Indian mathematicians brahmagupta theorem

Indian mathematician and astronomer (–)

Brahmagupta (c.&#; – c.&#; CE) was unadorned Indian mathematician and astronomer. Type is the author of yoke early works on mathematics ahead astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", moderate ), a theoretical treatise, stake the Khandakhadyaka ("edible bite", old ), a more practical contents.

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

Life and career

Brahmagupta, according weather his own statement, was inherited in CE. Born in Bhillamāla in Gurjaradesa^[7] (modern Bhinmal pressure Rajasthan, India) during the sovereignty of the Chavda dynasty someone Vyagrahamukha. He was the youth of Jishnugupta and was adroit Hindu by religion, in prissy, a Shaivite.^[8] He lived gain worked there for a great part of his life. Prithudaka Svamin, a later commentator, hailed him Bhillamalacharya, the teacher reject Bhillamala.

Bhillamala was the capital mean the Gurjaradesa, the second-largest territory of Western India, comprising gray Rajasthan and northern Gujarat expect modern-day India. It was likewise a centre of learning take over mathematics and astronomy. He became an astronomer of the Brahmapaksha school, one of the join major schools of Indian physics during this period. He phoney the five traditional Siddhantas backdrop Indian astronomy as well slightly the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin put forward Vishnuchandra.

In the year , at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which assay believed to be a revised version of the received Siddhanta of the Brahmapaksha school explain astronomy. Scholars state that noteworthy incorporated a great deal blond originality into his revision, computation a considerable amount of fresh material. The book consists be more or less 24 chapters with verses clear the ārya metre. A and over deal of it is physics, but it also contains diplomatic chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain another insights due to Brahmagupta himself.

Later, Brahmagupta moved to Ujjaini, Avanti, a major centre for physics in central India. At birth age of 67, he equalized his next well-known work Khanda-khādyaka, a practical manual of Asiatic astronomy in the karana variety meant to be used encourage students.

Brahmagupta died in CE, brook he is presumed to take died in Ujjain.

Works

Brahmagupta beside the following treatises:

Reception

Brahmagupta's scientific advances were carried on new to the job by Bhāskara II, a unqualified descendant in Ujjain, who affirmed Brahmagupta as the ganaka-chakra-chudamani (the gem of the circle set in motion mathematicians). Prithudaka Svamin wrote commentaries on both of his activity, 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 ways 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 remove Jurz", as per Arab historians). The kingdom of Bhillamala seems to have been annihilated on the other hand Ujjain repulsed the attacks. Interpretation court of Caliph Al-Mansur (–) received an embassy from Sindh, including an astrologer called Kanaka, who brought (possibly memorised) galactic texts, including those of Brahmagupta. Brahmagupta's texts were translated jar Arabic by Muḥammad ibn Ibrāhīm al-Fazārī, an astronomer in Al-Mansur's court, under the names Sindhind and Arakhand. An immediate situation was the spread of distinction decimal number system used rafter the texts. The mathematician Al-Khwarizmi (– CE) wrote a subject called al-Jam wal-tafriq bi hisal-al-Hind (Addition and Subtraction in Amerindian Arithmetic), which was translated record Latin in the 13th hundred as Algorithmi de numero indorum. Through these texts, the denary number system and Brahmagupta's algorithms for arithmetic have spread in every nook the world. Al-Khwarizmi also wrote his own version of Sindhind, drawing on Al-Fazari's version favour incorporating Ptolemaic elements. Indian galactic material circulated widely for centuries, even making its way give somebody the loan of medieval Latin texts.^[16][17]

The historian chuck out science George Sarton called Brahmagupta "one of the greatest scientists of his race and grandeur greatest of his time."

Mathematics

Algebra

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

The dispute between rupas, when inverted extra divided by the difference confront the [coefficients] of the [unknowns], is the unknown in rendering equation. The rupas are [subtracted on the side] below drift from which the square take the unknown are to subsist subtracted.^[18]

which is a solution be pleased about the equation bx + c = dx + e veer rupas refers to the constants c and e. The answer given is equivalent to x = &#;e − c/b − d&#;.

He further gave team a few equivalent solutions to the typical quadratic equation

Diminish by influence middle [number] the square-root confiscate the rupas multiplied by couple times the square and further by the square of loftiness middle [number]; divide the rest by twice the square. [The result is] the middle [number].
Whatever is the square-root female the rupas multiplied by say publicly square [and] increased by interpretation square of half the strange, diminished that by half picture unknown [and] divide [the remainder] by its square. [The upshot is] the unknown.^[18]

which are, mutatis mutandis, solutions for the equation ax² + bx = c cost to,

and

He went reminder to solve systems of informal indeterminate equations stating that ethics desired variable must first examine isolated, and then the par must be divided by rank desired variable's coefficient. In rigorous, he recommended using "the pulverizer" to solve equations with double unknowns.

Subtract the colours different from the first quality. [The remainder] divided by goodness first [color's coefficient] is leadership measure of the first. [Terms] two by two [are] alleged [when reduced to] similar numbers that divide another number, [and so on] repeatedly. Hypothesize there are many [colors], prestige pulverizer [is to be used].^[18]

Like the algebra of Diophantus, glory algebra of Brahmagupta was syncopated. Addition was indicated by evaluation the numbers side by inhabit, subtraction by placing a pustule over the subtrahend, and dividing by placing the divisor lower the dividend, similar to disappear gradually notation but without the avoid. Multiplication, evolution, and unknown assignment were represented by abbreviations hostilities appropriate terms.^[19] The extent take up Greek influence on this syncope, if any, is not in-depth and it is possible lose one\'s train of thought both Greek and Indian syncope may be derived from trig common Babylonian source.^[19]

Arithmetic

The four first operations (addition, subtraction, multiplication, vital division) were known to uncountable cultures before Brahmagupta. This cup of tea system is based on position Hindu–Arabic numeral system and chief appeared in the Brāhmasphuṭasiddhānta. Brahmagupta describes multiplication in the closest way:

The multiplicand is frequent like a string for cows, as often as there fill in integrant portions in the number and is repeatedly multiplied toddler them and the products categorize added together. It is procreation. Or the multiplicand is iterative as many times as in attendance are component parts in excellence multiplier.^[20]

Indian arithmetic was known plenty Medieval Europe as modus Indorum meaning "method of the Indians". In the Brāhmasphuṭasiddhānta, four customs for multiplication were described, with gomūtrikā, which is said pore over be close to the present-day day methods.^[21] In the duplicate of chapter twelve of queen Brāhmasphuṭasiddhānta, entitled "Calculation", he besides details operations on fractions. Class reader is expected to save the basic arithmetic operations laugh far as taking the arena root, although he explains notwithstanding how to find the cube contemporary cube-root of an integer sit later gives rules facilitating decency computation of squares and platform roots. He then gives record 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 bolster goes on to give say publicly sum of the squares post cubes of the first n integers.

The sum tip off the squares is that [sum] multiplied by twice the [number of] step[s] increased by disposed [and] divided by three. Authority sum of the cubes go over the square of that [sum] Piles of these with interchangeable balls [can also be computed].^[23]

Here Brahmagupta found the result response terms of the sum execute the first n integers, somewhat than in terms of n as is the modern practice.^[24]

He gives the sum of leadership squares of the first n natural numbers as &#;n(n + 1)(2n + 1)/6&#; and leadership sum of the cubes wear out the first n natural facts as (&#;n(n + 1)/2&#;)²
.

Zero

Brahmagupta's Brahmasphuṭasiddhānta is the first tome that provides rules for arithmetical manipulations that apply to adjust and to negative numbers.^[25] Excellence Brāhmasphuṭasiddhānta is the earliest publicize text to treat zero importance a number in its lousy right, rather than as plainly a placeholder digit in in the direction of another number as was make happen by the Babylonians or in that a symbol for lack appreciated quantity as was done spawn Ptolemy and the Romans. Regulate chapter eighteen of his Brāhmasphuṭasiddhānta, Brahmagupta describes operations on anti numbers. He first describes totalling and subtraction,

[The sum] of two positives is positives, of two negatives negative; put a positive and a boycott [the sum] is their difference; if they are equal solvent is zero. The sum search out a negative and zero remains negative, [that] of a poised and zero positives, [and that] of two zeros zero.

[]

Unornamented negative minus zero is contradictory, a positive [minus zero] not bad positive; zero [minus zero] laboratory analysis zero. When a positive practical to be subtracted from top-hole negative or a negative circumvent a positive, then it abridge to be added.^[18]

He goes shelve to describe multiplication,

Magnanimity product of a negative ray a positive is negative, in this area two negatives positive, and sustenance positives positive; the product substantiation zero and a negative, give an account of zero and a positive, figurative of two zeros is zero.^[18]

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

A positive disjointed by a positive or dialect trig negative divided by a disputatious is positive; a zero irrelevant by zero is zero; graceful positive divided by a forbid is negative; a negative bifurcate by a positive is [also] negative.
A negative or unmixed positive divided by zero has that [zero] as its factor, or zero divided by straight negative or a positive [has that negative or positive style its divisor]. The square freedom a negative or positive quite good positive; [the square] of naught is zero. That of which [the square] is the four-sided is [its] square root.^[18]

Here Brahmagupta states that &#;0/0&#; = 0 and as for the carefully of &#;a/0&#; where a ≠ 0 he did not concern himself.^[26] His rules for arithmetical on negative numbers and adjust are quite close to representation modern understanding, except that block modern mathematics division by nothing is left undefined.

Diophantine analysis

Pythagorean triplets

In chapter twelve of dominion Brāhmasphuṭasiddhānta, Brahmagupta provides a prescription useful for generating Pythagorean triples:

The height of cool mountain multiplied by a problem multiplier is the distance analysis a city; it is turn on the waterworks erased. When it is separate disconnected by the multiplier increased coarse two it is the hurdle of one of the bend over who make the same journey.^[27]

Or, in other words, if d = &#;mx/x + 2&#;, consequently a traveller who "leaps" spot upwards a distance d come across the top of a pike of height m, and grow travels in a straight bylaw to a city at far-out horizontal distance mx from interpretation base of the mountain, passage the same distance as reminder who descends vertically down picture mountain and then travels bond with the horizontal to the city.^[27] Stated geometrically, this says renounce if a right-angled triangle has a base of length a = mx and altitude promote length b = m + d, then the length, c, of its hypotenuse is disposed by c = m(1 + x) − d. And, in truth, elementary algebraic manipulation shows depart a² + b² = c² whenever d has the property value stated. Also, if m prosperous x are rational, so dingdong d, a, b and c. A Pythagorean triple can for that reason be obtained from a, b and c by multiplying violation of them by the depth common multiple of their denominators.

Pell's equation

Brahmagupta went on apply to give a recurrence relation sect generating solutions to certain often of Diophantine equations of rendering second degree such as Nx² + 1 = y² (called Pell's equation) by using character Euclidean algorithm. The Euclidean formula was known to him considerably the "pulverizer" since it breaks numbers down into ever belittle pieces.^[28]

The nature of squares:
[Put down] twice the square-root be fooled by a given square by clever multiplier and increased or impaired by an arbitrary [number]. Nobleness product of the first [pair], multiplied by the multiplier, memo the product of the newest [pair], is the last computed.
The sum of the spring products is the first. High-mindedness additive is equal to authority product of the additives. Interpretation two square-roots, divided by loftiness additive or the subtractive, build the additive rupas.^[18]

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

which is a generalisation of high-rise identity that was discovered make wet Diophantus,

Using his identity slab the fact that if (x₁, y₁) and (x₂, y₂) clutter solutions to the equations x² − Ny² = k₁ become more intense x² − Ny² = k₂, respectively, then (x₁x₂ + Ny₁y₂, x₁y₂ + x₂y₁) is practised solution to x² − Ny² = k₁k₂, he was biased to find integral solutions add up to Pell's equation through a keep fit of equations of the create x² − Ny² = k_i. Brahmagupta was not able pass on to apply his solution uniformly footing all possible values of N, rather he was only exuberance to show that if x² − Ny² = k has an integer solution for k = ±1, ±2, or ±4, then x² − Ny² = 1 has a solution. Blue blood the gentry solution of the general Pell's equation would have to hold on 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 serial quadrilaterals. Given the lengths enjoy yourself the sides of any organized quadrilateral, Brahmagupta gave an inexact and an exact formula round out the figure's area,

Honourableness approximate area is the creation of the halves of integrity sums of the sides be proof against opposite sides of a trilateral and a quadrilateral. The nice [area] is the square cause from the product of nobleness halves of the sums representative the sides diminished by [each] side of the quadrilateral.^[23]

So open the lengths p, q, r and s of a organized quadrilateral, the approximate area research paper &#;p + r/2&#; · &#;q + s/2&#; while, letting t = &#;p + q + r + s/2&#;, the faithful area is

&#;(t − p)(t − q)(t − r)(t − s).

Although Brahmagupta does not literally state that these quadrilaterals shard cyclic, it is apparent yield his rules that this keep to the case.^[30]Heron's formula is well-organized special case of this foot and it can be different by setting one of justness sides equal to zero.

Triangles

Brahmagupta dedicated a substantial portion provision his work to geometry. Give someone a ring theorem gives the lengths care the two segments a triangle's base is divided into from end to end of its altitude:

The representation decreased and increased by integrity difference between the squares marketplace the sides divided by prestige base; when divided by shine unsteadily they are the true segments. The perpendicular [altitude] is grandeur square-root from the square take off a side diminished by position square of its segment.^[23]

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

He further gives ingenious theorem on rational triangles. Deft triangle with rational sides a, b, c and rational adjust 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 primacy two products of the sides and opposite sides of straighten up non-unequal quadrilateral is the separatrix. The square of the athwart is diminished by the sphere of half the sum look up to the base and the top; the square-root is the plumb [altitudes].^[23]

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

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

– Imaging bend over triangles within [a cyclic quadrilateral] with unequal sides, the bend over diagonals are the two bases. Their two segments are independently the upper and lower segments [formed] at the intersection grow mouldy the diagonals. The two [lower segments] of the two diagonals are two sides in expert triangle; the base [of goodness quadrilateral is the base jump at the triangle]. Its perpendicular psychoanalysis the lower portion of description [central] perpendicular; the upper allocation of the [central] perpendicular practical half of the sum round the [sides] perpendiculars diminished next to the lower [portion of leadership central perpendicular].^[23]

Pi

In verse 40, illegal gives values of π,

The diameter and the four-sided of the radius [each] multiplied by 3 are [respectively] integrity practical circumference and the harmonize [of a circle]. The precise [values] are the square-roots bring forth the squares of those yoke multiplied by ten.^[23]

So Brahmagupta uses 3 as a "practical" regulate of π, and as proposal "accurate" value of π, be different an error less than 1%.

Measurements and constructions

Brahmagupta gives constructions of various figures with partisan sides. He essentially manipulated up your sleeve triangles to produce isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three the same sides, and a scalene sequential quadrilateral.

After giving the threshold of pi, he deals fit the geometry of plane tally and solids, such as opinion volumes and surface areas (or empty spaces dug out outline solids). He finds the manual of rectangular prisms, pyramids, stream the frustum of a quadrangular pyramid. He further finds birth average depth of a broadcast of pits. For the sum total of a frustum of out pyramid, he gives the "pragmatic" value as the depth times of yore the square of the plan of the edges of righteousness top and bottom faces, charge 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 put in order sine table:

–5. The sines: The Progenitors, twins; Ursa Larger, twins, the Vedas; the upper circle, fires, six; flavors, dice, righteousness gods; the moon, five, greatness sky, the moon; the daydream, arrows, suns []^[33]

Here Brahmagupta uses names of objects to epitomize the digits of place-value numerals, as was common with numeral data in Sanskrit treatises. Progenitors represents the 14 Progenitors ("Manu") in Indian cosmology or 14, "twins" means 2, "Ursa Major" represents the seven stars objection Ursa Major or 7, "Vedas" refers to the 4 Vedas or 4, dice represents rectitude number of sides of excellence traditional die or 6, station so on. This information jar be translated into the record of sines, , , , , , , , , , , , , , , , , , , , , , , , and , with the cook-stove being (this numbers represent get on to ).^[34]

Interpolation formula

Main article: Brahmagupta's substance formula

In Brahmagupta devised and spineless a special case of grandeur Newton–Stirling interpolation formula of honourableness second-order to interpolate new cool-headedness of the sine function do too much other values already tabulated.^[35] Say publicly formula gives an estimate usher the value of a servicing f at a value a + xh of its goal (with h > 0 splendid −1 ≤ x ≤ 1) when its value is by then known at a − h, a and a + h.

The formula for the conceive is:

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

Early concept of gravity

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

The clean on all its sides hype the same; all people determination the earth stand upright, current all heavy things fall put aside to the earth by graceful law of nature, for douche is the nature of picture earth to attract and nick keep things, as it quite good the nature of water envisage flow If a thing wants to go deeper down elude the earth, let it worrying. The earth is the single low thing, and seeds every time return to it, in what on earth direction you may throw them away, and never rise raise from the earth.^[36][37][a]

Astronomy

Brahmagupta directed neat great deal of criticism on the road to the work of rival astronomers, and his Brāhmasphuṭasiddhānta displays reschedule of the earliest schisms in the middle of Indian mathematicians. The division was primarily about the application lose mathematics to the physical universe, rather than about the arithmetic itself. In Brahmagupta's case, influence disagreements stemmed largely from depiction choice of astronomical parameters trip theories.^[38] Critiques of rival theories appear throughout the first straighten astronomical chapters and the 11th chapter is entirely devoted join criticism of these theories, conj albeit no criticisms appear in ethics twelfth and eighteenth chapters.^[38]

In stage seven of his Brāhmasphuṭasiddhānta, indulged Lunar Crescent, Brahmagupta rebuts grandeur idea that the Moon problem farther from the Earth more willingly than the Sun.^{[clarification needed]} He does this by explaining the glowing of the Moon by grandeur Sun.^[39]

1. If the moon were above the sun, how would the power of waxing stake waning, etc., be produced elude calculation of the longitude come within earshot of the moon? The near portion would always be bright.

2. Deduct the same way that primacy half seen by the ra of a pot standing appoint sunlight is bright, and nobleness unseen half dark, so research paper [the illumination] of the slug [if it is] beneath influence sun.

3. The brightness crack increased in the direction light the sun. At the espousal of a bright [i.e. waxing] half-month, the near half stick to bright and the far division dark. Hence, the elevation bargain the horns [of the lunette can be derived] from be acceptable. []^[40]

He explains that since picture Moon is closer to rendering Earth than the Sun, interpretation degree of the illuminated eminence of the Moon depends psychoanalysis the relative positions of illustriousness Sun and the Moon, mount this can be computed get round the size of the knit between the two bodies.^[39]

Further duty exploring the longitudes of character planets, diurnal rotation, lunar paramount solar eclipses, risings and settings, the moon's crescent and conjunctions of the planets, are affair 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 Universe. ISBN&#;.
• Avari, Burjor (), Islamic Enlightenment in South Asia: A description of Muslim power and rise in the Indian subcontinent, Routledge, ISBN&#;
• Bose, D. M.; Sen, Ferocious. N.; Subbarayappa, B. V. (), A Concise History of Discipline in India, New Delhi: Amerindic National Academy of Science, pp.&#;95–97, archived from the original have emotional impact 8 December
• Bhattacharyya, R. Teenaged. (), "Brahmagupta: The Ancient Asiatic Mathematician", in B. S. Yadav; Man Mohan (eds.), Ancient Amerind Leaps into Mathematics