love bassi project on scientists of mathematics

8/14/2019 love bassi project on scientists of mathematics

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HISTORY AND THEIR INVENTIONS.......

1.Thales

2.Carl friedrich Gauss

3.Pythagoras

4.Aryabhata

5.Sir Isaac newton

6.Pierre simon Laplace

7.Muhammad ibn musa- al-khwarizmi


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THALES:-

Life

Thales lived around the mid 620s mid 540s BC and was born in the city ofMiletus. Miletus

was an ancient Greek Ionian city on the western coast ofAsia Minor(in what is today the

Aydin Province ofTurkey) near the mouth of the Maeander River

The dates of Thales' life are not known precisely. The time of his life is roughly established

by a few dateable events mentioned in the sources and an estimate of his length of life.

According to Herodotus, Thales once predicted a solar eclipse which has been determined by

modern methods to have been on May 28, 585 BC.[3]Diogenes Lartius quotes the chronicle

ofApollodorus as saying that Thales died at 78 in the 58th Olympiad (548545), and

Sosicrates as reporting that he was 90 at his death.

As mentioned, according to tradition, Thales was born in Miletus, Asia Minor. Diogenes

Laertius states that ("according to Herodotus and Douris and Democritus") his parents were

Examyes and Cleobuline, Phoeniciannobles. Giving another opinion, he ultimately connectsThales' family line back toPhoenician prince Cadmus. Diogenes also reports two other

stories, one that he married and had a son, Cybisthus orCybisthon, or adopted his nephew of

the same name. The second is that he never married, telling his mother as a young man that it

was too early to marry, and as an older man that it was too late. A much earlier source -

Plutarch - tells the following story: Solon who visited Thales asked him the reason which

kept him single. Thales answered that he did not like the idea of having to worry about

children. Nevertheless, several years later Thales anxious for family adopted his nephew

Cybisthus

INVENTIONS:-
http://en.wikipedia.org/wiki/Miletushttp://en.wikipedia.org/wiki/Miletushttp://en.wikipedia.org/wiki/Miletushttp://en.wikipedia.org/wiki/Ioniahttp://en.wikipedia.org/wiki/Asia_Minorhttp://en.wikipedia.org/wiki/Asia_Minorhttp://en.wikipedia.org/wiki/Aydin_Provincehttp://en.wikipedia.org/wiki/Turkeyhttp://en.wikipedia.org/wiki/Maeander_Riverhttp://en.wikipedia.org/wiki/Herodotushttp://en.wikipedia.org/wiki/Thales#cite_note-2http://en.wikipedia.org/wiki/Diogenes_La%C3%ABrtiushttp://en.wikipedia.org/wiki/Apollodorushttp://en.wikipedia.org/wiki/Olympiadhttp://en.wikipedia.org/wiki/Sosicrateshttp://en.wikipedia.org/wiki/Miletushttp://en.wikipedia.org/wiki/Anatoliahttp://en.wikipedia.org/wiki/Phoeniciahttp://en.wikipedia.org/wiki/Phoeniciahttp://en.wikipedia.org/wiki/Phoeniciahttp://en.wikipedia.org/wiki/Phoeniciahttp://en.wikipedia.org/wiki/Cadmushttp://en.wikipedia.org/w/index.php?title=Cybisthus&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Cybisthon&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Cybisthon&action=edit&redlink=1http://en.wikipedia.org/wiki/Plutarchhttp://en.wikipedia.org/wiki/Solonhttp://en.wikipedia.org/w/index.php?title=Cybisthus&action=edit&redlink=1http://en.wikipedia.org/wiki/Ioniahttp://en.wikipedia.org/wiki/Asia_Minorhttp://en.wikipedia.org/wiki/Aydin_Provincehttp://en.wikipedia.org/wiki/Turkeyhttp://en.wikipedia.org/wiki/Maeander_Riverhttp://en.wikipedia.org/wiki/Herodotushttp://en.wikipedia.org/wiki/Thales#cite_note-2http://en.wikipedia.org/wiki/Diogenes_La%C3%ABrtiushttp://en.wikipedia.org/wiki/Apollodorushttp://en.wikipedia.org/wiki/Olympiadhttp://en.wikipedia.org/wiki/Sosicrateshttp://en.wikipedia.org/wiki/Miletushttp://en.wikipedia.org/wiki/Anatoliahttp://en.wikipedia.org/wiki/Phoeniciahttp://en.wikipedia.org/wiki/Phoeniciahttp://en.wikipedia.org/wiki/Cadmushttp://en.wikipedia.org/w/index.php?title=Cybisthus&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Cybisthon&action=edit&redlink=1http://en.wikipedia.org/wiki/Plutarchhttp://en.wikipedia.org/wiki/Solonhttp://en.wikipedia.org/w/index.php?title=Cybisthus&action=edit&redlink=1http://en.wikipedia.org/wiki/Miletus


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Geometry

Thales understood similar triangles and right triangles, and what is more, used that

knowledge in practical ways. The story is told in DL (loc. cit.) that he measured the height ofthepyramidsby their shadows at the moment when his own shadow was equal to his height.

A right triangle with two equal legs is a 45-degree right triangle, all of which are similar. The

length of the pyramids shadow measured from the center of the pyramid at that moment

must have been equal to its height.

This story reveals that he was familiar with the Egyptian seqt, or seked, defined by Problem

57 of the Rhind papyrus as the ratio of the run to the rise of a slope, which is currently the

cotangent function oftrigonometry. It characterizes the angle of rise.

Our cotangents require the same units for run and rise, but the papyrus uses cubits for rise and

palms for run, resulting in different (but still characteristic) numbers. Since there were 7palms in a cubit, the seqt was 7 times the cotangent.

Thales' Theorem :

To use an example often quoted in modern reference works, suppose the base of a pyramid is

140 cubits and the angle of rise 5.25 seqt. The Egyptians expressed their fractions as the sum

of fractions, but the decimals are sufficient for the example. What is the rise in cubits? The

run is 70 cubits, 490 palms. X, the rise, is 490 divided by 5.25 or 93 1/3 cubits. These figures

sufficed for the Egyptians and Thales. We would go on to calculate the cotangent as 70

divided by 93 1/3 to get 3/4 or .75 and looking that up in a table of cotangents find that the

angle of rise is a few minutes over 53 degrees.

Whether the ability to use the seqt, which preceded Thales by about 1000 years, means thathe was the first to define trigonometry is a matter of opinion. More practically Thales used
http://en.wikipedia.org/wiki/Similar_triangleshttp://en.wikipedia.org/wiki/Right_trianglehttp://en.wikipedia.org/wiki/Right_trianglehttp://en.wikipedia.org/wiki/Pyramidshttp://en.wikipedia.org/wiki/Pyramidshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Cotangenthttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Cubitshttp://en.wikipedia.org/wiki/Palm_(unit)http://en.wikipedia.org/wiki/Intercept_theoremhttp://en.wikipedia.org/wiki/File:Thales_theorem_1.pnghttp://en.wikipedia.org/wiki/File:Thales_theorem_1.pnghttp://en.wikipedia.org/wiki/Similar_triangleshttp://en.wikipedia.org/wiki/Right_trianglehttp://en.wikipedia.org/wiki/Pyramidshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Slopehttp://en.wikipedia.org/wiki/Cotangenthttp://en.wikipedia.org/wiki/Trigonometryhttp://en.wikipedia.org/wiki/Cubitshttp://en.wikipedia.org/wiki/Palm_(unit)http://en.wikipedia.org/wiki/Intercept_theorem


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the same method to measure the distances of ships at sea, said Eudemus as reported by

Proclus (in Euclidem). According to Kirk & Raven (reference cited below), all you need

for this feat is three straight sticks pinned at one end and knowledge of your altitude. One

stick goes vertically into the ground. A second is made level. With the third you sight the ship

and calculate the seqt from the height of the stick and its distance from the point of insertion

to the line of sight.

The seqt is a measure of the angle. Knowledge of two angles (the seqt and a right angle) and

an enclosed leg (the altitude) allows you to determine by similar triangles the second leg,

which is the distance. Thales probably had his own equipment rigged and recorded his own

seqts, but that is only a guess.

Thales Theoremis stated in another article. (Actually there are two theorems called Theorem

of Thales, one having to do with a triangle inscribed in a circle and having the circle's

diameter as one leg, the other theorem being also called the intercept theorem.) In addition

Eudemus attributed to him the discovery that a circle isbisectedby its diameter, that the base

angles of an isosceles triangle are equal and that vertical angles are equal. It would be hard toimagine civilization without these theorems.

It is possible, of course, to question whether Thales really did discover these principles. On

the other hand, it is not possible to answer such doubts definitively. The sources are all that

we have, even though they sometimes contradict each other.

(The most we can say is that Thales knew these principles. There is no evidence for Thales

discovering these principles, and, based on the evidence, we cannot say that Thales

discovered these principles.)

Interpretations

In the long sojourn of philosophy on the earth there has existed hardly a philosopher or

historian of philosophy who did not mention Thales and try to characterize him in some way.

He is generally recognized as having brought something new to human thought.

Mathematics, astronomy and medicine already existed. Thales added something to these

different collections of knowledge to produce a universality, which, as far as writing tells us,

was not in tradition before, but resulted in a new field, science.

Ever since, interested persons have been asking what that new something is. Answers fall into

(at least) two categories, the theory and the method. Once an answer has been arrived at, thenext logical step is to ask how Thales compares to other philosophers, which leads to his

classification (rightly or wrongly).
http://en.wikipedia.org/wiki/Proclushttp://en.wikipedia.org/wiki/Thales'_theoremhttp://en.wikipedia.org/wiki/Thales'_theoremhttp://en.wikipedia.org/wiki/Intercept_theoremhttp://en.wikipedia.org/wiki/Intercept_theoremhttp://en.wikipedia.org/wiki/Eudemushttp://en.wikipedia.org/wiki/Bisectionhttp://en.wikipedia.org/wiki/Bisectionhttp://en.wikipedia.org/wiki/Proclushttp://en.wikipedia.org/wiki/Thales'_theoremhttp://en.wikipedia.org/wiki/Intercept_theoremhttp://en.wikipedia.org/wiki/Eudemushttp://en.wikipedia.org/wiki/Bisection


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Carl-Friedrich-Gauss

Johann Carl Friedrich Gauswas a child prodigy. There are many anecdotespertaining to his precocity while a toddler, and he made his first ground-breakingmathematical discoveries while still a teenager. He completed Disquisitiones

Arithmeticae, his magnum opus, in 1798 at the age of 21, though it would not bepublished until 1801. This work was fundamental in consolidating number theoryas a discipline and has shaped the field to the present day.

Carl Friedrich Gauss was born on April 30, 1777 in Braunschweig, in the Electorate of

Brunswick-Lneburg, now part ofLower Saxony,Germany, as the son of poor working-class

parents.[4] He was christened and confirmed in a Catholic church near the school he had

attended as a child.[5] There are several stories of his early genius. According to one, his gifts

became very apparent at the age of three when he corrected, mentally and without fault in his

calculations, an error his father had made on paper while calculating finances.

Another famous story has it that inprimary school his teacher, J.G. Bttner, tried to occupy

pupils by making them add a list ofintegers in arithmetic progression; as the story is most

often told, these were the numbers from 1 to 100. The young Gauss reputedly produced the

correct answer within seconds, to the astonishment of his teacher and his assistant Martin

Bartels. Gauss's presumed method was to realize that pairwise addition of terms from

opposite ends of the list yielded identical intermediate sums: 1 + 100 = 101, 2 + 99 = 101,3 + 98 = 101, and so on, for a total sum of 50 101 = 5050. However, the details of the story

are at best uncertain (see[6] for discussion of the original Wolfgang Sartorius von

Waltershausen source and the changes in other versions); some authors, such asJoseph

Rotman in his bookA first course in Abstract Algebra, question whether it ever happened.

INVENTIONS:-

o algebra (in algebra (mathematics): The fundamental theorem of algebra; in

algebra (mathematics): Prime factorization; in fundamental theorem of algebra)

o analysis (in analysis (mathematics): Arithmetization of analysis; in analysis(mathematics): Extension of analytic concepts to complex numbers )
http://en.wikipedia.org/wiki/Child_prodigyhttp://en.wikipedia.org/wiki/Disquisitiones_Arithmeticaehttp://en.wikipedia.org/wiki/Disquisitiones_Arithmeticaehttp://en.wikipedia.org/wiki/Masterpiecehttp://en.wikipedia.org/wiki/Braunschweighttp://en.wikipedia.org/wiki/Electorate_of_Brunswick-L%C3%BCneburghttp://en.wikipedia.org/wiki/Electorate_of_Brunswick-L%C3%BCneburghttp://en.wikipedia.org/wiki/Lower_Saxonyhttp://en.wikipedia.org/wiki/Germanyhttp://en.wikipedia.org/wiki/Germanyhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-3http://en.wikipedia.org/wiki/Confirmationhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-4http://en.wikipedia.org/wiki/Primary_schoolhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Arithmetic_progressionhttp://en.wikipedia.org/wiki/Johann_Christian_Martin_Bartelshttp://en.wikipedia.org/wiki/Johann_Christian_Martin_Bartelshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-5http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-5http://en.wikipedia.org/wiki/Wolfgang_Sartorius_von_Waltershausenhttp://en.wikipedia.org/wiki/Wolfgang_Sartorius_von_Waltershausenhttp://en.wikipedia.org/wiki/Joseph_Rotmanhttp://en.wikipedia.org/wiki/Joseph_Rotmanhttp://en.wikipedia.org/wiki/Joseph_Rotmanhttp://www.britannica.com/EBchecked/topic/14885/algebra/231072/The-fundamental-theorem-of-algebra#ref762359http://www.britannica.com/EBchecked/topic/14885/algebra/231080/Fundamental-concepts-of-modern-algebra#ref829409http://www.britannica.com/EBchecked/topic/222211/fundamental-theorem-of-algebra#ref1034220http://www.britannica.com/EBchecked/topic/22486/analysis/247693/Rebuilding-the-foundations#ref848270http://www.britannica.com/EBchecked/topic/22486/analysis/218293/Extension-of-analytic-concepts-to-complex-numbers#ref736757http://www.britannica.com/EBchecked/topic/22486/analysis/218293/Extension-of-analytic-concepts-to-complex-numbers#ref736757http://en.wikipedia.org/wiki/File:Carl_Friedrich_Gauss.jpghttp://en.wikipedia.org/wiki/Child_prodigyhttp://en.wikipedia.org/wiki/Disquisitiones_Arithmeticaehttp://en.wikipedia.org/wiki/Disquisitiones_Arithmeticaehttp://en.wikipedia.org/wiki/Masterpiecehttp://en.wikipedia.org/wiki/Braunschweighttp://en.wikipedia.org/wiki/Electorate_of_Brunswick-L%C3%BCneburghttp://en.wikipedia.org/wiki/Electorate_of_Brunswick-L%C3%BCneburghttp://en.wikipedia.org/wiki/Lower_Saxonyhttp://en.wikipedia.org/wiki/Germanyhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-3http://en.wikipedia.org/wiki/Confirmationhttp://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-4http://en.wikipedia.org/wiki/Primary_schoolhttp://en.wikipedia.org/wiki/Integerhttp://en.wikipedia.org/wiki/Arithmetic_progressionhttp://en.wikipedia.org/wiki/Johann_Christian_Martin_Bartelshttp://en.wikipedia.org/wiki/Johann_Christian_Martin_Bartelshttp://en.wikipedia.org/wiki/Carl_Friedrich_Gauss#cite_note-5http://en.wikipedia.org/wiki/Wolfgang_Sartorius_von_Waltershausenhttp://en.wikipedia.org/wiki/Wolfgang_Sartorius_von_Waltershausenhttp://en.wikipedia.org/wiki/Joseph_Rotmanhttp://en.wikipedia.org/wiki/Joseph_Rotmanhttp://www.britannica.com/EBchecked/topic/14885/algebra/231072/The-fundamental-theorem-of-algebra#ref762359http://www.britannica.com/EBchecked/topic/14885/algebra/231080/Fundamental-concepts-of-modern-algebra#ref829409http://www.britannica.com/EBchecked/topic/222211/fundamental-theorem-of-algebra#ref1034220http://www.britannica.com/EBchecked/topic/22486/analysis/247693/Rebuilding-the-foundations#ref848270http://www.britannica.com/EBchecked/topic/22486/analysis/218293/Extension-of-analytic-concepts-to-complex-numbers#ref736757http://www.britannica.com/EBchecked/topic/22486/analysis/218293/Extension-of-analytic-concepts-to-complex-numbers#ref736757


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o arithmetic (in arithmetic; in arithmetic: Fundamental theory)

o construction of regular n-gon (in Pierre de Fermat (French mathematician):

Work on theory of numbers; in Euclidean geometry: Regular polygons )

o foundations of mathematics (infoundations of mathematics: Number

systems)

o Gaussian curvature (in differential geometry: Curvature of surfaces)o modular arithmetic (in modular arithmetic)

o non-Euclidean geometry (in geometry (mathematics): Non-Euclidean

geometries)

o normal distribution (innormal distribution (statistics))

o number theory (in number theory (mathematics):Disquisitiones

Arithmeticae; in mathematics: The theory of numbers)

o numerical analysis (in numerical analysis (mathematics): Historical

background)

o prime numbers (inprime number theorem (mathematics))

o probability (in Chebyshevs inequality (mathematics))

o statistical mathematics (inprobability and statistics (mathematics): The

spread of statistical mathematics)

o theory of equations (in mathematics: Theory of equations)

Non-Euclidean geometries

The Enlightenment was not so preoccupied with analysis as to completely ignore the problem

of Euclids fifth postulate. In 1733 Girolamo Saccheri(16671733), a Jesuit professor of

mathematics at the University of Pavia, Italy, substantially advanced the age-old discussion

by setting forth the alternatives in great clarity and detail before declaring that he had

cleared Euclid of every defect (Euclides ab Omni Naevo Vindicatus, 1733). Euclids fifthpostulate runs: If a straight line falling on two straight lines makes the interior angles on the

same side less than two right angles, the straight lines, if produced indefinitely, will meet on

that side on which are the angles less than two right angles. Saccheri took up the

quadrilateral ofOmar Khayyam (10481131), who started with two parallel linesAB andDC,

formed the sides by drawing linesAD andBCperpendicular toAB, and then considered three

hypotheses for the internal angles at CandD: to be right, obtuse, or acute (seefigure

). The first possibility gives Euclidean geometry. Saccheri devoted himself to

proving that the obtuse and the acute alternatives both end in contradictions, which would

thereby eliminate the need for an explicit parallel postulate.

On the way to this spurious demonstration, Saccheri established several theorems of non-

Euclidean geometryfor example, that according to whether the right, obtuse, or acute

hypothesis is true, the sum of the angles of a triangle respectively equals, exceeds, or falls

short of 180. He then destroyed the obtuse hypothesis by an argument that depended upon

allowing lines to increase in length indefinitely. If this is disallowed, the hypothesis of the

obtuse angle produces a system equivalent to standard spherical geometry, the geometry of

figures drawn on the surface of a sphere.
http://www.britannica.com/EBchecked/topic/34730/arithmetic#ref390225http://www.britannica.com/EBchecked/topic/34730/arithmetic/24751/Fundamental-theory#ref390256http://www.britannica.com/EBchecked/topic/34730/arithmetic/24751/Fundamental-theory#ref390256http://www.britannica.com/EBchecked/topic/204668/Pierre-de-Fermat/2277/Work-on-theory-of-numbers#ref5135http://www.britannica.com/EBchecked/topic/204668/Pierre-de-Fermat/2277/Work-on-theory-of-numbers#ref5135http://www.britannica.com/EBchecked/topic/194901/Euclidean-geometry/235568/Regular-polygons#ref828284http://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics/35444/Number-systems#ref412136http://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics/35444/Number-systems#ref412136http://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics/35444/Number-systems#ref412136http://www.britannica.com/EBchecked/topic/162938/differential-geometry/235557/Curvature-of-surfaces#ref828832http://www.britannica.com/EBchecked/topic/920687/modular-arithmetic#ref790651http://www.britannica.com/EBchecked/topic/229851/geometry/217502/Non-Euclidean-geometries#ref727435http://www.britannica.com/EBchecked/topic/229851/geometry/217502/Non-Euclidean-geometries#ref727435http://www.britannica.com/EBchecked/topic/418227/normal-distribution#ref843490http://www.britannica.com/EBchecked/topic/418227/normal-distribution#ref843490http://www.britannica.com/EBchecked/topic/422325/number-theory/233903/Number-theory-in-the-19th-century#ref796454http://www.britannica.com/EBchecked/topic/422325/number-theory/233903/Number-theory-in-the-19th-century#ref796454http://www.britannica.com/EBchecked/topic/422325/number-theory/233903/Number-theory-in-the-19th-century#ref796454http://www.britannica.com/EBchecked/topic/422325/number-theory/233903/Number-theory-in-the-19th-century#ref796454http://www.britannica.com/EBchecked/topic/369194/mathematics/66020/The-theory-of-numbers#ref536421http://www.britannica.com/EBchecked/topic/369194/mathematics/66020/The-theory-of-numbers#ref536421http://www.britannica.com/EBchecked/topic/422388/numerical-analysis/235497/Historical-background#ref364041http://www.britannica.com/EBchecked/topic/422388/numerical-analysis/235497/Historical-background#ref364041http://www.britannica.com/EBchecked/topic/476362/prime-number-theorem#ref790669http://www.britannica.com/EBchecked/topic/108218/Chebyshevs-inequality#ref843741http://www.britannica.com/EBchecked/topic/477493/probability/248194/The-spread-of-statistical-mathematics#ref849487http://www.britannica.com/EBchecked/topic/477493/probability/248194/The-spread-of-statistical-mathematics#ref849487http://www.britannica.com/EBchecked/topic/369194/mathematics/66013/Theory-of-equations#ref536360http://www.britannica.com/EBchecked/topic/515215/Girolamo-Saccherihttp://www.britannica.com/EBchecked/topic/515215/Girolamo-Saccherihttp://www.britannica.com/EBchecked/topic/428267/Omar-Khayyamhttp://www.britannica.com/EBchecked/topic-art/229851/57049/Quadrilateral-of-Omar-Khayyam-Omar-Khayyam-constructed-the-quadrilateral-shownhttp://www.britannica.com/EBchecked/topic-art/229851/57049/Quadrilateral-of-Omar-Khayyam-Omar-Khayyam-constructed-the-quadrilateral-shownhttp://www.britannica.com/EBchecked/topic/559649/spherical-geometryhttp://www.britannica.com/EBchecked/topic-art/229851/57049/Quadrilateral-of-Omar-Khayyam-Omar-Khayyam-constructed-the-quadrilateral-shownhttp://www.britannica.com/EBchecked/topic/34730/arithmetic#ref390225http://www.britannica.com/EBchecked/topic/34730/arithmetic/24751/Fundamental-theory#ref390256http://www.britannica.com/EBchecked/topic/204668/Pierre-de-Fermat/2277/Work-on-theory-of-numbers#ref5135http://www.britannica.com/EBchecked/topic/204668/Pierre-de-Fermat/2277/Work-on-theory-of-numbers#ref5135http://www.britannica.com/EBchecked/topic/194901/Euclidean-geometry/235568/Regular-polygons#ref828284http://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics/35444/Number-systems#ref412136http://www.britannica.com/EBchecked/topic/369221/foundations-of-mathematics/35444/Number-systems#ref412136http://www.britannica.com/EBchecked/topic/162938/differential-geometry/235557/Curvature-of-surfaces#ref828832http://www.britannica.com/EBchecked/topic/920687/modular-arithmetic#ref790651http://www.britannica.com/EBchecked/topic/229851/geometry/217502/Non-Euclidean-geometries#ref727435http://www.britannica.com/EBchecked/topic/229851/geometry/217502/Non-Euclidean-geometries#ref727435http://www.britannica.com/EBchecked/topic/418227/normal-distribution#ref843490http://www.britannica.com/EBchecked/topic/422325/number-theory/233903/Number-theory-in-the-19th-century#ref796454http://www.britannica.com/EBchecked/topic/422325/number-theory/233903/Number-theory-in-the-19th-century#ref796454http://www.britannica.com/EBchecked/topic/369194/mathematics/66020/The-theory-of-numbers#ref536421http://www.britannica.com/EBchecked/topic/422388/numerical-analysis/235497/Historical-backgroun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Pythagoras

Life And History:-

Pythagoras was born on Samos, the Greek island in the eastern Aegean, and wealso learn that Pythagoras was the son of Mnesarchus.[7] His father was a gem-engraver or a merchant. His name led him to be associated with PythianApollo;

Aristippus explained his name by saying, "He spoke (agor-) the truth no less thandid the Pythian (Pyth-)," and Iamblichus tells the story that the Pythia prophesiedthat his pregnant mother would give birth to a man supremely beautiful, wise,and beneficial to humankind.[8] A late source gives his mother's name as Pythias.[9] As to the date of his birth, Aristoxenus stated that Pythagoras left Samos inthe reign ofPolycrates, at the age of 40, which would give a date of birth around570 BC.[10]

Pythagoras made influential contributions to philosophy and religious teaching inthe late 6th century BC. He is often revered as a great mathematician, mysticand scientist, and he is best known for the Pythagorean theorem which bears hisname. However, because legend and obfuscation cloud his work even more than
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myth by followers ofPlato over two centuries after the death of Pythagoras, mainly to bolster

the case for Platonic meta-physics, which resonate well with the ideas they attributed to

Pythagoras. This attribution has stuck, down the centuries up to modern times.[42]The earliest

known mention of Pythagoras's name in connection with the theorem occurred five centuries

after his death, in the writings ofCiceroand Plutarch.

ARYABHATA

Aryabhata mentions in the Aryabhatiyathat it was composed 3,600 years into the Kali Yuga,

when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476

CE.[1]

Aryabhata provides no information about his place of birth. The only information comes from

Bhskara I, who describes Aryabhata as makya, "one belonging to the amaka country."

While amaka was originally situated in the northwest of India, it is widely attested that,during the Buddha's time, a branch of the Amaka people settled in the region between the
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Narmada andGodavari rivers, in the South GujaratNorth Maharashtra region of central

India. Aryabhata is believed to have been born there.[1][3] However, early Buddhist texts

describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts

describe the Ashmakas as having foughtAlexander, which would put them further north.[3]

Work

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that

he lived there for some time.[4] Both Hindu and Buddhist tradition, as well asBhskara I (CE

629), identify Kusumapura as P aliputra , modern Patna.[1] A verse mentions that Aryabhata

was the head of an institution (kulapa) at Kusumapura, and, because the university ofNalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated

that Aryabhata might have been the head of the Nalanda university as well. [1] Aryabhata isalso reputed to have set up an observatory at the Sun temple inTaregana, Bihar

INVENTIONS:-

Place value system and zero

Theplace-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in

place in his work; he certainly did not use the symbol, but French mathematician Georges

Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place

holder for the powers of ten with null coefficients [10]

However, Aryabhata did not use the brahmi numerals. Continuing the Sanskritic tradition

from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities,

such as the table ofsines in amnemonicform.[11]

Pi as irrational

Aryabhata worked on the approximation forPi (), and may have come to the conclusion that

is irrational. In the second part of theAryabhatiyam (gaitapda 10), he writes:

chaturadhikam atamaaguam dvaistath sahasrmAyutadvayavikambhasysanno vrttapariaha."Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle

with a diameter of 20,000 can be approached."

This implies that the ratio of the circumference to the diameter is ((4+100)8+62000)/20000

= 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word sanna (approaching), to mean that not only isthis an approximation but that the value is incommensurable (orirrational). If this is correct,

it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in

1761 by Lambert).[12]
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After Aryabhatiya was translated intoArabic(ca. 820 CE) this approximation was mentioned

in Al-Khwarizmi's book on algebra.[3]

Mensuration and trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half-side is the area." [13]

Aryabhata discussed the concept ofsine in his work by the name ofardha-jya. Literally, itmeans "half-chord". For simplicity, people started calling itjya. When Arabic writers

translated his works from Sanskrit into Arabic, they referred it asjiba. However, in Arabic

writings, vowels are omitted, and it was abbreviated asjb. Later writers substituted it with

jiab, meaning "cove" or "bay." (In Arabic,jiba is a meaningless word.) Later in the 12th

century, when Gherardo of Cremonatranslated these writings from Arabic into Latin, he

replaced the Arabicjiab with its Latin counterpart,sinus, which means "cove" or "bay". And

after that, thesinus becamesine in English.[14]

Indeterminate equations

A problem of great interest to Indian mathematicianssince ancient times has been to find

integer solutions to equations that have the form ax + b = cy, a topic that has come to be

known as diophantine equations. This is an example from Bhaskara's commentary on

Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 asthe remainder when divided by 9, and 1 as the remainder when divided by7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In

general, diophantine equations, such as this, can be notoriously difficult. They were discussed

extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800

BCE. Aryabhata's method of solving such problems is called the kuaka () method.Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves arecursive algorithm for writing the original factors in smaller numbers. Today this algorithm,

elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantineequations and is often referred to as the Aryabhata algorithm.[15]The diophantine equations

are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method

and earlier work in the Sulvasutras.

Algebra

InAryabhatiya Aryabhata provided elegant results for the summation ofseriesof squares and

cubes:[16]
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and

Sir Isaac Newton

Life nd History:-

Isaac Newton was born on 4 January 1643 [OS: 25 December 1642][1] atWoolsthorpe Manor in Woolsthorpe-by-Colsterworth, a hamlet in the county ofLincolnshire. At the time of Newton's birth, England had not adopted theGregorian calendar and therefore his date of birth was recorded as Christmas

Day, 25 December 1642. Newton was born three months after the death of hisfather, a prosperous farmer also named Isaac Newton. Born prematurely, he was
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a small child; his mother Hannah Ayscough reportedly said that he could have fitinside a quart mug ( 1.1 litre). From this information, it can be estimated thathe was born roughly 11 to 15 weeks early [original research?]. When Newton was three,his mother remarried and went to live with her new husband, the ReverendBarnabus Smith, leaving her son in the care of his maternal grandmother,

Margery Ayscough. The young Isaac disliked his stepfather and held someenmity towards his mother for marrying him, as revealed by this entry in a list ofsins committed up to the age of 19: "Threatening my father and mother Smith toburn them and the house over them."[10]

INVENTIONS:-

Mathematics

Newton's mathematical work has been said "to distinctly advance every branch ofmathematics then studied".[16] Newton's early work on the subject usually referred to as

fluxions or calculus is seen, for example, in a manuscript of October 1666, now published

among Newton's mathematical papers.[17] A related subject of his mathematical work was

infinite series. Newton's manuscript "De analysi per aequationes numero terminorum

infinitas" ("On analysis by equations infinite in number of terms") was sent byIsaac Barrow

to John Collins in June 1669: in August 1669 Barrow identified its author to Collins as "Mr

Newton, a fellow of our College, and very young ... but of an extraordinary genius and

proficiency in these things".[18] Newton later became involved in a dispute with Leibniz over

priority in the development of infinitesimal calculus. Most modern historians believe that

Newton and Leibniz developed infinitesimal calculusindependently, although with very

different notations. Occasionally it has been suggested that Newton published almost nothing

about it until 1693, and did not give a full account until 1704, while Leibniz began publishing

a full account of his methods in 1684. (Leibniz's notation and "differential Method",

nowadays recognized as much more convenient notations, were adopted by continental

European mathematicians, and after 1820 or so, also by British mathematicians.) Such a

suggestion, however, omits to notice the content of calculus which critics of Newton's time

and modern times have pointed out in Book 1of Newton'sPrincipia itself (published 1687)and in its forerunner manuscripts, such asDe motu corporum in gyrum ("On the motion of

bodies in orbit"), of 1684. ThePrincipiais not written in the language of calculus either as

we know it or as Newton's (later) 'dot' notation would write it. But Newton's work extensively

uses an infinitesimal calculus in geometric form, based on limiting values of the ratios ofvanishing small quantities: in thePrincipia itself Newton gave demonstration of this underthe name of 'the method of first and last ratios'[19]and explained why he put his expositions in

this form,[20]remarking also that 'hereby the same thing is performed as by the method of

indivisibles'. Because of this content thePrincipia has been called "a book dense with thetheory and application of the infinitesimal calculus" in modern times [21] and "lequel est

presque tout de ce calcul" ('nearly all of it is of this calculus') in Newton's time.[22]Newton's

use of methods involving "one or more orders of the infinitesimally small" is present in

Newton'sDe Motu Corporum in Gyrum of 1684[23] and in his papers on motion "during the

two decades preceding 1684"

Newton did not enter Cambridge specifically to study mathematics, but inhis first year he bought and studied Euclid, Descartes, Kepler and most
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important John Wallis'Arithmetica Infinitorum which concerned infiniteseries treated geometrically. He also later read Galileo and Fermat.

His first original work, in 1665, aged 23, concerned infinite (power) series.In particular, he proved the binomial theorem

(1 + x)r = 1 + rx + (r(r-1)/2)x2 +...,

(his notation was quite different). This had long been known for integralr, but Newton proved it for rational, positive or negative r, for which it isa power series; for example, he found power series expansions of1/root(1-x2), 1/(1+x2) , etc and their derivatives and antiderivatives bytermwise differentiation. He simply regarded power series as polynomialsof infinite degree, and did not consider convergence. His intuition guidedhim in avoiding divergent series. Thus he was able to find power series forsin, cos, tan, arcsin, arccos, arctan and ln (1+x).

The method of fluxions Over time, Newton produced three differentfoundations for his calculus, but there is no doubt that the one he used forhis discoveries, and his most popular presentation, was to look on a curveas the path of a moving particle, so the first and second derivativesalways exist and represent velocity and acceleration. Both x and y = f(x)are fluents or flowing quantities, and dot x and dot y are their fluxionsor rates of change with respect to time. So the slope of a curve would bedot y/ dot x , what we would call the parametric representation of thederivative. Similarly he has dot dot y for the fluxion of dot y and y' for the

fluent whose fluxion is y , ie the antiderivative of y . The existence ofthese functions is justified by the existence of instantaneous velocities.

The second method, which Newton used to actually compute thederivatives of various functions was the " little o " notation, exactly as inFermat, so o is an infinitesimal or infinitely small quantity.

The third is the " method of first and last ratios " which is similar to ourcurrent ideas of limits. Newton intended this method the replaceExhaustion as a logical foundation for his calculus.

Newton was slow to publish. He was persuaded by Edmund Halley, hisfriend, promoter and Professor at Oxford, later Astronomer Royal, topublish his first and most important publication, Principia Mathematica ,the Mathematical Principles of Natural Philosophy. This contains acomplete development of calculus as well as dynamics and theirapplication to astronomy. In particular it contains Newton's three laws ofmotion, the Law of Gravitation (particles of mass m and M at distance dattract each other with force mM/d2)),

Isaac Newton explained the workings of the universe through mathematics. He

formulated laws of motion and gravitation. These laws are math formulas that explain

how objects move when a force acts on them. Isaac published his most famous book,


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Principia, in 1687 while he was a mathematics professor at Trinity College, Cambridge.

In the Principia, Isaac explained three basic laws that govern the way objects move. He

then described his idea, or theory, about gravity. Gravity is the force that causes things

to fall down. If a pencil falls off a desk, it will land on the floor, not the ceiling. In his

book Isaac also used his laws to show that the planets revolve around the suns in orbits

that are oval, not round.

Pierre-Simon Laplace:-

LIFE and history:-
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Pierre-Simon, marquis de Laplace (23 March 1749 5 March 1827) was aFrenchmathematician and astronomer whose work was pivotal to thedevelopment ofmathematical astronomy and statistics. He summarized andextended the work of his predecessors in his five volume Mcanique Cleste(Celestial Mechanics) (17991825). This work translated the geometric study of

classical mechanics to one based on calculus, opening up a broader range ofproblems. In statistics, the so-called Bayesian interpretation of probability wasmainly developed by Lapl

He is remembered as one of the greatest scientists of all time, sometimes referred to as a

FrenchNewtonorNewton of France, with a phenomenal natural mathematical faculty

superior to any of his contemporaries.[2]

He became a count of the First French Empire in 1806 and was named amarquisin 1817,

after the Bourbon Restoration.

Inventions:-

Analytic theory of probabilities

In 1812, Laplace issued his Thorie analytique des probabilits in which he laid down manyfundamental results in statistics. In 1819, he published a popular account of his work on

probability. This book bears the same relation to the Thorie des probabilits that theSystme du monde does to the Mchanique cleste.[6]

Probability-generating function

The method of estimating the ratio of the number of favourable cases, compared to the whole

number of possible cases, had been previously indicated by Laplace in a paper written in

1779. It consists of treating the successive values of any functionas the coefficients in the

expansion of another function, with reference to a different variable. The latter is therefore

called theprobability-generating function of the former. Laplace then shows how, by means

ofinterpolation, these coefficients may be determined from the generating function. Next he

attacks the converse problem, and from the coefficients he finds the generating function; this

is effected by the solution of a finite difference equation.[6]

Least squares

This treatise includes an exposition of the method of least squares, a remarkable testimony to

Laplace's command over the processes of analysis. The method of least squares for the

combination of numerous observations had been given empirically by Carl Friedrich Gauss

(around 1794) and Legendre (in 1805), but the fourth chapter of this work contains a formal

proof of it, on which the whole of the theory of errors has been since based. This was affected

only by a most intricate analysis specially invented for the purpose, but the form in which it ispresented is so meagre and unsatisfactory that, in spite of the uniform accuracy of the results,
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it was at one time questioned whether Laplace had actually gone through the difficult work

he so briefly and often incorrectly indicates.[6]

Inductive probability

While he conducted much research inphysics, another major theme of his life's endeavourswasprobability theory. In hisEssai philosophique sur les probabilits (1814), Laplace set out

a mathematical system ofinductive reasoning based onprobability, which we would today

recognise as Bayesian. He begins the text with a series of principles of probability, the first

six being:

1) Probability is the ratio of the "favored events" to the total possible events.

2) The probability of all possible events are equal, or we must find another unit of

probabilistic measurement which will commensurate the measurement of the probability of

all possible events.

3) For independent events, the probability of the occurrence of all is the probability of each

multiplied together.

4) For events not independent, the probability of event B following event A (or event A

causing B) is the probability of A multiplied by the probability that A and B both occur.

5) The probability that A will occur, given B has occurred, is the probability of A divided by

the probability of B.

6) Three corollaries are given for the sixth principle, which amount to Bayesian probability.Where event exhausts the list of possible causes for event B,

Pr(B) =Pr(A1,A2,...An). Then .

One well-known formula arising from his system is the rule of succession, given as principle

seven. Suppose that some trial has only two possible outcomes, labeled "success" and

"failure". Under the assumption that little or nothing is known a priori about the relative

plausibilities of the outcomes, Laplace derived a formula for the probability that the next trial

will be a success.

wheres is the number of previously observed successes and n is the total number of observedtrials. It is still used as an estimator for the probability of an event if we know the event

space, but only have a small number of samples.

The rule of succession has been subject to much criticism, partly due to the example which

Laplace chose to illustrate it. He calculated that the probability that the sun will rise

tomorrow, given that it has never failed to in the past, was
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where dis the number of times the sun has risen in the past. This result has been derided asabsurd, and some authors have concluded that all applications of the Rule of Succession are

absurd by extension. However, Laplace was fully aware of the absurdity of the result;

immediately following the example, he wrote, "But this number [i.e., the probability that the

sun will rise tomorrow] is far greater for him who, seeing in the totality of phenomena the

principle regulating the days and seasons, realizes that nothing at the present moment can

arrest the course of it."[22]

Muhammad ibn musa al-khwarizmi

LIFE and HISTORY:-

Ab Abdallh Muammad ibn Ms al-Khwrizm[1] (c. 780, Khwrizm[2][3][4] c. 850) was aPersian[5] [2] [6] mathematician, astronomerand geographer, ascholarin the

House of Wisdomin Baghdad.

HisKitab al-Jabr wa-l-Muqabalapresented the first systematic solution oflinearand

quadratic equations. He is considered the founder ofalgebra,[7] a credit he shares with

Diophantus. In the twelfth century, Latin translations ofhis workon the Indian numerals,

introduced the decimalpositional number systemto the Western world.[4] He revised

Ptolemy's Geography and wrote on astronomy and astrology.
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His contributions had a great impact on language. "Algebra" is derived from al-jabr, one of

the two operations he used to solvequadratic equations.Algorismandalgorithm stem from

Algoritmi, the Latin form of his name.[8] His name is the origin of (Spanish)guarismo[9] and

of (Portuguese) algarismo, both meaning digit.

Few details of al-Khwrizm's life are known with certainty, even his birthplace is unsure.His name may indicate that he came from Khwarezm (Khiva), then in Greater Khorasan,

which occupied the eastern part of thePersian Empire, now Xorazm ProvinceinUzbekistan.

Abu Rayhan Biruni calls the people of Khwarizm "a branch of the Persian tree".[10]

Al-Tabarigave his name as Muhammad ibn Musa al-Khwrizm al-Majousi al-Katarbali

(Arabic: ). The epithetal-Qutrubbullicould indicate he might instead have come from Qutrubbul (Qatrabbul)[11], a viticulture

district nearBaghdad. However, Rashed[12] points out that:

There is no need to be an expert on the period or a philologist to see that al-

Tabari's second citation should read Muhammad ibn Msa al-Khwrizmand al-Majsi al-Qutrubbulli, and that there are two people (al-Khwrizm and al-Majsial-Qutrubbulli) between whom the letter wa [Arabic for the article and] hasbeen omitted in an early copy. This would not be worth mentioning if a series oferrors concerning the personality of al-Khwrizm, occasionally even the originsof his knowledge, had not been made. Recently, G. J. Toomer with naiveconfidence constructed an entire fantasy on the error which cannot be deniedthe merit of amusing the reader.

Regarding al-Khwrizm's religion, Toomer writes:

Another epithet given to him by al-abar, "al-Majs," would seem to indicatethat he was an adherent of the old Zoroastrian religion. This would still havebeen possible at that time for a man of Iranian origin, but the pious preface to al-Khwrizm'sAlgebra shows that he was an orthodox Muslim, so al-abar'sepithet could mean no more than that his forebears, and perhaps he in his youth,had been Zoroastrians

Inventions:-

Algebra

Al-Kitb al-mukhtaar f isb al-jabr wa-l-muqbala (Arabic: The Compendious Book on Calculation by Completion and

Balancing) is a mathematical book written approximately 830 CE. The book was written

with the encouragement of the Caliph Al-Ma'mun as a popular work on calculation and is

replete with examples and applications to a wide range of problems in trade, surveying and

legal inheritance[15]. The term algebrais derived from the name of one of the basic operations

with equations (al-jabr) described in this book. The book was translated in Latin asLiber

algebrae et almucabala by Robert of Chester(Segovia, 1145) hence "algebra", and also by

Gerard of Cremona. A unique Arabic copy is kept at Oxford and was translated in 1831 by F.

Rosen. A Latin translation is kept in Cambridge.

[16]
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The al-jabris considered the foundational text of modern algebra. It provided an exhaustive

account of solving polynomial equations up to the second degree,[17]and introduced the

fundamental methods of "reduction" and "balancing", referring to the transposition of

subtracted terms to the other side of an equation, that is, the cancellation of like terms on

opposite sides of the equation.[18]

Al-Khwrizm's method of solving linear and quadratic equations worked by first reducing

the equation to one of six standard forms (where b and c are positive integers)

squares equal roots (ax2 = bx) squares equal number (ax2 = c) roots equal number (bx= c) squares and roots equal number (ax2 + bx= c) squares and number equal roots (ax2 + c = bx) roots and number equal squares (bx+ c = ax2)

by dividing out the coefficient of the square and using the two operations al-abr(Arabic: restoring or completion) and al-muqbala ("balancing"). Al-abr is the process ofremoving negative units, roots and squares from the equation by adding the same quantity to

each side. For example,x2 = 40x 4x2 is reduced to 5x2 = 40x. Al-muqbala is the process ofbringing quantities of the same type to the same side of the equation. For example,x2 + 14 =

x + 5 is reduced tox2 + 9 =x.

The above discussion uses modern mathematical notation for the types of problems which the

book discusses. However, in Al-Khwrizm's day, most of this notation had not yet been

invented, so he had to use ordinary text to present problems and their solutions. For example,

for one problem he writes, (from an 1831 translation)

"If some one say: "You divide ten into two parts: multiply the one by itself; it willbe equal to the other taken eighty-one times." Computation: You say, ten lessthing, multiplied by itself, is a hundred plus a square less twenty things, and thisis equal to eighty-one things. Separate the twenty things from a hundred and asquare, and add them to eighty-one. It will then be a hundred plus a square,which is equal to a hundred and one roots. Halve the roots; the moiety is fiftyand a half. Multiply this by itself, it is two thousand five hundred and fifty and aquarter. Subtract from this one hundred; the remainder is two thousand fourhundred and fifty and a quarter. Extract the root from this; it is forty-nine and a

half. Subtract this from the moiety of the roots, which is fifty and a half. Thereremains one, and this is one of the two parts."[15]

In modern notation this process, with 'x' the "thing" (shay') or "root", is given by the steps,

(10 x)2 = 81x

x2 + 100 = 101x

Let the roots of the equation be 'p' and 'q'. Then ,pq = 100 and
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So a root is given by

Several authors have also published texts under the name ofKitb al-abr wa-l-muqbala,includingAb anfa al-Dnawar, Ab Kmil Shuj ibn Aslam, AbMuammad al-Adl, Ab Ysuf al-Mi, 'Abd al-Hamd ibn Turk, Sind ibnAl, Sahl ibn Bir, and arafaddn al-s.

J. J. O'Conner and E. F. Robertson wrote in theMacTutor History of Mathematics archive:

"Perhaps one of the most significant advances made by Arabic mathematics

began at this time with the work of al-Khwarizmi, namely the beginnings ofalgebra. It is important to understand just how significant this new idea was. Itwas a revolutionary move away from the Greek concept of mathematics whichwas essentially geometry. Algebra was a unifying theory which allowed rationalnumbers, irrational numbers, geometrical magnitudes, etc., to all be treated as"algebraic objects". It gave mathematics a whole new development path somuch broader in concept to that which had existed before, and provided avehicle for future development of the subject. Another important aspect of theintroduction of algebraic ideas was that it allowed mathematics to be applied toitself in a way which had not happened before."[19]

R. Rashed and Angela Armstrong write:

"Al-Khwarizmi's text can be seen to be distinct not only from the Babyloniantablets, but also from Diophantus'Arithmetica. It no longer concerns a series ofproblems to be resolved, but an exposition which starts with primitive terms inwhich the combinations must give all possible prototypes for equations, whichhenceforward explicitly constitute the true object of study. On the other hand,the idea of an equation for its own sake appears from the beginning and, onecould say, in a generic manner, insofar as it does not simply emerge in thecourse of solving a problem, but is specifically called on to define an infinite classof problems."[20]

Arithmetic

Al-Khwrizm's second major work was on the subject of arithmetic, which survived in a

Latin translation but was lost in the original Arabic. The translation was most likely done in

the twelfth century by Adelard of Bath, who had also translated the astronomical tables in

1126.

The Latin manuscripts are untitled, but are commonly referred to by the first two words with

which they start:Dixit algorizmi ("So said al-Khwrizm"), orAlgoritmi de numero Indorum("al-Khwrizm on the Hindu Art of Reckoning"), a name given to the work by Baldassarre
http://en.wikipedia.org/wiki/Al-Dinawarihttp://en.wikipedia.org/wiki/Al-Dinawarihttp://en.wikipedia.org/wiki/Ab%C5%AB_K%C4%81mil_Shuj%C4%81_ibn_Aslamhttp://en.wikipedia.org/wiki/'Abd_al-Ham%C4%ABd_ibn_Turkhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/MacTutor_History_of_Mathematics_archivehttp://en.wikipedia.org/wiki/MacTutor_History_of_Mathematics_archivehttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Muhammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB#cite_note-18http://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Diophantushttp://en.wikipedia.org/wiki/Arithmeticahttp://en.wikipedia.org/wiki/Problemhttp://en.wikipedia.org/wiki/Expository_writinghttp://en.wikipedia.org/wiki/Muhammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB#cite_note-19http://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Arabic_languagehttp://en.wikipedia.org/wiki/Arabic_languagehttp://en.wikipedia.org/wiki/Adelard_of_Bathhttp://en.wikipedia.org/wiki/Baldassarre_Boncompagnihttp://en.wikipedia.org/wiki/Al-Dinawarihttp://en.wikipedia.org/wiki/Ab%C5%AB_K%C4%81mil_Shuj%C4%81_ibn_Aslamhttp://en.wikipedia.org/wiki/'Abd_al-Ham%C4%ABd_ibn_Turkhttp://en.wikipedia.org/wiki/Sharaf_al-D%C4%ABn_al-T%C5%ABs%C4%ABhttp://en.wikipedia.org/wiki/MacTutor_History_of_Mathematics_archivehttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Rational_numbershttp://en.wikipedia.org/wiki/Irrational_numberhttp://en.wikipedia.org/wiki/Muhammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB#cite_note-18http://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Babylonian_mathematicshttp://en.wikipedia.org/wiki/Diophantushttp://en.wikipedia.org/wiki/Arithmeticahttp://en.wikipedia.org/wiki/Problemhttp://en.wikipedia.org/wiki/Expository_writinghttp://en.wikipedia.org/wiki/Muhammad_ibn_M%C5%ABs%C4%81_al-Khw%C4%81rizm%C4%AB#cite_note-19http://en.wikipedia.org/wiki/Latinhttp://en.wikipedia.org/wiki/Arabic_languagehttp://en.wikipedia.org/wiki/Adelard_of_Bathhttp://en.wikipedia.org/wiki/Baldassarre_Boncompagni


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Boncompagniin 1857. The original Arabic title was possibly Kitb al-Jam wa-l-tafrqbi-isb al-Hind[21] ("The Book of Addition and Subtraction According to the HinduCalculation")[22]

Al-Khwarizmi's work on arithmetic was responsible for introducing theArabic numerals,

based on the Hindu-Arabic numeral systemdeveloped in Indian mathematics, to the Westernworld. The term "algorithm" is derived from the algorism, the technique of performing

arithmetic with Hindu-Arabic numerals developed by al-Khwarizmi. Both "algorithm" and

"algorism" are derived from theLatinized forms of al-Khwarizmi's name,Algoritmi and

Algorismi, respectively.
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