Ancient Indian Mathematics. Part Three.

Numerals and the decimal number system[edit]

It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear.

The earliest extant script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system.

The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.

There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."

A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. According to Plofker 2009, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left. The earliest reference employing object numbers is a ca. 269 CE Sanskrit text, Yavanajātaka (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.[53] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.

It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. According to Plofker 2009,

These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."

Bakhshali Manuscript

The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit" in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously dated—as early as the "early centuries of the Christian era" and as late as between the 9th and 12th century CE.The 7th century CE is now considered a plausible date, albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work."

The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi) and algebra (simultaneous linear equations and quadratic equations), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."[55] Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following:

One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant.

The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.

Classical Period (400–1600)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā). This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika[61] (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 CE)—of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.

Fifth and sixth centuries

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry.[citation needed] Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.

This ancient text uses the following as trigonometric functions for the first time: Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). It also contains the early uses of tangent,.secant.

.

Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.

Chhedi calendar

This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).

Aryabhata I

Aryabhata (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:

Quadratic equations

Trigonometry

The value of π, correct to 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:

Trigonometry: (See also : Aryabhata's sine table)

Introduced the trigonometric functions.

Defined the sine (jya) as the modern relationship between half an angle and half a chord.

Defined the cosine (kojya).

Defined the versine (utkrama-jya).

Defined the inverse sine (otkram jya).

Gave methods of calculating their approximate numerical values.

Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.

Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.

Spherical trigonometry.

Arithmetic: Continued fractions. Algebra:

Solutions of simultaneous quadratic equations.

Whole number solutions of linear equations by a method equivalent to the modern method.

General solution of the indeterminate linear equation .

Mathematical astronomy:

Accurate calculations for astronomical constants, such as the:

Solar eclipse.

Lunar eclipse.

The formula for the sum of the cubes, which was an important step in the development of integral calculus.

Varahamihira

Varahamihira (505–587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:

\sin^2(x) + \cos^2(x) = 1

\sin(x)=\cos\left(\frac{\pi}{2}-x\right)

\frac{1-\cos(2x)}{2}=\sin^2(x)

Seventh and eighth centuries[edit]

Brahmagupta's theorem states that AF = FD.

In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:

Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).

Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by

A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \,

where s, the semiperimeter, given by s=\frac{a+b+c+d}{2}.

Brahmagupta's Theorem on rational triangles: A triangle with rational sides a, b, c and rational area is of the form:

a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w)

for some rational numbers u, v, and w .

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers and is considered the first systematic treatment of the subject. The rules (which included a + 0 = \ a and a \times 0 = 0 ) were all correct, with one exception: \frac{0}{0} = 0 . Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:

\ ax^2+bx=c

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.

This is equivalent to: x = \frac{\sqrt{4ac+b^2}-b}{2a}

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,

\ x^2-Ny^2=1,

where N is a nonsquare integer. He did this by discovering the following identity:

Brahmagupta's Identity: \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 which was a generalisation of an earlier identity of Diophantus: Brahmagupta used his identity to prove the following lemma:

Lemma (Brahmagupta): If x=x_1,\ \ y=y_1 \ \ is a solution of \ \ x^2 - Ny^2 = k_1, and, x=x_2, \ \ y=y_2 \ \ is a solution of \ \ x^2 - Ny^2 = k_2, , then:

x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \ is a solution of \ x^2-Ny^2=k_1k_2

He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem:

Theorem (Brahmagupta): If the equation \ x^2 - Ny^2 =k has an integer solution for any one of \ k=\pm 4, \pm 2, -1 then Pell's equation:

\ x^2 -Ny^2 = 1

also has an integer solution. Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: Example (Brahmagupta): Find integers \ x,\ y\ such that: \ x^2 - 92y^2=1

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was:

\ x=1151, \ y=120

Bhaskara I

Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced:

Solutions of indeterminate equations.

A rational approximation of the sine function.

A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.

Ninth to twelfth centuries.

Virasena

Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which:

Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order.

The same concept for base 3 (trakacheda) and base 4 (caturthacheda).

Virasena also gave:

The derivation of the volume of a frustum by a sort of infinite procedure.

It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.

Mahavira

Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:

Zero

Squares

Cubes

square roots, cube roots, and the series extending beyond these

Plane geometry

Solid geometry

Problems relating to the casting of shadows

Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle.

Mahavira also:

Asserted that the square root of a negative number did not exist

Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.

Solved cubic equations.

Solved quartic equations.

Solved some quintic equations and higher-order polynomials.

Gave the general solutions of the higher order polynomial equations:

\ ax^n = q

a \frac{x^n - 1}{x - 1} = p

Solved indeterminate quadratic equations.

Solved indeterminate cubic equations.

Solved indeterminate higher order equations

Numerals and the decimal number system[edit]

It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe. The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers. However, how, when, and where the first decimal place value system was invented is not so clear.

The earliest extant script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially not based on a place-value system.

The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE. A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate. Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.

There are older textual sources, although the extant manuscript copies of these texts are from much later dates. Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE. Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred." Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."

A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books. Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier. According to Plofker 2009, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon). So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left. The earliest reference employing object numbers is a ca. 269 CE Sanskrit text, Yavanajātaka (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.[53] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.

It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE. According to Plofker 2009,

These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."

Bakhshali Manuscript

The oldest extant mathematical manuscript in South Asia is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit" in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE. The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously dated—as early as the "early centuries of the Christian era" and as late as between the 9th and 12th century CE.The 7th century CE is now considered a plausible date, albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work."

The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi) and algebra (simultaneous linear equations and quadratic equations), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."[55] Many of its problems are of a category known as 'equalisation problems' that lead to systems of linear equations. One example from Fragment III-5-3v is the following:

One merchant has seven asava horses, a second has nine haya horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant.

The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers.

Classical Period (400–1600)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā). This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika[61] (literally panca, "five," siddhānta, "conclusion of deliberation", dated 575 CE)—of five earlier works, Surya Siddhanta, Romaka Siddhanta, Paulisa Siddhanta, Vasishtha Siddhanta and Paitamaha Siddhanta, which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries.

Fifth and sixth centuries

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry.[citation needed] Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.

This ancient text uses the following as trigonometric functions for the first time: Sine (Jya). Cosine (Kojya). Inverse sine (Otkram jya). It also contains the early uses of tangent,.secant.

.

Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East.

Chhedi calendar

This Chhedi calendar (594) contains an early use of the modern place-value Hindu-Arabic numeral system now used universally (see also Hindu-Arabic numerals).

Aryabhata I

Aryabhata (476–550) wrote the Aryabhatiya. He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:

Quadratic equations

Trigonometry

The value of π, correct to 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:

Trigonometry: (See also : Aryabhata's sine table)

Introduced the trigonometric functions.

Defined the sine (jya) as the modern relationship between half an angle and half a chord.

Defined the cosine (kojya).

Defined the versine (utkrama-jya).

Defined the inverse sine (otkram jya).

Gave methods of calculating their approximate numerical values.

Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.

Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.

Spherical trigonometry.

Arithmetic: Continued fractions. Algebra:

Solutions of simultaneous quadratic equations.

Whole number solutions of linear equations by a method equivalent to the modern method.

General solution of the indeterminate linear equation .

Mathematical astronomy:

Accurate calculations for astronomical constants, such as the:

Solar eclipse.

Lunar eclipse.

The formula for the sum of the cubes, which was an important step in the development of integral calculus.

Varahamihira

Varahamihira (505–587) produced the Pancha Siddhanta (The Five Astronomical Canons). He made important contributions to trigonometry, including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating sine and cosine functions:

\sin^2(x) + \cos^2(x) = 1

\sin(x)=\cos\left(\frac{\pi}{2}-x\right)

\frac{1-\cos(2x)}{2}=\sin^2(x)

Seventh and eighth centuries[edit]

Brahmagupta's theorem states that AF = FD.

In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations). Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:

Brahmagupta's theorem: If a cyclic quadrilateral has diagonals that are perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalisation of Heron's formula), as well as a complete description of rational triangles (i.e. triangles with rational sides and rational areas).

Brahmagupta's formula: The area, A, of a cyclic quadrilateral with sides of lengths a, b, c, d, respectively, is given by

A = \sqrt{(s-a)(s-b)(s-c)(s-d)} \,

where s, the semiperimeter, given by s=\frac{a+b+c+d}{2}.

Brahmagupta's Theorem on rational triangles: A triangle with rational sides a, b, c and rational area is of the form:

a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w)

for some rational numbers u, v, and w .

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers and is considered the first systematic treatment of the subject. The rules (which included a + 0 = \ a and a \times 0 = 0 ) were all correct, with one exception: \frac{0}{0} = 0 . Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:

\ ax^2+bx=c

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.

This is equivalent to: x = \frac{\sqrt{4ac+b^2}-b}{2a}

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,

\ x^2-Ny^2=1,

where N is a nonsquare integer. He did this by discovering the following identity:

Brahmagupta's Identity: \ (x^2-Ny^2)(x'^2-Ny'^2) = (xx'+Nyy')^2 - N(xy'+x'y)^2 which was a generalisation of an earlier identity of Diophantus: Brahmagupta used his identity to prove the following lemma:

Lemma (Brahmagupta): If x=x_1,\ \ y=y_1 \ \ is a solution of \ \ x^2 - Ny^2 = k_1, and, x=x_2, \ \ y=y_2 \ \ is a solution of \ \ x^2 - Ny^2 = k_2, , then:

x=x_1x_2+Ny_1y_2,\ \ y=x_1y_2+x_2y_1 \ \ is a solution of \ x^2-Ny^2=k_1k_2

He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem:

Theorem (Brahmagupta): If the equation \ x^2 - Ny^2 =k has an integer solution for any one of \ k=\pm 4, \pm 2, -1 then Pell's equation:

\ x^2 -Ny^2 = 1

also has an integer solution. Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: Example (Brahmagupta): Find integers \ x,\ y\ such that: \ x^2 - 92y^2=1

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was:

\ x=1151, \ y=120

Bhaskara I

Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced:

Solutions of indeterminate equations.

A rational approximation of the sine function.

A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.

Ninth to twelfth centuries.

Virasena

Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which:

Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two, but differs on other numbers, more closely resembling the 2-adic order.

The same concept for base 3 (trakacheda) and base 4 (caturthacheda).

Virasena also gave:

The derivation of the volume of a frustum by a sort of infinite procedure.

It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.

Mahavira

Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain mathematicians, lived in the 9th century and was patronised by the Rashtrakuta king Amoghavarsha. He wrote a book titled Ganit Saar Sangraha on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:

Zero

Squares

Cubes

square roots, cube roots, and the series extending beyond these

Plane geometry

Solid geometry

Problems relating to the casting of shadows

Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle.

Mahavira also:

Asserted that the square root of a negative number did not exist

Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.

Solved cubic equations.

Solved quartic equations.

Solved some quintic equations and higher-order polynomials.

Gave the general solutions of the higher order polynomial equations:

\ ax^n = q

a \frac{x^n - 1}{x - 1} = p

Solved indeterminate quadratic equations.

Solved indeterminate cubic equations.

Solved indeterminate higher order equations

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