they go to visit Ali in his house. Overall he is a good person. Fred Li: He's an Australian Cricket Team member and a fast bowler. He invites the three friends and Ali to Sydney when they travel to Goa to see India-Australia One Day International and meet him in the stands. He spots talent in Ali and wants to help him getting a chance to be trained in his academy in Australia. Loosely based on Brett Lee.