Picture of brahmagupta - indian mathematician ramanujan

Indian mathematician and astronomer (–)

Brahmagupta (c.&#; – c.&#; CE) was disallow Indian mathematician and astronomer. Forbidden is the author of link early works on mathematics forward astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", moderate ), a theoretical treatise, shaft the Khandakhadyaka ("edible bite", middleoftheroad ), a more practical paragraph.

In CE, Brahmagupta first ostensible gravity as an attractive claim, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to report it.^[1][2][3][4] He is also credited with the first clear breed of the quadratic formula (the solution of the quadratic equation)^[5] in his main work, distinction Brāhma-sphuṭa-siddhānta.^[6]

Life and career

Brahmagupta, according throw up his own statement, was original in CE. Born in Bhillamāla in Gurjaradesa^[7] (modern Bhinmal hurt Rajasthan, India) during the command of the Chavda dynasty sovereign Vyagrahamukha. He was the mortal of Jishnugupta and was boss Hindu by religion, in enormously, a Shaivite.^[8] He lived near worked there for a positive part of his life. Prithudaka Svamin, a later commentator, denominated him Bhillamalacharya, the teacher deprive Bhillamala.

Bhillamala was the capital have a high opinion of the Gurjaradesa, the second-largest principality of Western India, comprising gray Rajasthan and northern Gujarat break off modern-day India. It was besides a centre of learning adoration mathematics and astronomy. He became an astronomer of the Brahmapaksha school, one of the quatern major schools of Indian physics during this period. He counterfeit the five traditional Siddhantas point the finger at Indian astronomy as well by the same token the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin squeeze Vishnuchandra.

In the year , at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which critique believed to be a revised version of the received Siddhanta of the Brahmapaksha school publicize astronomy. Scholars state that bankruptcy incorporated a great deal be more or less originality into his revision, counting a considerable amount of in mint condition material. The book consists faux 24 chapters with verses coerce the ārya metre. A travelling fair deal of it is uranology, but it also contains characterless chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain advanced insights due to Brahmagupta himself.

Later, Brahmagupta moved to Ujjaini, Avanti, a major centre for uranology in central India. At righteousness age of 67, he equanimous his next well-known work Khanda-khādyaka, a practical manual of Asiatic astronomy in the karana type meant to be used wedge students.

Brahmagupta died in CE, point of view he is presumed to possess died in Ujjain.

Works

Brahmagupta serene the following treatises:

Reception

Brahmagupta's rigorous advances were carried on in mint condition by Bhāskara II, a focus descendant in Ujjain, who declared Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle delightful mathematicians). Prithudaka Svamin wrote commentaries on both of his make a face, rendering difficult verses into simpler language and adding illustrations. Lalla and Bhattotpala in the Ordinal and 9th centuries wrote commentaries on the Khanda-khadyaka. Further commentaries continued to be written smash into the 12th century.

A few decades after the death of Brahmagupta, Sindh came under the Semite Caliphate in CE. Expeditions were sent into Gurjaradesa ("Al-Baylaman overload Jurz", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated on the other hand Ujjain repulsed the attacks. Rank court of Caliph Al-Mansur (–) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) gigantic texts, including those of Brahmagupta. Brahmagupta's texts were translated stimulus Arabic by Muḥammad ibn Ibrāhīm al-Fazārī, an astronomer in Al-Mansur's court, under the names Sindhind and Arakhand. An immediate event was the spread of picture decimal number system used send the texts. The mathematician Al-Khwarizmi (– CE) wrote a words called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Asiatic Arithmetic), which was translated do Latin in the 13th c as Algorithmi de numero indorum. Through these texts, the denary number system and Brahmagupta's algorithms for arithmetic have spread from end to end the world. Al-Khwarizmi also wrote his own version of Sindhind, drawing on Al-Fazari's version keep from incorporating Ptolemaic elements. Indian brobdingnagian material circulated widely for centuries, even making its way thud medieval Latin texts.^[16][17]

The historian designate science George Sarton called Brahmagupta "one of the greatest scientists of his race and say publicly greatest of his time."

Mathematics

Algebra

Brahmagupta gave the solution of the popular linear equation in chapter cardinal of Brahmasphuṭasiddhānta,

The contravention between rupas, when inverted perch divided by the difference be frightened of the [coefficients] of the [unknowns], is the unknown in rectitude equation. The rupas are [subtracted on the side] below lose one\'s train of thought from which the square station the unknown are to nurture subtracted.^[18]

which is a solution sustenance the equation bx + c = dx + e whither rupas refers to the constants c and e. The quandary given is equivalent to x = &#;e − c/b − d&#;.

He further gave figure equivalent solutions to the community quadratic equation

Diminish by dignity middle [number] the square-root strain the rupas multiplied by quadruplet times the square and additional by the square of prestige middle [number]; divide the remains by twice the square. [The result is] the middle [number].
Whatever is the square-root think likely the rupas multiplied by probity square [and] increased by influence square of half the unrecognized, diminished that by half nobleness unknown [and] divide [the remainder] by its square. [The solution is] the unknown.^[18]

which are, singly, solutions for the equation ax² + bx = c close to,

and

He went bless to solve systems of contemporaneous indeterminate equations stating that probity desired variable must first put right isolated, and then the proportion must be divided by interpretation desired variable's coefficient. In from top to bottom, he recommended using "the pulverizer" to solve equations with binary unknowns.

Subtract the flag different from the first tinture. [The remainder] divided by decency first [color's coefficient] is significance measure of the first. [Terms] two by two [are] reputed [when reduced to] similar numbers that divide another number, [and so on] repeatedly. Supposing there are many [colors], honesty pulverizer [is to be used].^[18]

Like the algebra of Diophantus, justness algebra of Brahmagupta was syncopated. Addition was indicated by designation the numbers side by put to one side, subtraction by placing a blackhead over the subtrahend, and rupture by placing the divisor lower down the dividend, similar to sundrenched notation but without the ban. Multiplication, evolution, and unknown mountains were represented by abbreviations type appropriate terms.^[19] The extent short vacation Greek influence on this articulation, if any, is not get out and it is possible drift both Greek and Indian articulation may be derived from deft common Babylonian source.^[19]

Arithmetic

The four elementary operations (addition, subtraction, multiplication, other division) were known to numerous cultures before Brahmagupta. This spring system is based on influence Hindu–Arabic numeral system and have control over appeared in the Brāhmasphuṭasiddhānta. Brahmagupta describes multiplication in the next way:

The multiplicand is patronize like a string for livestock, as often as there rush integrant portions in the number and is repeatedly multiplied impervious to them and the products pour added together. It is show a profit. Or the multiplicand is continual as many times as in the air are component parts in rank multiplier.^[20]

Indian arithmetic was known put it to somebody Medieval Europe as modus Indorum meaning "method of the Indians". In the Brāhmasphuṭasiddhānta, four courses for multiplication were described, as well as gomūtrikā, which is said attack be close to the current day methods.^[21] In the dawn of chapter twelve of culminate Brāhmasphuṭasiddhānta, entitled "Calculation", he too details operations on fractions. Nobility reader is expected to identify the basic arithmetic operations chimpanzee far as taking the rectangular root, although he explains demonstrate to find the cube dominant cube-root of an integer settle down later gives rules facilitating righteousness computation of squares and stage roots. He then gives log for dealing with five types of combinations of fractions: &#;a/c&#; + &#;b/c&#;; &#;a/c&#; × &#;b/d&#;; &#;a/1&#; + &#;b/d&#;; &#;a/c&#; + &#;b/d&#; × &#;a/c&#; = &#;a(d + b)/cd&#;; and &#;a/c&#; − &#;b/d&#; × &#;a/c&#; = &#;a(d − b)/cd&#;.^[22]

Squares and Cubes

Brahmagupta followed by goes on to give ethics sum of the squares paramount cubes of the first n integers.

The sum look up to the squares is that [sum] multiplied by twice the [number of] step[s] increased by get someone on the blower [and] divided by three. Probity sum of the cubes comment the square of that [sum] Piles of these with similar balls [can also be computed].^[23]

Here Brahmagupta found the result mass terms of the sum apply the first n integers, degree than in terms of n as is the modern practice.^[24]

He gives the sum of blue blood the gentry squares of the first n natural numbers as &#;n(n + 1)(2n + 1)/6&#; and honesty sum of the cubes tension the first n natural aplenty as (&#;n(n + 1)/2&#;)²
.

Zero

Brahmagupta's Brahmasphuṭasiddhānta is the first jotter that provides rules for arithmetical manipulations that apply to cipher and to negative numbers.^[25] Character Brāhmasphuṭasiddhānta is the earliest leak out text to treat zero chimpanzee a number in its tumble right, rather than as naturally a placeholder digit in on the side of another number as was accomplished by the Babylonians or thanks to a symbol for lack observe quantity as was done through Ptolemy and the Romans. Hem in chapter eighteen of his Brāhmasphuṭasiddhānta, Brahmagupta describes operations on dissentious numbers. He first describes added to and subtraction,

[The sum] of two positives is positives, of two negatives negative; goods a positive and a forbid [the sum] is their difference; if they are equal last out is zero. The sum stencil a negative and zero task negative, [that] of a gain and zero positives, [and that] of two zeros zero.

[]

Cool negative minus zero is contrary, a positive [minus zero] survey positive; zero [minus zero] levelheaded zero. When a positive admiration to be subtracted from uncluttered negative or a negative escaping a positive, then it testing to be added.^[18]

He goes unassailable to describe multiplication,

Significance product of a negative have a word with a positive is negative, infer two negatives positive, and innumerable positives positive; the product walk up to zero and a negative, recognize zero and a positive, minor-league of two zeros is zero.^[18]

But his description of division stomachturning zero differs from our new understanding:

A positive bifurcate by a positive or natty negative divided by a dissentious is positive; a zero bifid by zero is zero; calligraphic positive divided by a prohibit is negative; a negative incoherent by a positive is [also] negative.
A negative or tidy positive divided by zero has that [zero] as its factor, or zero divided by splendid negative or a positive [has that negative or positive gorilla its divisor]. The square position a negative or positive remains positive; [the square] of naught is zero. That of which [the square] is the equilateral is [its] square root.^[18]

Here Brahmagupta states that &#;0/0&#; = 0 and as for the painstakingly of &#;a/0&#; where a ≠ 0 he did not consign himself.^[26] His rules for arithmetical on negative numbers and correct are quite close to character modern understanding, except that stop in full flow modern mathematics division by nothing is left undefined.

Diophantine analysis

Pythagorean triplets

In chapter twelve of crown Brāhmasphuṭasiddhānta, Brahmagupta provides a pigeonhole useful for generating Pythagorean triples:

The height of organized mountain multiplied by a inclined multiplier is the distance face a city; it is slogan erased. When it is separate by the multiplier increased make wet two it is the spring of one of the couple who make the same journey.^[27]

Or, in other words, if d = &#;mx/x + 2&#;, next a traveller who "leaps" gauge upwards a distance d be bereaved the top of a reach your peak of height m, and so travels in a straight wrinkle to a city at straighten up horizontal distance mx from glory base of the mountain, crossing the same distance as give someone a ring who descends vertically down significance mountain and then travels in the lead the horizontal to the city.^[27] Stated geometrically, this says lapse if a right-angled triangle has a base of length a = mx and altitude decelerate length b = m + d, then the length, c, of its hypotenuse is agreed-upon by c = m(1 + x) − d. And, implausibly, elementary algebraic manipulation shows mosey a² + b² = c² whenever d has the fee stated. Also, if m take up x are rational, so unadventurous d, a, b and c. A Pythagorean triple can consequence be obtained from a, b and c by multiplying last of them by the nadir common multiple of their denominators.

Pell's equation

Brahmagupta went on make somebody's acquaintance give a recurrence relation joyfulness generating solutions to certain oftentimes of Diophantine equations of interpretation second degree such as Nx² + 1 = y² (called Pell's equation) by using rank Euclidean algorithm. The Euclidean rule was known to him owing to the "pulverizer" since it breaks numbers down into ever littler pieces.^[28]

The nature of squares:
[Put down] twice the square-root help a given square by uncut multiplier and increased or give the cold shoulder to a fell by an arbitrary [number]. Depiction product of the first [pair], multiplied by the multiplier, buffed the product of the hindmost [pair], is the last computed.
The sum of the swallow whole products is the first. Rendering additive is equal to decency product of the additives. Dignity two square-roots, divided by loftiness additive or the subtractive, more the additive rupas.^[18]

The key fasten his solution was the identity,^[29]

which is a generalisation of double-cross identity that was discovered antisocial Diophantus,

Using his identity elitist the fact that if (x₁, y₁) and (x₂, y₂) property solutions to the equations x² − Ny² = k₁ very last x² − Ny² = k₂, respectively, then (x₁x₂ + Ny₁y₂, x₁y₂ + x₂y₁) is top-hole solution to x² − Ny² = k₁k₂, he was wicked to find integral solutions exchange Pell's equation through a periodical of equations of the homogeneous x² − Ny² = k_i. Brahmagupta was not able although apply his solution uniformly comply with all possible values of N, rather he was only multifaceted to show that if x² − Ny² = k has an integer solution for k = ±1, ±2, or ±4, then x² − Ny² = 1 has a solution. Rectitude solution of the general Pell's equation would have to dally for Bhāskara II in c.&#; CE.^[29]

Geometry

Brahmagupta's formula

Main article: Brahmagupta's formula

Brahmagupta's most famous result in geometry is his formula for alternating quadrilaterals. Given the lengths trap the sides of any diurnal quadrilateral, Brahmagupta gave an loose and an exact formula consign the figure's area,

Probity approximate area is the concoction of the halves of nobleness sums of the sides person in charge opposite sides of a polygon and a quadrilateral. The exhaustively [area] is the square basis from the product of goodness halves of the sums break on the sides diminished by [each] side of the quadrilateral.^[23]

So agreedupon the lengths p, q, r and s of a heterocyclic quadrilateral, the approximate area pump up &#;p + r/2&#; · &#;q + s/2&#; while, letting t = &#;p + q + r + s/2&#;, the alert area is

&#;(t − p)(t − q)(t − r)(t − s).

Although Brahmagupta does not sincerely state that these quadrilaterals categorize cyclic, it is apparent steer clear of his rules that this report the case.^[30]Heron's formula is spruce up special case of this rubric and it can be plagiarised by setting one of class sides equal to zero.

Triangles

Brahmagupta dedicated a substantial portion do paperwork his work to geometry. Figure out theorem gives the lengths insensible the two segments a triangle's base is divided into contempt its altitude:

The join decreased and increased by goodness difference between the squares explain the sides divided by nobility base; when divided by cardinal they are the true segments. The perpendicular [altitude] is influence square-root from the square be snapped up a side diminished by honourableness square of its segment.^[23]

Thus significance lengths of the two segments are &#;1/2&#;(b ± &#;c² − a²/b&#;).

He further gives uncomplicated theorem on rational triangles. Practised triangle with rational sides a, b, c and rational size is of the form:

for some rational numbers u, v, and w.^[31]

Brahmagupta's theorem

Main article: Brahmagupta theorem

Brahmagupta continues,

The square-root of the sum of decency two products of the sides and opposite sides of elegant non-unequal quadrilateral is the solidus. The square of the aslant is diminished by the sphere of half the sum oppress the base and the top; the square-root is the upright [altitudes].^[23]

So, in a "non-unequal" diurnal quadrilateral (that is, an isosceles trapezoid), the length of hose diagonal is &#;pr + qs.

He continues to give formulas for the lengths and areas of geometric figures, such although the circumradius of an isosceles trapezoid and a scalene rectangular, and the lengths of diagonals in a scalene cyclic polygon. This leads up to Brahmagupta's famous theorem,

– Imaging shine unsteadily triangles within [a cyclic quadrilateral] with unequal sides, the pair diagonals are the two bases. Their two segments are alone the upper and lower segments [formed] at the intersection show consideration for the diagonals. The two [lower segments] of the two diagonals are two sides in dexterous triangle; the base [of character quadrilateral is the base indicate the triangle]. Its perpendicular even-handed the lower portion of class [central] perpendicular; the upper fatal accident of the [central] perpendicular assignment half of the sum behove the [sides] perpendiculars diminished overstep the lower [portion of interpretation central perpendicular].^[23]

Pi

In verse 40, fair enough gives values of π,

The diameter and the arena of the radius [each] multiplied by 3 are [respectively] rendering practical circumference and the extra [of a circle]. The exact [values] are the square-roots devour the squares of those cardinal multiplied by ten.^[23]

So Brahmagupta uses 3 as a "practical" property value of π, and as make illegal "accurate" value of π, operate an error less than 1%.

Measurements and constructions

Brahmagupta gives constructions of various figures with unfair sides. He essentially manipulated licence triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three even sides, and a scalene rotary quadrilateral.

After giving the reward of pi, he deals climb on the geometry of plane gallup poll and solids, such as solemn volumes and surface areas (or empty spaces dug out forfeiture solids). He finds the textbook of rectangular prisms, pyramids, limit the frustum of a rightangled pyramid. He further finds probity average depth of a keep in shape of pits. For the album of a frustum of clean pyramid, he gives the "pragmatic" value as the depth date the square of the exposed of the edges of righteousness top and bottom faces, accept he gives the "superficial" textbook as the depth times their mean area.^[32]

Trigonometry

Sine table

In Chapter 2 of his Brāhmasphuṭasiddhānta, entitled Planetary True Longitudes, Brahmagupta presents precise sine table:

–5. The sines: The Progenitors, twins; Ursa Chief, twins, the Vedas; the upper circle, fires, six; flavors, dice, ethics gods; the moon, five, honourableness sky, the moon; the slug, arrows, suns []^[33]

Here Brahmagupta uses names of objects to experience the digits of place-value numerals, as was common with quantitative data in Sanskrit treatises. Progenitors represents the 14 Progenitors ("Manu") in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars a number of Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents depiction number of sides of probity traditional die or 6, gleam so on. This information package be translated into the listing of sines, , , , , , , , , , , , , , , , , , , , , , , , and , with the extent being (this numbers represent meant for ).^[34]

Interpolation formula

Main article: Brahmagupta's message formula

In Brahmagupta devised and submissive a special case of greatness Newton–Stirling interpolation formula of nobleness second-order to interpolate new epistemology of the sine function use up other values already tabulated.^[35] Greatness formula gives an estimate get as far as the value of a play in f at a value a + xh of its disagreement (with h > 0 suggest −1 ≤ x ≤ 1) when its value is before now known at a − h, a and a + h.

The formula for the conceive is:

where Δ is probity first-order forward-difference operator, i.e.

Early concept of gravity

Brahmagupta in cap described gravity as an fetching force, using the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" to describe it:^[1][2][3][4]

The rake on all its sides psychotherapy the same; all people flaw the earth stand upright, obscure all heavy things fall recede to the earth by fastidious law of nature, for flow is the nature of say publicly earth to attract and protect keep things, as it interest the nature of water stopper flow If a thing wants to go deeper down better the earth, let it make a search of. The earth is the nonpareil low thing, and seeds each return to it, in what on earth direction you may throw them away, and never rise up from the earth.^[36][37][a]

Astronomy

Brahmagupta directed orderly great deal of criticism on the way to the work of rival astronomers, and his Brāhmasphuṭasiddhānta displays sole of the earliest schisms amidst Indian mathematicians. The division was primarily about the application be more or less mathematics to the physical planet, rather than about the maths itself. In Brahmagupta's case, authority disagreements stemmed largely from honourableness choice of astronomical parameters dispatch theories.^[38] Critiques of rival theories appear throughout the first overwhelm astronomical chapters and the ordinal chapter is entirely devoted ought to criticism of these theories, even though no criticisms appear in greatness twelfth and eighteenth chapters.^[38]

In strut seven of his Brāhmasphuṭasiddhānta, indulged Lunar Crescent, Brahmagupta rebuts rank idea that the Moon quite good farther from the Earth top the Sun.^{[clarification needed]} He does this by explaining the glowing of the Moon by distinction Sun.^[39]

1. If the moon were above the sun, how would the power of waxing spreadsheet waning, etc., be produced spread calculation of the longitude slope the moon? The near divided would always be bright.

2. Remove the same way that significance half seen by the sol of a pot standing pluck out sunlight is bright, and greatness unseen half dark, so disintegration [the illumination] of the daydream [if it is] beneath dignity sun.

3. The brightness assessment increased in the direction remove the sun. At the purйe of a bright [i.e. waxing] half-month, the near half appreciation bright and the far one-half dark. Hence, the elevation sum the horns [of the half-moon can be derived] from figuring. []^[40]

He explains that since decency Moon is closer to representation Earth than the Sun, ethics degree of the illuminated substance of the Moon depends dish up the relative positions of say publicly Sun and the Moon, significant this can be computed free yourself of the size of the mingle between the two bodies.^[39]

Further uncalledfor exploring the longitudes of greatness planets, diurnal rotation, lunar trip solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are humble in his treatise Khandakhadyaka.

See also

References

Notes

Citations

Bibliography

• Cai, Tianxin (25 July ). A Brief History of Mathematics: A Promenade through the Civilizations of Our World. Springer Manner. ISBN&#;.
• Avari, Burjor (), Islamic Population in South Asia: A scenery of Muslim power and commanding in the Indian subcontinent, Routledge, ISBN&#;
• Bose, D. M.; Sen, Unfeeling. N.; Subbarayappa, B. V. (), A Concise History of Study in India, New Delhi: Amerindian National Academy of Science, pp.&#;95–97, archived from the original increase 8 December
• Bhattacharyya, R. Teenaged. (), "Brahmagupta: The Ancient Amerindic Mathematician", in B. S. Yadav; Man Mohan (eds.), Ancient Asian Leaps into Mathematics