Brahmagupta

Brahmagupta (listen (help·info)) (born c. 598 CE, died c. 668 CE) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.

Brahmagupta
Born	c. 598 CE
Died	c. 668 CE
Residence	Bhillamāla Ujjain
Known for	Zero Modern number system Brahmagupta's theorem Brahmagupta's identity Brahmagupta's problem Brahmagupta-Fibonacci identity Brahmagupta's interpolation formula Brahmagupta's formula
Scientific career
Fields	Mathematics, astronomy

Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in elliptic verse in Sanskrit, as was common practice in Indian mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived.

Life and careerEdit

Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala (modern Bhinmal) during the reign of the Chapa dynasty ruler, Vyagrahamukha. He was the son of Jishnugupta and was a Shaivite by religion. Even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Multan or Abu region.

Bhillamala, called pi-lo-mo-lo by Xuanzang, was the apparent capital of the Gurjaradesa, the second largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional siddhanthas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.

In the year 628, at an age of 30, he composed the Brāhmasphuṭasiddhānta (the improved treatise of Brahma) which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he incorporated a great deal of originality to his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.

Later, Brahmagupta moved to Ujjain, which was also a major centre for astronomy. At the age of 67, he composed his next well known work Khanda-khādyaka, a practical manual of Indian astronomy in the karana category meant to be used by students.

Brahmagupta lived beyond 665 CE. He is believed to have died in Ujjain.

MathematicsEdit

AlgebraEdit

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,

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

which is a solution for the equation bx + c = dx + e equivalent to x = e − c/b − d, where rupas refers to the constants c and e. He further gave two equivalent solutions to the general quadratic equation

18.44. Diminish by the middle [number] the square-root of the rupas multiplied 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 the rupas multiplied 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.

which are, respectively, solutions for the equation ax² + bx = c equivalent to,

${\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}}$

and

${\displaystyle x={\frac {{\sqrt {ac+{\tfrac {b^{2}}{4}}}}-{\tfrac {b}{2}}}{a}}.}$

He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable's coefficient. 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].

Like the algebra of Diophantus, 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.The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source.

ArithmeticEdit

The 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 his Brahmasphutasiddhanta, entitled Calculation, 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: 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.

GeometryEdit

Brahmagupta's formulaEdit

Diagram for reference

Main article: Brahmagupta's formula

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. 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.

So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area is p + r/2 · q + s/2 while, letting t = p + q + r + s/2, the exact area is

√(t − p)(t − q)(t − r)(t − s).

Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero.

TrianglesEdit

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.

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

He further gives a theorem on rational triangles. A triangle with rational sides a, b, cand rational area is of the form:

${\displaystyle a={\frac {1}{2}}\left({\frac {u^{2}}{v}}+v\right),\ \ b={\frac {1}{2}}\left({\frac {u^{2}}{w}}+w\right),\ \ c={\frac {1}{2}}\left({\frac {u^{2}}{v}}-v+{\frac {u^{2}}{w}}-w\right)}$

for some rational numbers u, v, and w.

Brahmagupta's theoremEdit

Main article: Brahmagupta theorem

Brahmagupta's theorem states that AF = 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].

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is √pr + qs.

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 Brahmagupta's famous theorem,

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].

PiEdit

In verse 40, he gives values of π,

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.

So Brahmagupta uses 3 as a "practical" value of π, and ${\displaystyle {\sqrt {10}}\approx 3.1622\ldots }$ as an "accurate" value of π. The error in this "accurate" value is less than 1%.

TrigonometryEdit

Sine tableEdit

In Chapter 2 of his Brahmasphutasiddhanta, entitled Planetary 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 [...]

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.

Interpolation formulaEdit

Main article: Brahmagupta's interpolation formula

In 665 Brahmagupta devised and used a special case of the Newton–Stirling interpolation formula of the second-order to interpolate new values of the sine function from other values already tabulated.The formula gives an estimate for the value of a function f at a value a + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when its value is already known at a − h, a and a + h.

The formula for the estimate is:

${\displaystyle f(a+xh)\approx f(a)+x\left({\frac {\Delta f(a)+\Delta f(a-h)}{2}}\right)+{\frac {x^{2}\Delta ^{2}f(a-h)}{2!}}.}$

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

${\displaystyle \Delta f(a)\ {\stackrel {\mathrm {def} }{=}}\ f(a+h)-f(a).}$