Great Mathematicians

[temp1994]

Calculus[edit]
His work, theSiddhānta Shiromani, is an astronomical treatise and contains many theories not found in earlier works.[citation needed]Preliminary concepts ofinfinitesimal calculusandmathematical analysis, along with a number of results intrigonometry,differential calculusandintegral calculusthat are found in the work are of particular interest.​

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus.[17]It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement.[citation needed]Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.[18]​

• There is evidence of an early form ofRolle's theoremin his work
  • If
    
    then
    
    for some
    
    with
    
    then
    
    , thereby finding the derivative of sine, although he never developed the notion of derivatives.[19]
    • Bhaskara uses this result to work out the position angle of theecliptic, a quantity required for accurately predicting the time of an eclipse.
  • In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than atruti, or a1⁄33750of a second, and his measure of velocity was expressed in this infinitesimal unit of time.
  • He was aware that when a variable attains the maximum value, itsdifferentialvanishes.
  • He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.[citation needed]In this result, there are traces of the generalmean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle's theorem. The mean value theorem was later found byParameshvarain the 15th century in theLilavati Bhasya, a commentary on Bhaskara'sLilavati.
  Madhava(1340–1425) and theKerala Schoolmathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development ofcalculusin India.

• Astronomy[edit]
  Using an astronomical model developed byBrahmaguptain the 7th century, Bhaskara accurately defined many astronomical quantities, including, for example, the length of thesidereal year, the time that is required for the Earth to orbit the Sun, as 365.2588 days which is the same as in Suryasiddhanta.[citation needed]The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes.[citation needed]

• His mathematical astronomy textSiddhanta Shiromaniis written in two parts: the first part on mathematical astronomy and the second part on thesphere.

• The twelve chapters of the first part cover topics such as:

• • Meanlongitudesof theplanets.
  • True longitudes of the planets.
  • The three problems ofdiurnal rotation.
  • Syzygies.
  • Lunar eclipses.
  • Solar eclipses.
  • Latitudesof the planets.
  • Sunrise equation
  • TheMoon'screscent.
  • Conjunctionsof the planets with each other.
  • Conjunctions of the planets with the fixedstars.
  • The paths of the Sun and Moon.
  The second part contains thirteen chapters on the sphere. It covers topics such as:

• • Praise of study of the sphere.
  • Nature of the sphere.
  • Cosmographyandgeography.
  • Planetarymean motion.
  • Eccentricepicyclicmodel of the planets.
  • Thearmillary sphere.
  • Spherical trigonometry.
  • Ellipsecalculations.[citation needed]
  • First visibilities of the planets.
  • Calculating the lunar crescent.
  • Astronomical instruments.
  • Theseasons.
  • Problems of astronomical calculations.
  Engineering[edit]
  The earliest reference to aperpetual motionmachine date back to 1150, when Bhāskara II described awheelthat he claimed would run forever.[20]

• Bhāskara II used a measuring device known asYasti-yantra. This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale.[21]

"

​

​

 

[temp1994]

Brahmagupta - Wikipedia, the free encyclopedia​

"
Algebra
Brahmagupta gave the solution of the generallinear equationin chapter eighteen ofBrahmasphutasiddhanta,​

The difference betweenrupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. Therupasare [subtracted on the side] below that from which the square and the unknown are to be subtracted.[7]​

which is a solution for the equation

equivalent to

, whererupasrefers to the constantscande. He further gave two equivalent solutions to the generalquadratic equation​

18.44. Diminish by the middle [number] the square-root of therupasmultiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number].​

18.45. Whatever is the square-root of therupasmultiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.[7]​

which are, respectively, solutions for the equation

equivalent to,​

and​

He went on to solve systems of simultaneousindeterminate equationsstating that the desired variable must first be isolated, and then the equation must be divided by the desired variable'scoefficient. In particular, he recommended using "the pulverizer" to solve equations with multiple unknowns.​

18.51. Subtract the colors different from the first color. [The remainder] divided by the first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [is to be used].[7]​

Like the algebra ofDiophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.[8]The extent of Greek influence on thissyncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.[8]​

​

​

 

[temp1994]

Arithmetic
Four fundamental operations (addition, subtraction, multiplication and division) were known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the multiplication as thus “The multiplicand is repeated like a string for cattle, as often as there are integrant portions in the multiplier and is repeatedly multiplied by them and the products are added together. It is multiplication. Or the multiplicand is repeated as many times as there are component parts in the multiplier”.Indian arithmetic was known in Medieval Europe as "Modus Indoram" meaning method of the Indians. In Brahmasphutasiddhanta, Multiplication was named Gomutrika. In the beginning of chapter twelve of hisBrahmasphutasiddhanta, entitledCalculation, Brahmagupta details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions,

,

,

,

, and

​

​

​

 

[temp1994]

Series
Brahmagupta then goes on to give the sum of the squares and cubes of the firstnintegers.​

12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]."]Brahmagupta - Wikipedia, the free encyclopedia[11][/URL]​

Here Brahmagupta found the result in terms of thesumof the firstnintegers, rather than in terms ofnas is the modern practice."]Brahmagupta - Wikipedia, the free encyclopedia[12][/URL]​

He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)².​

Zero["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Brahmagupta'sBrahmasphuṭasiddhantais the first book that mentions zero as a number[[url]http://en.wikipedia.org/wiki/Wikipedia:Citation_neededcitation needed[/URL]], hence Brahmagupta is considered the first to formulate the concept of[url]http://en.wikipedia.org/wiki/Zerozero[/URL]. He gave rules of using zero with negative and positive numbers. Zero plus a positive number is the positive number and negative number plus zero is a negative number etc. TheBrahmasphutasiddhantais the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the[url]http://en.wikipedia.org/wiki/BabyloniansBabylonians[/URL]or as a symbol for a lack of quantity as was done by[url]http://en.wikipedia.org/wiki/PtolemyPtolemy[/URL]and the[url]http://en.wikipedia.org/wiki/Ancient_RomeRomans[/URL]. In chapter eighteen of hisBrahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction,​

18.30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero.​

[...]​

18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added."]Brahmagupta - Wikipedia, the free encyclopedia[7][/URL]
He goes on to describe multiplication,​

18.33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero."]Brahmagupta - Wikipedia, the free encyclopedia[7][/URL]​

But his description of[url]http://en.wikipedia.org/wiki/Division_by_zerodivision by zero[/URL]differs from our modern understanding,​

18.34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative.​

18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root."]Brahmagupta - Wikipedia, the free encyclopedia[7][/URL]​

Here Brahmagupta states that

and as for the question of

where

he did not commit himself."]Brahmagupta - Wikipedia, the free encyclopedia[13][/URL]His rules for[url]http://en.wikipedia.org/wiki/Arithmeticarithmetic[/URL]on[url]http://en.wikipedia.org/wiki/Negative_numbernegative numbers[/URL]and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left[url]http://en.wikipedia.org/wiki/Defined_and_undefinedundefined[/URL].​

Diophantine analysis["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Pythagorean triples["]Cannot find section - Wikipedia, the free encyclopedia"]Cannot find section - Wikipedia, the free encyclopedia[/url]edit[/URL]]
In chapter twelve of hisBrahmasphutasiddhanta, Brahmagupta provides a formula useful for generating[url]http://en.wikipedia.org/wiki/Pythagorean_triplePythagorean triples[/URL]:​

12.39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey."]Brahmagupta - Wikipedia, the free encyclopedia[14][/URL]​

Or, in other words, ifd = mx/(x + 2), then a traveller who "leaps" vertically upwards a distanced from the top of a mountain of heightm, and then travels in a straight line to a city at a horizontal distancemxfrom the base of the mountain, travels the same distance as one who descends vertically down the mountain and then travels along the horizontal to the city."]Brahmagupta - Wikipedia, the free encyclopedia[14][/URL]Stated geometrically, this says that if a right-angled triangle has a base of lengtha = mxand altitude of lengthb = m + d, then the length,c, of its hypotenuse is given byc = m (1+x) – d. And, indeed, elementary algebraic manipulation shows thata2+ b2= c2wheneverd has the value stated. Also, ifmandxare rational, so ared, a, bandc. A Pythagorean triple can therefore be obtained froma, bandc by multiplying each of them by the[url]http://en.wikipedia.org/wiki/Least_common_multipleleast common multiple[/URL]of their[url]http://en.wikipedia.org/wiki/Denominatordenominators[/URL].​

Pell's equation["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as

(called[url]http://en.wikipedia.org/wiki/Pell's_equationPell's equation[/URL]) by using the[url]http://en.wikipedia.org/wiki/Euclidean_algorithmEuclidean algorithm[/URL]. The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces."]Brahmagupta - Wikipedia, the free encyclopedia[15][/URL]​

​

​

 

[temp1994]

The nature of squares:​

18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.​

18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additiverupas."]Brahmagupta - Wikipedia, the free encyclopedia[7][/URL]​

The key to his solution was the identity,"]Brahmagupta - Wikipedia, the free encyclopedia[16][/URL]​

which is a generalization of an identity that was discovered by[url]http://en.wikipedia.org/wiki/DiophantusDiophantus[/URL],​

Using his identity and the fact that if

and

are solutions to the equations

and

, respectively, then

is a solution to

, he was able to find integral solutions to the Pell's equation through a series of equations of the form

. Unfortunately, Brahmagupta was not able to apply his solution uniformly for all possible values ofN, rather he was only able to show that if

has an integer solution for k = ±1, ±2, or ±4, then

has a solution. The solution of the general Pell's equation would have to wait for[url]http://en.wikipedia.org/wiki/Bhaskara_IIBhaskara II[/URL]in c. 1150 CE."]Brahmagupta - Wikipedia, the free encyclopedia[16][/URL]​

Geometry["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Brahmagupta's formula["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
[url]http://en.wikipedia.org/wiki/File:Brahmaguptas_formula.svg

[/URL]
Diagram for reference
Main article:[url]http://en.wikipedia.org/wiki/Brahmagupta's_formulaBrahmagupta's formula[/URL]
Brahmagupta's most famous result in geometry is his[url]http://en.wikipedia.org/wiki/Brahmagupta's_formulaformula[/URL]for[url]http://en.wikipedia.org/wiki/Cyclic_quadrilateralscyclic quadrilaterals[/URL]. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area,​

12.21. The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral. The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral."]Brahmagupta - Wikipedia, the free encyclopedia[11][/URL]​

So given the lengthsp,q,randsof a cyclic quadrilateral, the approximate area is

while, letting

, the exact area is​

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case."]Brahmagupta - Wikipedia, the free encyclopedia[17][/URL][url]http://en.wikipedia.org/wiki/Heron's_formulaHeron's formula[/URL]is a special case of this formula and it can be derived by setting one of the sides equal to zero.​

Triangles["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Brahmagupta dedicated a substantial portion of his work to geometry. One theorem gives the lengths of the two segments a triangle's base is divided into by its altitude:​

12.22. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment."]Brahmagupta - Wikipedia, the free encyclopedia[11][/URL]​

Thus the lengths of the two segments are

.​

He further gives a theorem on[url]http://en.wikipedia.org/wiki/Rational_trianglesrational triangles[/URL]. A triangle with rational sidesa,b,cand rational area is of the form:​

for some rational numbersu,v, andw."]Brahmagupta - Wikipedia, the free encyclopedia[18][/URL]​

Brahmagupta's theorem["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Main article:[url]http://en.wikipedia.org/wiki/Brahmagupta_theoremBrahmagupta theorem[/URL]
[url]http://en.wikipedia.org/wiki/File:Brahmaguptra's_theorem.svg

[/URL]​

​

​

 

[temp1994]

Brahmagupta's theorem states thatAF=FD.
Brahmagupta continues,​

12.23. The square-root of the sum of the two products of the sides and opposite sides of a non-unequal quadrilateral is the diagonal. The square of the diagonal is diminished by the square of half the sum of the base and the top; the square-root is the perpendicular [altitudes]."]Brahmagupta - Wikipedia, the free encyclopedia[11][/URL]​

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles[url]http://en.wikipedia.org/wiki/Trapezoidtrapezoid[/URL]), the length of each diagonal is

.​

He continues to give formulas for the lengths and areas of geometric figures, such as the circumradius of an isosceles trapezoid and a scalene quadrilateral, and the lengths of diagonals in a scalene cyclic quadrilateral. This leads up to[url]http://en.wikipedia.org/wiki/Brahmagupta's_theoremBrahmagupta's famous theorem[/URL],​

12.30-31. Imaging two triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals are the two bases. Their two segments are separately the upper and lower segments [formed] at the intersection of the diagonals. The two [lower segments] of the two diagonals are two sides in a triangle; the base [of the quadrilateral is the base of the triangle]. Its perpendicular is the lower portion of the [central] perpendicular; the upper portion of the [central] perpendicular is half of the sum of the [sides] perpendiculars diminished by the lower [portion of the central perpendicular]."]Brahmagupta - Wikipedia, the free encyclopedia[11][/URL]​

Pi["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
In verse 40, he gives values of[url]http://en.wikipedia.org/wiki/Piπ[/URL],​

​

​

12.40. The diameter and the square of the radius [each] multiplied by 3 are [respectively] the practical circumference and the area [of a circle]. The accurate [values] are the square-roots from the squares of those two multiplied by ten."]Brahmagupta - Wikipedia, the free encyclopedia[11][/URL]​

So Brahmagupta uses 3 as a "practical" value ofπ, and

as an "accurate" value ofπ.​

Measurements and constructions["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
In some of the verses before verse 40, Brahmagupta gives constructions of various figures with arbitrary sides. He essentially manipulated right triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and a scalene cyclic quadrilateral.​

After giving the value of pi, he deals with the geometry of plane figures and solids, such as finding volumes and surface areas (or empty spaces dug out of solids). He finds the volume of rectangular prisms, pyramids, and the frustum of a square pyramid. He further finds the average depth of a series of pits. For the volume of a[url]http://en.wikipedia.org/wiki/Frustumfrustum[/URL]of a pyramid, he gives the "pragmatic" value as the depth times the square of the mean of the edges of the top and bottom faces, and he gives the "superficial" volume as the depth times their mean area."]Brahmagupta - Wikipedia, the free encyclopedia[19][/URL]​

Trigonometry["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
Sine table["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
In Chapter 2 of hisBrahmasphutasiddhanta, entitledPlanetary True Longitudes, Brahmagupta presents a sine table:​

2.2-5. The sines: The Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six; flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows, suns [...]"]Brahmagupta - Wikipedia, the free encyclopedia[20][/URL]​

Here Brahmagupta uses names of objects to represent the digits of place-value numerals, as was common with numerical data in Sanskrit treatises. Progenitors represents the 14 Progenitors ("Manu") in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars of Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents the number of sides of the tradition die or 6, and so on. This information can be translated into the list of sines, 214, 427, 638, 846, 1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270."]Brahmagupta - Wikipedia, the free encyclopedia[21][/URL]​

Interpolation formula["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
See main article:[url]http://en.wikipedia.org/wiki/Brahmagupta's_interpolation_formulaBrahmagupta's interpolation formula[/URL]​

In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to[url]http://en.wikipedia.org/wiki/Interpolationinterpolate[/URL]new values of the[url]http://en.wikipedia.org/wiki/Trigonometric_functionsine[/URL]function from other values already tabulated."]Brahmagupta - Wikipedia, the free encyclopedia[22][/URL]The formula gives an estimate for the value of a function

at a valuea + xhof its argument (withh > 0 and −1 ≤ x ≤ 1) when its value is already known ata − h, aanda + h.​

The formula for the estimate is:​

where Δ is the first-order forward-[url]http://en.wikipedia.org/wiki/Difference_operatordifference operator[/URL], i.e.​

Astronomy["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
It was through theBrahmasphutasiddhantathat the Arabs learned of Indian astronomy."]Brahmagupta - Wikipedia, the free encyclopedia[23][/URL]Edward Saxhau stated that "Brahmagupta, it was he who taught Arabs astronomy","]Brahmagupta - Wikipedia, the free encyclopedia[24][/URL]The famous[url]http://en.wikipedia.org/wiki/AbbasidAbbasid[/URL]caliph[url]http://en.wikipedia.org/wiki/Al-MansurAl-Mansur[/URL](712–775) founded[url]http://en.wikipedia.org/wiki/BaghdadBaghdad[/URL], which is situated on the banks of the[url]http://en.wikipedia.org/wiki/TigrisTigris[/URL], and made it a center of learning. The caliph invited a scholar of[url]http://en.wikipedia.org/wiki/UjjainUjjain[/URL], by the name of Kankah, in 770 CE. Kankah used theBrahmasphutasiddhantato explain the Hindu system of arithmetic astronomy.[url]http://en.wikipedia.org/wiki/Muhammad_al-FazariMuhammad al-Fazari[/URL]translated Brahmugupta's work into Arabic upon the request of the caliph.​

In chapter seven of hisBrahmasphutasiddhanta, entitledLunar Crescent, Brahmagupta rebuts the idea that the Moon is farther from the Earth than the Sun, an idea which is maintained in scriptures. He does this by explaining the illumination of the Moon by the Sun."]Brahmagupta - Wikipedia, the free encyclopedia[25][/URL]​

7.1. If the moon were above the sun, how would the power of waxing and waning, etc., be produced from calculation of the [longitude of the] moon? the near half [would be] always bright.​

7.2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.​

7.3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...]"]Brahmagupta - Wikipedia, the free encyclopedia[26][/URL]
He explains that since the Moon is closer to the Earth than the Sun, the degree of the illuminated part of the Moon depends on the relative positions of the Sun and the Moon, and this can be computed from the size of the angle between the two bodies."]Brahmagupta - Wikipedia, the free encyclopedia[25][/URL]​

Some of the important contributions made by Brahmagupta in astronomy are: methods for calculating the position of heavenly bodies over time ([url]http://en.wikipedia.org/wiki/Ephemerisephemerides[/URL]), their rising and setting,[url]http://en.wikipedia.org/wiki/Conjunction_(astronomy)conjunctions[/URL], and the calculation of solar and lunar[url]http://en.wikipedia.org/wiki/Eclipseeclipses[/URL]."]Brahmagupta - Wikipedia, the free encyclopedia[27][/URL]Brahmagupta criticized the[url]http://en.wikipedia.org/wiki/PuranicPuranic[/URL]view that the Earth was flat or hollow. Instead, he observed that the Earth and heaven were spherical.​

"​

Madhava of Sangamagrama - Wikipedia, the free encyclopedia​

"​

Contributions["]Cannot find section - Wikipedia, the free encyclopediaedit[/URL]]
If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by[url]http://en.wikipedia.org/wiki/James_Gregory_(mathematician)James Gregory[/URL]in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term)."]Madhava of Sangamagrama - Wikipedia, the free encyclopedia[10][/URL]This implies that the limit nature of the infinite series was quite well understood by him. Thus, Madhava may have invented the ideas underlying[url]http://en.wikipedia.org/wiki/Infinite_seriesinfinite series[/URL]expansions of functions,[url]http://en.wikipedia.org/wiki/Power_seriespower series[/URL],[url]http://en.wikipedia.org/wiki/Trigonometric_seriestrigonometric series[/URL], and rational approximations of infinite series."]Madhava of Sangamagrama - Wikipedia, the free encyclopedia[11][/URL]​

However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars.​

​

​

 

[temp1994]

TheHindu–Arabic numeral system or Hindu numeral system is a positional decimal numeral system, nowadays the most common symbolic representation of numbers in the world. It was invented between the 1st and 4th centuries by Hindu mathematicians.

Hindu–Arabic numeral system - Wikipedia, the free encyclopedia

​

​

​

 

[The Great One]

Hindu–Arabic numeral system - Wikipedia, the free encyclopedia

View attachment 141044 ​

​

​

What exactly is the 'Arabic' part in Hindu-Arabic numeral system?

 

[Shark2]

What exactly is the 'Arabic' part in Hindu-Arabic numeral system?

The Arabs spread the Indian numeral system to Europe.

 

[temp1994]

What exactly is the 'Arabic' part in Hindu-Arabic numeral system?

Nothing.

That is the point. Europeans call these as Arabic numerals because they learned decimal system from Arabs. Medieval period was pretty shitty for every one, whether Indians, Chinese, or Europeans. Of these Europeans use to go out of their way to be ignorant.

This is case with whole of Abstract mathematics, whether it is Algebra, Trigonometry, or primitive calculus. India was an intellectual giant in Mathematics, metallurgy,Philosophy field ( until Islamic invasion ).

Here is complete list of Indian mathematicians.

List of Indian mathematicians - Wikipedia, the free encyclopedia

I could not post works of all mathematicians due to 10 image/post limit on this forum.( most consist of complex equations ).

 

[The Great One]

The Arabs spread the Indian numeral system to Europe.

So they're like Blue Dart?

Nothing.

That is the point. Europeans call these as Arabic numerals because they learned decimal system from Arabs. Medieval period was pretty shitty for every one, whether Indians, Chinese, or Europeans. Of these Europeans use to go out of their way to be ignorant.

This is case with whole of Abstract mathematics, whether it is Algebra, Trigonometry, or primitive calculus. India was a intellectual giant in Mathematics, metallurgy,Philosophy field ( until Islamic invasion ).

Here is complete list of Indian mathematicians.

List of Indian mathematicians - Wikipedia, the free encyclopedia

I could not post works of all mathematicians due to 10 image/post limit on this forum.( most consist of complex equations ).

Good to know

 

[temp1994]

So they're like Blue Dart?

Most of the Arab mathematical works are translation from some other language ( Predominantly sanskrit ).Islamic Sciences flourished for a very short period of 250 years before Mathematics and Science were deemed heretical and evil by a fatwah of Al-ghazali.

​

This is the reason why there is not a single muslim mathematician from India in pre 1947 era.


Europeans learned these from Arabs, so they attribute their genesis to Arabs.

 

[INDIC]

Aryabhatta and Brahmagupta.

 

[Narsimha Commando]

Proud to be an Indian for our great ancestors.

 

[Gauss]

Euclid (C. 325-265 B.C.E)

Euclid was the author of the most successful mathematics book ever written, the Elements, which appeared in over 1000 different editions from ancient to modern times. Little is known about Euclid’s life, other than that he taught at the famous academy at Alexandria. Apparently Euclid did not stress applications. When a student asked what he would get by learning geometry, Euclid explained that knowledge was worth acquiring for its own sake and told his servant to give the student a coin “since he must make a profit from what he learns.”