biography of brahmagupta in 300 words

Brahmagupta Biography

Born In: Bhinmal

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Died At Age: 72

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Died on: 670

You wanted to know

When did brahmagupta live.

Brahmagupta lived during the 7th century CE.

What is Brahmagupta known for?

Brahmagupta is known for his contributions to mathematics, particularly in the field of algebra and astronomy.

What is Brahmagupta's most famous work?

Brahmagupta's most famous work is the Brahmasphutasiddhanta, which is a mathematical treatise.

What are some of the key ideas introduced by Brahmagupta in mathematics?

Brahmagupta introduced the concept of zero, negative numbers, and methods for solving quadratic equations.

How did Brahmagupta's work influence the development of mathematics?

Brahmagupta's work laid the foundation for the development of algebra and trigonometry in India and had a significant impact on the field of mathematics globally.

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Brahmagupta was a pioneering mathematician and astronomer from ancient India, known for his significant contributions to the fields of mathematics and astronomy.

He is credited with introducing the concept of zero as a number and developing rules for arithmetic operations involving zero, laying the foundation for modern mathematical notation.

Brahmagupta's work also included advancements in algebra, geometry, and trigonometry, furthering the understanding of complex mathematical concepts during his time.

In addition to his mathematical achievements, Brahmagupta made important contributions to astronomy, particularly in the study of planetary motion and the calculation of eclipses.

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Brahmagupta.

When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero.

A debt minus zero is a debt. A fortune minus zero is a fortune. Zero minus zero is a zero. A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multipliedby zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt.

Positive or negative numbers when divided by zero is a fraction the zero as denominator. Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.

Brahmagupta perhaps used the method of continued fractions to find the integral solution of an indeterminate equation of the type a x + c = b y ax + c = by a x + c = b y .

Five hundred drammas were loaned at an unknown rate of interest, The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten mounths to 78 drammas. Give the rate of interest.

References ( show )

• D Pingree, Biography in Dictionary of Scientific Biography ( New York 1970 - 1990) . See THIS LINK .
• Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Brahmagupta
• H T Colebrooke, Algebra, with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhaskara (1817) .
• G Ifrah, A universal history of numbers : From prehistory to the invention of the computer ( London, 1998) .
• S S Prakash Sarasvati, A critical study of Brahmagupta and his works : The most distinguished Indian astronomer and mathematician of the sixth century A.D. ( Delhi, 1986) .
• S P Arya, On the Brahmagupta- Bhaskara equation, Math. Ed. 8 (1) (1991) , 23 - 27 .
• G S Bhalla, Brahmagupta's quadrilateral, Math. Comput. Ed. 20 (3) (1986) , 191 - 196 .
• B Chatterjee, Al-Biruni and Brahmagupta, Indian J. History Sci. 10 (2) (1975) , 161 - 165 .
• B Datta, Brahmagupta, Bull. Calcutta Math. Soc. 22 (1930) , 39 - 51 .
• K Elfering, Die negativen Zahlen und die Rechenregeln mit ihnen bei Brahmagupta, in Mathemata, Boethius Texte Abh. Gesch. Exakt. Wissensch. XII ( Wiesbaden, 1985 , 83 - 86 .
• R C Gupta, Brahmagupta's formulas for the area and diagonals of a cyclic quadrilateral, Math. Education 8 (1974) , B 33 -B 36 .
• R C Gupta, Brahmagupta's rule for the volume of frustum-like solids, Math. Education 6 (1972) , B 117 -B 120 .
• R C Gupta, Munisvara's modification of Brahmagupta's rule for second order interpolation, Indian J. Hist. Sci. 14 (1) (1979) , 66 - 72 .
• S Jha, A critical study on 'Brahmagupta and Mahaviracharya and their contributions in the field of mathematics', Math. Ed. ( Siwan ) 12 (4) (1978) , 66 - 69 .
• S C Kak, The Brahmagupta algorithm for square rooting, Ganita Bharati 11 (1 - 4) (1989) , 27 - 29 .
• T Kusuba, Brahmagupta's sutras on tri- and quadrilaterals, Historia Sci. 21 (1981) , 43 - 55 .
• P K Majumdar, A rationale of Brahmagupta's method of solving ax + c = by, Indian J. Hist. Sci. 16 (2) (1981) , 111 - 117 .
• J Pottage, The mensuration of quadrilaterals and the generation of Pythagorean triads : a mathematical, heuristical and historical study with special reference to Brahmagupta's rules, Arch. History Exact Sci. 12 (1974) , 299 - 354 .
• E R Suryanarayan, The Brahmagupta polynomials, Fibonacci Quart. 34 (1) (1996) , 30 - 39 .

Additional Resources ( show )

Other pages about Brahmagupta:

• See Brahmagupta on a timeline
• Astronomy: The Structure of the Solar System
• Heinz Klaus Strick biography

Other websites about Brahmagupta:

• Dictionary of Scientific Biography
• Encyclopaedia Britannica
• MathSciNet Author profile

Honours ( show )

Honours awarded to Brahmagupta

• Popular biographies list Number 4

Cross-references ( show )

• History Topics: A chronology of π
• History Topics: A history of Zero
• History Topics: An overview of Indian mathematics
• History Topics: Infinity
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• History Topics: Quadratic, cubic and quartic equations
• History Topics: The Arabic numeral system
• History Topics: The trigonometric functions
• Student Projects: Indian Mathematics - Redressing the balance: Chapter 10
• Student Projects: Indian Mathematics - Redressing the balance: Chapter 11
• Student Projects: Indian Mathematics - Redressing the balance: Chapter 12
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Brahmagupta

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Brahmagupta (born 598—died c. 665, possibly Bhillamala [modern Bhinmal], Rajasthan , India) was one of the most accomplished of the ancient Indian astronomers. He also had a profound and direct influence on Islamic and Byzantine astronomy .

Brahmagupta was an orthodox Hindu, and his religious views, particularly the Hindu yuga system of measuring the ages of mankind, influenced his work. He severely criticized Jain cosmological views and other heterodox ideas, such as the view of Aryabhata (born 476) that the Earth is a spinning sphere, a view that was widely disseminated by Brahmagupta’s contemporary and rival Bhaskara I .

Brahmagupta’s fame rests mostly on his Brahma-sphuta-siddhanta (628; “Correctly Established Doctrine of Brahma”), an astronomical work that he probably wrote while living in Bhillamala, then the capital of the Gurjara-Pratihara dynasty . It was translated into Arabic in Baghdad about 771 and had a major impact on Islamic mathematics and astronomy. Late in his life, Brahmagupta wrote Khandakhadyaka (665; “A Piece Eatable”), an astronomical handbook that employed Aryabhata’s system of starting each day at midnight.

In addition to expounding on traditional Indian astronomy in his books, Brahmagupta devoted several chapters of Brahma-sphuta-siddhanta to mathematics. In chapters 12 and 18 in particular, he laid the foundations of the two major fields of Indian mathematics , pati-ganita (“mathematics of procedures,” or algorithms ) and bija-ganita (“mathematics of seeds,” or equations), which roughly correspond to arithmetic (including mensuration) and algebra, respectively. Chapter 12 is simply named “Mathematics,” probably because the “basic operations,” such as arithmetic operations and proportions, and the “practical mathematics,” such as mixture and series, treated there occupied the major part of the mathematics of Brahmagupta’s milieu . He stressed the importance of these topics as a qualification for a mathematician, or calculator ( ganaka ). Chapter 18, “Pulverizer,” is named after the first topic of the chapter, probably because no particular name for this area (algebra) existed yet.

Among his major accomplishments, Brahmagupta defined zero as the result of subtracting a number from itself and gave rules for arithmetical operations among negative numbers (“debts”) and positive numbers (“property”), as well as surds. He also gave partial solutions to certain types of indeterminate equations of the second degree with two unknown variables. Perhaps his most famous result was a formula for the area of a cyclic quadrilateral (a four-sided polygon whose vertices all reside on some circle) and the length of its diagonals in terms of the length of its sides. He also gave a valuable interpolation formula for computing sines.

Brahmagupta Biography

Table of Contents

Brahmagupta (c. 598 – c. 668 CE) was an Indian mathematician and astronomer. He wrote two significant works: the Brāhmasphuṭasiddhānta (BSS) in 628 CE, which is a theoretical text, and the Khaṇḍakhādyaka in 665 CE, a more practical guide.

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In 628 CE, Brahmagupta first described gravity as an attractive force, using the Sanskrit term “ gurutvākarṣaṇam (गुरुत्वाकर्षणम्) ” to explain it. He is also credited with the first clear description of the quadratic formula in his main work, the Brāhmasphuṭasiddhānta.

Also Check: Aryabhatta Biography

Who is Brahmagupta?

Brahmagupta was an ancient Indian mathematician and astronomer who lived from 598 to 668 CE. He resided in Bhillamāla in Gurjaradesa, which is now Bhinmal in Rajasthan, India. Born to Jishnugupta and a follower of Hinduism, Brahmagupta spent most of his life in this region.

Brahmagupta is considered one of the most influential mathematicians of his era. His contributions span algebra, arithmetic, and geometry. He is best known for his works, “ Brahmasphutasiddhanta ” and “ Khandakhadyaka ,” comprehensive treatises on mathematics and astronomy.

Brahmagupta was the first mathematician to develop formulas for the area of a cyclic quadrilateral, now known as the Brahmagupta formula . He also provided guidelines for calculating with zero. His works, written in Sanskrit verse, have had a lasting impact on the field of mathematics.

Bhillamala was the capital of Gurjaradesa, a region in what is now southern Jaipur and north Gujarat. It was an important center for arithmetic and astronomical research. During this period, Brahmagupta became a prominent astronomer of the Brahmaraksha tradition, one of India’s four major astronomical schools. Brahmagupta mathematician , introduced a lot of new ideas and information into his work.

Brahmagupta books, divided into 24 sections and containing 1008 Arya poems, covers various mathematical topics such as arithmetic, trigonometry, geometry, and algorithms. Many of these concepts are credited to Brahmagupta himself. Brahmagupta studied the writings of notable scholars like Aryabhata I, Pradyumna, Latadeva, Varahamihira, Srisena, Simha, and Vijayanandan, along with Vishnuchandra and the five traditional Indian astrological Siddhantas. His work, including the famous Brahmagupta formula, has made significant contributions to mathematics.

Brahmagupta Biography: Early life

Brahmagupta was born in 598 CE. He lived in Bhillamala, now Bhinmal, in Rajasthan, during the reign of the Chavda dynasty ruler, Vyagrahamukha. Brahmagupta, known as a Bhillamalacharya or the teacher from Bhillamala, was dedicated to discovering new concepts. Bhillamala was the capital of Gurjaradesa, a significant region in West India, which included parts of modern southern Rajasthan and northern Gujarat. It was also a center for mathematics and astronomy studies.

Brahmagupta studied the five classic Siddhantas of Indian astronomy and the works of other astronomers like Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin, and Vishnuchandra. At the age of 30, Brahmagupta authored the Brahmasphutasiddhanta , a revised version of the Siddhanta of the Brahmapaksha school of astronomy.

Brahmagupta book contains significant teachings in mathematics, including algebra, geometry, trigonometry, and algorithms, featuring new concepts credited to Brahmagupta himself. At 67, he wrote Khandakhadyaka, a practical guide to Indian astronomy for students. This Brahmagupta information highlights his contributions as a renowned Brahmagupta mathematician . The Brahmagupta formula, which he developed, remains a significant part of his legacy.

Contribution of Brahmagupta in Mathematics

Brahmagupta, an influential Indian mathematician, established the properties of the number zero, which were crucial for the advancement of mathematics and science. Here are some key contributions by Brahmagupta:

• When we subtract a number from itself, we get zero.
• Dividing any number by zero results in zero.
• Zero divided by zero equals zero.
• Brahmagupta developed a formula for solving quadratic problems.
• He estimated the value of pi as 3.162, slightly higher than the true value of 3.14.
• He calculated that the Earth is closer to the moon than to the sun.
• He discovered a formula for finding the area of any four-sided shape with corners touching the interior of a circle, known as the Brahmagupta formula .
• Brahmagupta determined that a year is 365 days, 6 hours, 12 minutes, and 9 seconds long.
• He mentioned the concept of gravity, stating that bodies fall towards the Earth because it attracts them, similar to how water flows.
• Adding a negative number to another negative number gives a negative result.
• Subtracting a negative number from a positive number is like adding the two numbers.
• Multiplying two negative numbers results in a positive number.

Contributions Brahmagupta to Science and Astrology

Brahmagupta, a renowned Indian mathematician, made significant contributions to science and astrology. He argued that the Earth and the universe are spherical, not flat. He was the first to use mathematics to predict the positions of planets and the timings of lunar and solar eclipses. These findings were major scientific advancements at the time. Brahmagupta also calculated the length of the solar year to be 365 days, 5 minutes, and 19 seconds, very close to the current measurement of 365 days, 5 hours, and 19 seconds. He mentioned gravity, explaining, “Bodies fall toward the Earth because it is in the Earth’s nature to attract them, just as water attracts objects.”

Brahmasphutasiddhanta by Brahmagupta

At 30, Brahmagupta wrote his most famous work, the Brahmasphutasiddhanta, in 628 AD. Brahmagupta book , meaning “The Correctly Established Doctrine of Brahma,” is divided into twenty-five chapters and contains 1008 Sanskrit verses. It includes many of his original studies and calculations.

While much of the Brahmagupta books focuses on astronomy, it also covers a wide range of mathematical topics such as algorithms, trigonometry, geometry, and algebra. The book explains the importance of zero, rules for working with positive and negative numbers, and formulas for solving linear and quadratic equations. Brahmagupta also reinforced his belief that the Earth is spherical, countering the prevalent flat Earth theory of his time.

Contributions Brahmagupta Astronomy

Brahmagupta’s work in astronomy reached the Arabs through the Brahmasphutasiddhanta. In 770 AD, the Caliph of Baghdad, Al-Mansur, invited Kankah, a scholar from Ujjain, to explain Indian astronomical theories. Kankah used Brahmagupta’s book to teach the Hindu methods of mathematical astronomy. At the Caliph’s request, Muhammad al-Fazari translated Brahmagupta’s work into Arabic.

Brahmagupta made many contributions to astronomy, including methods for calculating the positions of celestial bodies, their rise and set times, and the prediction of lunar and solar eclipses. He explained that the moon’s illumination varies based on its position relative to the sun, which can be calculated using the angle between them. Brahmagupta also challenged the Puranic belief in a flat Earth, observing instead that both the Earth and the sky are round and that the Earth is in motion.

Achievements of Brahmagupta

Brahmagupta, an Indian mathematician and astronomer, made significant contributions to mathematics and astronomy. Here are some key achievements of Brahmagupta:

• Defining Zero : Brahmagupta defined the properties of zero, a critical development for mathematics and science. This had a profound impact on algebra and other mathematical fields.
• Quadratic Equations : He discovered a formula to solve quadratic equations, now known as the Brahmagupta formula .
• Trigonometry Formulas : Brahmagupta developed important trigonometric formulas, including those for sine and cosine.
• Value of Pi : He calculated the value of pi to be approximately 3.162, close to the actual value.
• Area of Quadrilaterals : He found a formula to calculate the area of any four-sided figure whose corners touch the inside of a circle, another important Brahmagupta formula.
• Length of a Year : Brahmagupta calculated the length of a year to be 365 days, 6 hours, 12 minutes, and 9 seconds.
• Earth’s Shape and Circumference : He proved that the Earth is a sphere and calculated its circumference to be around 36,000 km (22,500 miles).
• Astronomy : Brahmagupta made significant contributions to astronomy by developing methods for calculating the positions of planets and other celestial objects.

The achievements of Brahmagupta had a lasting influence on the study of mathematics and science in India and around the world. As a renowned Brahmagupta mathematician, his work continues to be celebrated for its impact on various scientific fields.

Why is Zero Important?

Brahmagupta, a pioneering Indian mathematician, introduced principles for mathematical operations involving zero and negative numbers in his book, Brahmasphutasiddhanta. This work was the first to define how zero and negative integers should be used in calculations. Zero is a crucial concept in mathematics and is fundamental to our number system.

Here are some reasons why zero is important:

• Arithmetic Operations: Zero plays a key role in arithmetic. Adding or subtracting zero from a number leaves the number unchanged. Multiplying any number by zero results in zero. Dividing by zero is undefined, emphasizing its unique role in mathematics.
• Placeholder: Zero acts as a placeholder in our number system, allowing us to represent numbers of different sizes. For example, without zero, writing the number 102 would be impossible.
• Calculus and Infinities: Zero is important in studying infinities. It serves as a starting point for the concept of “approaching zero,” which is essential in calculus for defining limits and derivatives.
• Symbolic Representation: Zero is also crucial in symbolic representation, where it is used as a placeholder, a coefficient, and a starting point for various mathematical functions.

Brahmagupta mathematician laid the foundation for these concepts with his Brahmagupta formula , highlighting the significance of zero in mathematics.

Brahmagupta Death

It is believed that Brahmagupta died between 660 and 670 CE, with many sources suggesting he lived until 668 CE. Brahmagupta is considered one of the greatest Indian mathematicians of all time. His contributions to mathematics and science have had a significant impact, establishing fundamental rules that help solve many mathematical problems today. The period around Brahmagupta death marks the end of a remarkable era of scientific advancement.

FAQs on Brahmagupta Biography

Where was brahmagupta born and when.

Brahmagupta was born in 597 AD in the town of Bhinmal, Rajasthan.

What is Brahmagupta's other name?

Brahmagupta was referred to by Bhaskara II, his successor at Ujjain, as the 'ganak-chakra-churamani,' meaning the gem of the circle of mathematicians.

What was Brahmagupta's occupation?

Brahmagupta was the superintendent of the observatory in Ujjain, a major center for ancient Indian mathematical astrology. Among Brahmagupta's books, the most well-known is the Brahmasphutasiddhanta, which covers both astronomy and mathematics.

What did Brahmagupta discover?

In chapter eighteen of his Brahmasphutasiddhanta, he provided a solution for the general linear equation. He also offered two equivalent solutions for the general quadratic equation. The Brahmasphutasiddhanta is the first book to outline rules for arithmetic operations involving zero and negative numbers. His most famous geometric result is the Brahmagupta formula for calculating the area of cyclic quadrilaterals.

When did Brahmagupta die?

Brahmagupta passed away in 668 AD.

What were Brahmagupta's contributions to mathematics?

He provided solutions for general linear and quadratic equations. His book, Brahmasphutasiddhanta, introduced rules for using zero and negative numbers in arithmetic. He is renowned for his geometric work, especially the Brahmagupta formula for cyclic quadrilaterals. Brahmagupta's work also touched on the concept of gravity, explaining that bodies fall towards the Earth due to its attraction, similar to how water flows.

Who Invented Zero Aryabhatta or Brahmagupta?

Aryabhata, a renowned Indian mathematician and astronomer from the 5th century AD, made important contributions to the development of mathematical ideas, including the concept of zero. While Brahmagupta later formalized the mathematical rules for zero, Aryabhata's work provided the foundation for these advancements.

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Brahmagupta

Brahmagupta is unique. He is the only scientist we have to thank for discovering the properties of precisely zero…

Brahmagupta was an Ancient Indian astronomer and mathematician who lived from 597 AD to 668 AD. He was born in the city of Bhinmal in Northwest India. His father, whose name was Jisnugupta, was an astrologer.

Although Brahmagupta thought of himself as an astronomer who did some mathematics, he is now mainly remembered for his contributions to mathematics.

Many of his important discoveries were written as poetry rather than as mathematical equations! Nevertheless, truth is truth, regardless of how it may be written.

Quick Guide to Brahmagupta

Brahmagupta:

• was the director of the astronomical observatory of Ujjain, the center of Ancient Indian mathematical astronomy.

• wrote four books about astronomy and mathematics, the most famous of which is Brahma-sphuta-siddhanta ( Brahma’s Correct System of Astronomy , or The Opening of the Universe .)

• said solving mathematical problems was something he did for pleasure.

• was the first person in history to define the properties of the number zero. Identifying zero as a number whose properties needed to be defined was vital for the future of mathematics and science.

• defined zero as the number you get when you subtract a number from itself.

• said that zero divided by any other number is zero.

• said dividing zero by zero produces zero. (Although, this seems reasonable, Brahmagupta actually got this one wrong. Mathematicians have now shown that zero divided by zero is undefined – it has no meaning. There really is no answer to zero divided by zero.)

• was the first person to discover the formula for solving quadratic equations.

• wrote that pi, the ratio of a circle’s circumference to its diameter, could usually be taken to be 3, but if accuracy were needed, then the square-root of 10 (this equals 3.162…) should be used. This is about 0.66 percent higher than the true value of pi.

• indicated that Earth is nearer the moon than the sun

• incorrectly said that Earth did not spin and that Earth does not orbit the sun. This, however, may have been for reasons of self-preservation. Opposing the Brahmins’ religious myths of the time would have been dangerous.

• produced a formula to find the area of any four-sided shape whose corners touch the inside of a circle. This actually simplifies to Heron’s formula for triangles.

• said the length of a year is 365 days 6 hours 12 minutes 9 seconds.

• calculated that Earth is a sphere of circumference around 36,000 km (22,500 miles).

Brahmagupta established rules for working with positive and negative numbers, such as:

• adding two negative numbers together always results in a negative number.

• subtracting a negative number from a positive number is the same as adding the two numbers.

• multiplying two negative numbers together is the same as multiplying two positive numbers.

• dividing a positive number by a negative, or a negative number by a positive results in a negative number.

Why is Zero Important?

Although it may seem obvious to us now that zero is a number, and obvious that we can produce it by subtracting a number from itself, and that dividing zero by a non-zero number gives an answer of zero, these results are not actually obvious.

The brilliant mathematicians of Ancient Greece, so far ahead of their time in many ways, had not been able to make this breakthrough. Neither had anyone else, until Brahmagupta came along!

It was a huge conceptual leap to see that zero is a number in its own right. Once this leap had been made, mathematics and science could make progress that would otherwise have been impossible.

Update September 14, 2017 Scientists at the University of Oxford have established that an Indian manuscript dated 200-400 AD is the first documented use of zero, as shown in the video below. Zero was invented before Brahmagupta’s era!

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Brahmagupta | Great Indian Mathematician

Table of contents.

22 September 2020                

Read time: 3 minutes

Introduction

In India, it is second nature to consult an astrologer who suggests an auspicious time of Muhurat for an important event. In other words, Astrology and Astronomy is a part of our life. Our calendar both the solar and the lunar calendar accurately calculate the festivals, moon phases, eclipses and many other happenings in not just our solar system, but also the cosmos or the Universe. What is impressive is that our ancient Astronomers and Astrologers used mathematics to calculate the auspicious timing for important events in our life.

One thing our ancient scientists were aware of was that there is an order/ logic in this huge expanse and vastness. It is this realization that led to the discoveries in Mathematics.

Who is Brahmagupta?

Brahmagupta ( 597- 668AD) was one such genius Astronomer - Mathematician. His father Jisnugupta was an Astrologer in the city of Bhinmal ( Rajasthan). Brahmagupta too considered himself an Astronomer however today he is remembered for his huge contributions to the field of Mathematics. By his admission, he did Mathematics or solved problems for pleasure!

Ujjain was the centre of Ancient Indian mathematical astronomy. Brahmagupta was the director of this centre. Brahmagupta wrote many textbooks for mathematics and astronomy while he was in Ujjain. These include ‘Durkeamynarda’ (672), ‘Khandakhadyaka’ (665), ‘Brahmasphutasiddhanta’ (628) and ‘Cadamakela’ (624). The ‘Brahmasphutasiddhanta’ meaning the ‘Corrected Treatise of Brahma’ is one of his well-known works.

Works of Brahmagupta

Brahmagupta, like all scholars in those times, wrote in elliptical verse.

Brahmasphutasiddhanta ((Brahma’s Correct System of Astronomy, or The Opening of the Universe.) written in 628 was his most famous work. This book has twenty-five chapters and a total of 1008 stanzas. Historians believe that the first ten were originally written by Brahmagupta because they are arranged like the typical mathematical astronomy texts in that period. It covers mean longitudes of the planets; true longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; the moon's shadow; conjunctions of the planets with each other; and conjunctions of the planets with the fixed stars.

The remaining fifteen chapters seem to form a second work which is a major addendum to the original treatise.

Brahmasphutasiddhanta is the earliest known text that treated zero as a number. The Greeks and Romans merely used symbols, and the Babylonians used a shell to represent nothing.

He gave the concept of positive numbers which he called wealth or dhan and negative numbers which he called debt or ऋण

 He wrote the rules as follows:

This was a revolution as most people dismissed the possibility of a negative number thereby proving that quadratic equations (of the type \(\rm{}x2 + 2 = 11,\) for example) could, in theory, have two possible solutions, one of which could be negative, because \(32 = 9\) and \(-32 = 9\) . Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables) and solving quadratic equations with two variables

An example from Brahmasphutasiddhanta

Five hundred drammas were loaned at an unknown rate of interest. The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten months to \(78\) drammas. Give the rate of interest.

  Brahmagupta Formula

Brahmagupta found the formula for cyclic quadrilaterals though he did not focus on the cyclic character of the figure. Given the lengths of the sides of any cyclic quadrilateral, Brahmagupta gave an approximate and an exact formula for the figure's area.

His second book The Khandakhadyaka - 665 AD has eight chapters. This book too details longitudes of the planets; the three problems of diurnal rotation; lunar eclipses; solar eclipses; risings and settings; the moon's crescent; and conjunctions of the planets.

What stands out as a mathematical genius in this work is the interpolation formula he uses to compute values of sines.

Brahmagupta achievements

Brahmagupta defined the properties of the number zero, which was crucial for the future of mathematics and science. Brahmagupta enumerated the properties of zero as:

   ★ When a number is subtracted from itself, we get a zero

   ★ Any number divided by zero will have the answer as zero

   ★ Zero divided by zero is equal to zero

Discovered the formula to solve quadratic equations.

Discovered the value of pi ( 3.162….) almost accurately. He put the value 0.66% higher than the true value. ( 3.14)

With calculations, he indicated that Earth is nearer to the moon than the sun.

Found a formula to calculate the area of any four-sided figure whose corners touch the inside of a circle.

Calculated the length of a year is 365 days 6 hours 12 minutes 9 seconds.

Brahmagupta talked about ‘gravity.’ To quote him, ‘Bodies fall towards the earth as it is in the nature of the earth to attract bodies, just as it is in the nature of water to flow.’

Proved that the Earth is a sphere and calculated its circumference to be around 36,000 km (22,500 miles).

Brahmagupta established rules for working with positive and negative numbers, such as:

  Ø Negative \(+\) Negative number \(=\) Negative number

  Ø Subtracting a Negative from a positive number is the same as adding the two numbers.

  Ø Negative X Negative number \(=\) Positive number.

  Ø Positive number ÷ Negative number \(=\) Negative number.

Indian philosophy reiterates that we are a small part of a Brahmand, the cosmos or the universe. This humbling knowledge was perhaps the basis of the concept of a zero or a void because it came from a culture that conceived and acknowledged the idea of the infinite. A symbol \((0)\) denoting “nothing” was a part of Indian culture. This becomes particularly relevant as it indicates a vibrant, philosophical culture that recognised the power of nothingness and thus actually recognised the power of Mathematics and its role in the order of nothingness.

Although Brahmagupta thought of himself as an astronomer who did some mathematics, he is now mainly remembered for his contributions to mathematics. He was honoured by the title given to him by a fellow scientist ‘ Ganita Chakra Chudamani’ which is translated as ‘The gem of the circle of mathematicians’.

Frequently Asked Questions (FAQs)

What did brahmagupta discover.

Brahmadutta has a lot to his credit:

• Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhānta.
• He gave two equivalent solutions to the general quadratic equation.
• Brahmagupta's Brahmasphuṭasiddhānta is the first book that provides rules for arithmetic manipulations that apply to zero and to negative numbers.
• Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals.

Where was brahmagupta born?

Brahmagupta was born in the city of Bhinmal, Rajasthan.

When was brahmagupta born?

Brahmagupta was born in 597 AD.

When did brahmagupta die?

Brahmagupta passed away in 668 AD.

Famous Mathematicians

List and Biographies of Great Mathematicians

Brahmagupta

According to himself, Brahmagupta was born in 598 CE and was the follower of Shaivism. During the rule of Chapa dynasty ruler, Vyagrahamukha, he lived in Bhillamala according to historian, yet there is no conclusive proof of that. Other sociologists believed he might have belonged to Multan region. Bhillamala was the capital of Gurjaradesa, currently known as Gujrat. It was the hub of all mathematical and astronomical learning. Bharmagupta assumed the position of an astronomer at Brahmapaksha school.

At a young age of 30, he wrote the improved treatise of Brahma called Brāhmasphuṭasiddhānta . It is speculated that it was the revision of the siddhanta he received from the school. He brought originality to the treatise by adding a great deal of new material to it. The book is written in arya-meter comprising 1008 verses and 24 chapters. An enormous amount of material is found on astronomy, while it also includes chapters on mathematics, trigonometry, algorithms and algebra.

After completing his work in Bhillamala, he moved to Ujjain which was also considered a chief location with respect to studies in astronomy. Aside from his revision of Brahma treatise, at the mature age of 67, he wrote another foremost work in mathematics entitled, Khanda-khādyaka . This text is a practical manual of Indian astronomy which is meant to guide students.

In his Brahma treatise, Brahmagupta criticized contemporary Indian astronomer on their different opinion. The rift between the mathematicians was created based on their varying ways of applying mathematics to physical world. Brahma had different views on astronomical parameters and theories. In his books he dedicated several chapters critiquing mathematical theories and their application.

There are numerous science historians who made testimony to his great scientific contribution. According to George Sarton, he was a great scientist of his race. In Medieval Europe Indian arithmetic was called “Modus Indoram” which means method of the Indians. He called multiplication gomutrika in his Brahmasphutasiddhanta . His work was further explored by Bhāskara II who held Brahmagupta at an elevated position for his immense contribution to mathematics. His work was further simplified and added illustrations to by Prithudaka Svamin. In addition to that his work was commented upon by Lalla and Bhattotpala in the eighth and ninth century. When Sindh was conquered by Arabs, his work was translated into Arabic by an astronomer, Muhammad al-Fazari which led to the use of decimal number system in written discourse.

Some of the major contribution to the field of astronomy by Brahmagupta are solar and lunar eclipse calculations and methods for calculating the position of heavenly bodies over time. Moreover, in a chapter titled Lunar Cresent he criticized the notion that the Moon is farther from the Earth than the Sun which was mentioned in Vedic scripture. He was of the view that the Moon is closer to the Earth than the Sun based on its power of waxing and waning. The illumination of the moon depends on the position and angle of Sunlight that hits the surface of the moon.

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(598–665?). One of the most accomplished of the ancient Indian astronomers was Brahmagupta. He also had a profound and direct influence on Islamic and Byzantine astronomy.

Brahmagupta was born in 598 an orthodox Hindu, and his religious views, particularly the Hindu yuga system of measuring the ages of humankind, influenced his work. He severely criticized Jain cosmological views and other non-conforming ideas, such as the view of Aryabhata I (born 476) that the Earth is a spinning sphere, a view that was widely disseminated by Brahmagupta’s contemporary and rival Bhaskara I .

Brahmagupta’s fame rests mostly on his Brahma-sphuta-siddhanta (628; “Correctly Established Doctrine of Brahma”), an astronomical work that he probably wrote while living in Bhillamala, then the capital of the Gurjara-Pratihara dynasty . It was translated into Arabic in Baghdad about 771 and had a major impact on Islamic mathematics and astronomy. Late in his life, Brahmagupta wrote Khandakhadyaka (665; “A Piece Eatable”), an astronomical handbook that employed Aryabhata’s system of starting each day at midnight.

In addition to expounding on traditional Indian astronomy in his books, Brahmagupta devoted several chapters of Brahma-sphuta-siddhanta to mathematics . In chapters 12 and 18 in particular, he laid the foundations of the two major fields of Indian mathematics, pati-ganita (“mathematics of procedures,” or algorithms) and bija-ganita (“mathematics of seeds,” or equations), which roughly correspond to arithmetic (including mensuration) and algebra, respectively. Chapter 12 is simply named “Mathematics,” probably because the “basic operations,” such as arithmetic operations and proportions, and the “practical mathematics,” such as mixture and series, treated there occupied the major part of the mathematics of Brahmagupta’s milieu. He stressed the importance of these topics as a qualification for a mathematician, or calculator ( ganaka ). Chapter 18, “Pulverizer,” is named after the first topic of the chapter, probably because no particular name for this area (algebra) existed yet.

Among his major accomplishments, Brahmagupta defined zero as the result of subtracting a number from itself and gave rules for arithmetical operations among negative numbers (“debts”) and positive numbers (“property”), as well as surds. He also gave partial solutions to certain types of indeterminate equations of the second degree with two unknown variables. Perhaps his most famous result was a formula for the area of a cyclic quadrilateral (a four-sided polygon whose vertices all reside on some circle) and the length of its diagonals in terms of the length of its sides. He also gave a valuable interpolation formula for computing sines. He died about 665, possibly in Bhillamala (modern Bhinmal).

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Mathematicians Math Lessons Square Roots Math Calculators Brahmagupta: Mathematician and Astronomer BRAHMAGUPTA: MATHEMATICIAN AND ASTRONOMER      (628; where means "concepts"), is based on a positional number system, and is the oldest known work where the ) appears in arithmetical operations. There, Brahmagupta establishes the rule - =0, and also considers the fractions /0, which he sets equal to 0 for =0 and otherwise calls , a term of uncertain meaning. ), including the extraction of roots and the solution of proportions, and eight measurements ( ). In this fine classification of mathematical procedures, he also listed four methods for multiplication, and five rules for reducing a rational expression to a single fraction. , . (van der Waerden, 1985, p. 14). , New York: Wiley, 1968. Lahore, India: Motilal Banarsi Das, 1935. London: Taylor and Francis, pp. 68-71, 1969. Berlin: Springer-Verlag, 1985. Berlin: Springer-Verlag, 1983. Brahmagupta (598–668 CE) The great 7th Century Indian mathematician and astronomer Brahmagupta wrote some important works on both mathematics and astronomy. He was from the state of Rajasthan of northwest India (he is often referred to as Bhillamalacarya, the teacher from Bhillamala), and later became the head of the astronomical observatory at Ujjain in central India. Most of his works are composed in elliptic verse, a common practice in Indian mathematics at the time, and consequently have something of a poetic ring to them. It seems likely that Brahmagupta’s works, especially his most famous text, the “Brahmasphutasiddhanta”, were brought by the 8th Century Abbasid caliph Al-Mansur to his newly founded centre of learning at Baghdad on the banks of the Tigris, providing an important link between Indian mathematics and astronomy and the nascent upsurge in science and mathematics in the Islamic world . In his work on arithmetic, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots. He also gave rules for dealing with five types of combinations of fractions. He gave the sum of the squares of the first n natural numbers as n ( n + 1)(2 n + 1) ⁄ 6 and the sum of the cubes of the first n natural numbers as ( n ( n + 1) ⁄ 2 ) ² . Brahmasphutasiddhanta – Treat Zero as a Number  Brahmagupta’s rules for dealing with zero and negative numbers Brahmagupta’s genius, though, came in his treatment of the concept of (then relatively new) the number zero. Although often also attributed to the 7th Century Indian mathematician Bhaskara I, his “Brahmasphutasiddhanta” is probably the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit as was done by the Babylonians , or as a symbol for a lack of quantity as was done by the Greeks and Romans . Brahmagupta established the basic mathematical rules for dealing with zero (1 + 0 = 1; 1 – 0 = 1; and 1 x 0 = 0), although his understanding of division by zero was incomplete (he thought that 1 ÷ 0 = 0). Almost 500 years later, in the 12th Century, another Indian mathematician, Bhaskara II, showed that the answer should be infinity, not zero (on the grounds that 1 can be divided into an infinite number of pieces of size zero), an answer that was considered correct for centuries. However, this logic does not explain why 2 ÷ 0, 7 ÷ 0, etc, should also be zero – the modern view is that a number divided by zero is actually “undefined” (i.e. it doesn’t make sense). Brahmagupta’s view of numbers as abstract entities, rather than just for counting and measuring, allowed him to make yet another huge conceptual leap which would have profound consequence for future mathematics. Previously, the sum 3 – 4, for example, was considered to be either meaningless or, at best, just zero. Brahmagupta, however, realized that there could be such a thing as a negative number, which he referred to as “debt” as a opposed to “property”. He expounded on the rules for dealing with negative numbers (e.g. a negative times a negative is a positive, a negative times a positive is a negative, etc). Furthermore, he pointed out, quadratic equations (of the type x 2 + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 3 2 = 9 and -3 2 = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657. Brahmagupta’s Theorem on cyclic quadrilaterals Brahmagupta’s Theorem on cyclic quadrilaterals Brahmagupta even attempted to write down these rather abstract concepts, using the initials of the names of colours to represent unknowns in his equations, one of the earliest intimations of what we now know as algebra. Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established √10 (3.162277) as a good practical approximation for π (3.141593), and gave a formula, now known as Brahmagupta’s Formula, for the area of a cyclic quadrilateral, as well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta’s Theorem. Ancient India’s Brahmagupta: Pioneering the Foundations of Mathematics and Astronomy Post author By Mala Chandrashekhar Post date October 13, 2023 No Comments on Ancient India’s Brahmagupta: Pioneering the Foundations of Mathematics and Astronomy Introduction In the annals of history, there are individuals whose contributions shape the course of human knowledge. Brahmagupta, an Indian mathematician and astronomer who lived during the 7th century CE, was one such luminary. His groundbreaking works, the “Brāhmasphuṭasiddhānta” and the “Khaṇḍakhādyaka,” not only revolutionized the fields of mathematics and astronomy but also laid the groundwork for countless future discoveries. In this blog post, we will delve into the life and accomplishments of Brahmagupta, highlighting his pioneering insights into gravity. Brahmagupta: A Brief Biography Born around 598 CE in the ancient city of Ujjain, Brahmagupta belonged to a lineage of esteemed mathematicians and astronomers. His father, Jisnugupta, was also a mathematician, and it is likely that Brahmagupta inherited his passion for these disciplines from his family. Brahmagupta’s genius became evident at a young age, and he made his mark as a prominent mathematician during the Gupta dynasty in India. The Brāhmasphuṭasiddhānta: A Theoretical Masterpiece In 628 CE, Brahmagupta authored his most renowned work, the “Brāhmasphuṭasiddhānta,” which translates to the “correctly established doctrine of Brahma.” This monumental text, consisting of 24 chapters, encompasses a wide range of mathematical and astronomical concepts. It was a significant departure from earlier works, as Brahmagupta introduced innovative ideas and theories that would influence scholars for centuries to come. One of his most notable contributions was his description of gravity as an attractive force. In this groundbreaking insight, Brahmagupta used the Sanskrit term “gurutvākarṣaṇam (गुरुत्वाकर्षणम्)” to articulate this concept. This was a remarkable precursor to Isaac Newton’s universal law of gravitation, which would not be formulated until over a millennium later. Brahmagupta’s work on algebra was equally pioneering. He introduced the rules for performing arithmetic operations with both positive and negative numbers, including zero. These foundational concepts were instrumental in shaping the development of algebraic notation and paved the way for future mathematical advances. The Khaṇḍakhādyaka: A Practical Guide Following the success of the “Brāhmasphuṭasiddhānta,” Brahmagupta continued to contribute to mathematics and astronomy. In 665 CE, he penned the “Khaṇḍakhādyaka,” which translates to “edible bite.” This text, in contrast to the theoretical nature of his earlier work, served as a more practical guide for astronomers and surveyors. It provided valuable insights into the calculation of various celestial phenomena, such as eclipses, planetary positions, and lunar and solar cycles. Legacy and Impact Brahmagupta’s contributions to mathematics and astronomy extended far beyond his time. His pioneering ideas and theories laid the foundation for the development of modern mathematics and the scientific understanding of celestial bodies. The concept of gravity as an attractive force, as described in the “Brāhmasphuṭasiddhānta,” was a remarkable precursor to Isaac Newton’s groundbreaking work in the 17th century. Furthermore, Brahmagupta’s work on algebraic rules and number systems significantly influenced subsequent mathematicians in India and the Islamic world. His legacy can be seen in the algebraic notation and techniques that we use today. Brahmagupta, a brilliant mathematician and astronomer of ancient India, left an indelible mark on the history of science. His theoretical treatise, the “Brāhmasphuṭasiddhānta,” and practical guide, the “Khaṇḍakhādyaka,” revolutionized our understanding of mathematics and astronomy. His description of gravity as an attractive force and his contributions to algebraic notation continue to inspire and shape the world of mathematics and science. Brahmagupta’s work serves as a testament to the enduring impact of knowledge and the brilliance of human innovation. By Mala Chandrashekhar Introducing Blogger Mala Chandrashekhar - A specialist academically trained in modern Western sciences, yet deeply enamored with India's timeless ethnic arts, crafts, and textiles. Her heart beats for the rich and glorious cultural and spiritual heritage of India, and she has dedicated her entire blog to spreading the immortal glories of ancient India worldwide. Through her simple yet impactful blog posts, Mala aims to reach every nook and corner of the globe, sharing India's beauty and wisdom with the world. But Mala doesn't stop at just sharing her own thoughts and ideas. She welcomes constructive criticisms and suggestions to improve her blog and make it even more impactful. And if you share her passion for India's culture and heritage, she extends a warm invitation for high-quality guest blog posts. Ready to dive into the world of India's ageless beauty? Follow Mala on LinkedIn, Twitter & Facebook and join her in spreading the magic of ancient India to the world. LinkedIn Profile: https://in.linkedin.com/in/mala-chandrashekhar-04095917a Twitter Handle: @MalaCShekhar Facebook Page: https://www.facebook.com/mala.chandrashekhar Leave a Reply Cancel reply Your email address will not be published. Required fields are marked * Save my name, email, and website in this browser for the next time I comment. Brahmagupta Biography Brahmagupta, an ancient Indian astronomer and mathematician, is best known for his groundbreaking work in the field of astronomy. His treatise, ‘Brāhmasphuṭasiddhānta’, not only had a profound impact on the development of astronomy in India but also influenced Islamic mathematics and astronomy. Despite being an orthodox Hindu, he was ahead of his time in realizing that the Earth is a sphere. Additionally, he was a highly revered mathematician and his book was the first to mention zero as a number and provide rules for its use with negative and positive numbers. Quick Facts Died At Age: 72 Died on: 670 Astronomers Mathematicians Childhood & Early Life Brahmagupta was born in 598 AD into an orthodox Shaivite Hindu family. His father’s name was Jishnugupta. It is generally believed that he was born in Ujjain. Not much is known about his early life. Education and Career As a young man, Brahmagupta studied astronomy extensively. He was well-read in the five traditional siddhanthas on Indian astronomy, and also studied the work of other ancient astronomers such as Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin, and Vishnuchandra. He became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during his era. Later Years Brahmagupta is believed to have lived and worked in Bhinmal in present-day Rajasthan, India, for a few years. The city was a center of learning for mathematics and astronomy, and he flourished as an astronomer in the intellectual atmosphere of the city. At the age of 30, he composed the theoretical treatise ‘Brāhmasphuṭasiddhānta’ (“Correctly established doctrine of Brahma”) in 628 AD. The work is thought to be a revised version of the received siddhanta of the Brahmapaksha school, incorporated with some of his own new material. Primarily a book of astronomy, it also contains several chapters on mathematics. Contributions Brahmagupta is credited with giving the most accurate early calculations of the length of the solar year. He also introduced new methods for solving quadratic equations and gave equations to solve systems of simultaneous indeterminate equations. In addition, he provided a formula useful for generating Pythagorean triples and gave a recurrence relation for generating solutions to certain instances of Diophantine equations. In mathematics, his contribution to geometry was especially significant. He gave formulas for the lengths and areas of various geometric figures, and his formula for cyclic quadrilaterals, now known as Brahmagupta’s formula, provides a way of calculating the area of any cyclic quadrilateral given the lengths of the sides. Major Works Brahmagupta’s treatise ‘Brāhmasphuṭasiddhānta’ is one of the first mathematical books to provide concrete ideas on positive numbers, negative numbers, and zero. The text also elaborated on the methods of solving linear and quadratic equations, rules for summing series, and a method for computing square roots. It also contained the first clear description of the quadratic formula. One of his later works was the treatise ‘Khaṇḍakhādyaka’ (meaning “edible bite; morsel of food”), written in 665 AD, which covered several topics on astronomy. Personal Life & Legacy The details regarding Brahmagupta’s family life are obscure. He is believed to have died sometime after 665 AD. Leave a Comment Cancel reply Save my name, email, and website in this browser for the next time I comment. Brahmagupta’s Contributions in Mathematics WARNING: unbalanced footnote start tag short code found. If this warning is irrelevant, please disable the syntax validation feature in the dashboard under General settings > Footnote start and end short codes > Check for balanced shortcodes. Unbalanced start tag short code found before: “p+r)⁄2 × (q+s)⁄2), whereas, the exact area is given by  √(t − p)(t − q)(t − r)(t − s), where t = (p+q+r+s)⁄2. Also, Heron’s formula is a special case of the Brahmagupta formula, which can be obtained by setting one side equal to zero. 8. Brahmagupta Theorem Brahmagupta theo…” 1. Properties of Zero According to him, zero is a number that is obtained, when a number is subtracted from itself. He also mentioned some properties of zero, where positive numbers are termed as fortunes and negative numbers are termed as debt. When zero is added to a number or subtracted from a number, the number remains unchanged; and a number multiplied by zero becomes zero. A debt minus zero is a debt. A fortune minus zero is a fortune. Zero minus zero is a zero. A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multiplied by zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt. He also tried to make some conclusions on the division by zero. For this he said, Positive or negative numbers when divided by zero is a fraction with zero as denominator. Zero divided by negative or positive numbers is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.” 2. Brahmagupta’s Method of Multiplication He proposed a method of multiplication, “gomutrika”, in his book “Brahmasphutasiddhanta”. The title of this method was translated by Ifrah as, “Like the trajectory of cow’s urine”. In the 12th chapter of his book, he also tried to explain the rules of simplifying 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. Let us try to multiply 315 by 306 with the help of the gomutrika method. Now multiply the 306 of the top row by the 3 in the top position of the left-hand column. Begin by 3 × 6=  18 , putting 8 below the 6 of the top row, carrying 1  in the usual way to get Now multiply the 306 of the second row by the 1 in the left-hand column writing the number in the line below the 918 but moving one place to the right Now multiply the 306 of the third row by the 5 in the left-hand column writing the number in the line below the 306 but moving one place to the right Now add the three numbers 91800 + 3060 + 1530 = 96390 is the required result. The second form of this method requires, first writing the second number on the right but with the order of the digits reversed as follows 306       5 306       1 306       4 In the third variant of this method, just write each number once but otherwise follows the second method. 3. Intermediate Equations Brahamgupta proposed some methods to solve equations of the type ax + by = c. According to Majumdar, Brahmgupta used continued fractions to solve such equations. He also tried to solve quadratic equations of the type ax² + c = y² and ax² – c = y². For example, for the equation 8x² + 1 = y² he obtained the solutions as (x, y)= ( 1 , 3 ) , ( 6 , 1 7 ) , ( 3 5 , 9 9 ) , ( 2 0 4 , 5 7 7 ) , ( 1 1 8 9 , 3 3 6 3 ) , . .He also solved 61x² + 1 = y²  having solution as x = 226153980, y = 1766319049 as its smallest solution. A sample of the types of problem solved by him is:- Five hundred drammas were loaned at an unknown rate of interest, The interest on the money for four months was loaned to another at the same rate of interest and amounted in ten mounths to  78 drammas. Give the rate of interest.” 4. Sum of Series He gave the sum of, a series of cubes and a series of squares for the first n natural numbers as follows: 1² + 2² +…….+n² = (n)(n+1)(2n+1)⁄6 1³ + 2³ +…….+n³ = (n(n+1)⁄2)² 5. Pythagorean triplets Brahmagupta in chapter 12, entitled “Calculation”, of his book, proposed a formula that was useful in generating Pythagorean triplets. He mentioned,  The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey.” If d= mx ⁄(x+2), then a traveller who leaps vertically upwards, a distance d from the top of the mountain of height m and, then covers a horizontal distance of mx from the base of the mountain, in a straight line, to the city, travels the same distance, as the one who descends vertically down the mountain and then travels along the horizontal line to the city. Geometrically this means, if a right-angled triangle has a base of length a = mx and altitude of length b = m + d, then the length, c, of its hypotenuse is given by c = m (1+x) – d. And, elementary algebraic manipulation shows, that a 2  + b 2  = c 2  whenever d  has the value stated. Also, if m and x are rational, so are d, a, b and c. A Pythagorean triple can therefore be obtained from a, b and c  by multiplying each of them by the least common multiple of their denominators. 6. Pell’s Equation Brahmagupta studied this equation 1000 years before Pell’s birth. Pell’s equation is of form, n x ² + 1 = y², which can also be written as y² – nx² = 1, where ‘n’ is an integer and we have to solve it for (x, y) integer solutions. Brahmagupta also provided a lemma, in which he stated that if ( a , b ) and ( c, d) are integer solutions of ‘Pell type equations’ of the form                    na²+ k = b²  and  nc² + k’ = d² then, (bc + ad, bd + nac)  and  (bc – ad, bd – nac) are both integer solutions of the ‘Pell type equation’ nx² + kk’ = y². Brahmagupta used the method of composition to find solutions for Pell’s equations. He composed (a, b) and (a, b) to get (2ab, b² + na²) as a solution to Pell’s equation. After getting (2ab, b² + na²) as a solution of the equation nx² + k² = y², he divided the x and y coordinates by k² which gave x = 2ab⁄k² and y = b² + na²⁄k², a solution of Pell’s equation of the form nx² + 1 = y². He then claimed, that with the help of the composition method one can generate many solutions to Pell’s equation. 7. Brahmagupta’s Formula Brahmagupta’s formula for the cyclic quadrilaterals is regarded as his most famous discovery in geometry. Given the sides of a cyclic quadrilateral, he provided an approximate and exact formula for the area of the cyclic quadrilateral. He mentioned,  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.” In the figure below, p, q, r, s are the sides of the cyclic quadrilateral. Its approximate area is given by ((p+r)⁄2 × (q+s)⁄2), whereas, the exact area is given by  √ (t − p)(t − q)(t − r)(t − s), where t = (p+q+r+s)⁄2. Also, Heron’s formula is a special case of the Brahmagupta formula, which can be obtained by setting one side equal to zero. 8. Brahmagupta Theorem Brahmagupta theorem states that, If a cyclic quadrilateral is orthodiagonal (i.e., has perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side.” Geometrically, this theorem means that, in a cyclic quadrilateral ABCD, diagonals AC and BD are perpendicular to each other. The intersection of AC and BD is M.  Drop the perpendicular from M to the line BC, calling the intersection point E. Let F be the intersection of the line EM and the side AD. Then, according to the theorem, F is the midpoint of side AD. Brahmagupta further extended his theory and claimed that,  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.” The above statement means that in an isosceles trapezoid having sides of length p, q, r, s, the length of the diagonal is given by √pr+qs. 9. Triangles A major portion of Brahmagupta’s work was dedicated to the study of geometry. One of his theorem about triangles states that, 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.” This theorem gives the length of two segments in which the base of a triangle is divided by its altitude, and the lengths are, 1 / 2 (b ± ( c 2  − a 2) / b ) . He also discussed rational triangles. A rational triangle with the rational area and sides a, b, c, are of the form: a = 1⁄2(u²⁄v + v), b = 1⁄2(u²⁄w + w), c = 1⁄2(u²⁄v – v + u²⁄w – w), for some rational numbers u, v, w. 10. Approximation of π Brahmagupta also tried to approximate the value of π and in stanza 40 of his book he mentioned,  The diameter and the square of the radius, each multiplied by 3 are the practical circumference and the area of a circle respectively. The accurate values are the square-roots from the squares of those two multiplied by ten.” He used √10 ≈ 3.1622….. approximated to 3, as an accurate value of π with an error of less than 1%. 11. Mensuration and Construction Brahmagupta illustrated the construction of several figures with arbitrary sides. He tried to construct figures such as isosceles triangles, scalene triangles, rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides, and scalene cyclic quadrilateral, mainly, with the help of right triangles. He also gave the volume and surface area of some figures, after estimating the value of π. He found the volume of rectangular prisms, pyramids and frustum of a square pyramid. He further proposed the average depth of a series of pits. 12. Trigonometry Brahmagupta, in chapter 2 of his book, provided a sine table. He wrote, The sines: The Progenitors, twins, Ursa Major, the Vedas, the gods, fires, flavors, dice, the moon, the sky, the moon, arrows, sun…..” He used the above objects to represent digits of place-value numerals. Progenitors represent 14 progenitors in Indian cosmology, 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 traditional die or 6, and so on. He gave the sine table with 3270 as radius and calculated 3270 sin(π⁄48). For 1 ≤ n ≤ 24, he got values as 214, 427, 638, 846, 1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 2459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, and 3270. 13. Interpolation formula Brahmagupta was the first who propose an interpolation formula using second-order difference. His Sanskrit verses on this formula were found in the Khandakadyaka work of Brahmagupta. Today, the Brahmagupta interpolation formula is known as Newton- Stirling interpolation formula. In his book, he termed the difference D r as the ‘gatakhanda’,  the difference D r+1 as the ‘bhogyakhanda’, ‘Vikala’ as the quantity in minutes by which the interval has been covered at the point of interpolation, which in modern notation denoted as a − x r . ‘Sphuta-bhogyakhanda’ is now known as   f r+1 − f r . A formula stated by him, for the computation of values of the sine table, having common interval (h) in the underlying base table as 900 minutes or 15 degrees, is given below. When translated these verses means, Multiply the ‘vikala’ by the half the difference of the ‘gatakhanda’ and the ‘bhogyakhanda’ and divide the product by 900. Add the result to half the sum of the ‘gatakhanda’ and the ‘bhogyakhanda’ if their half-sum is less than the ‘bhogyakhanda’, subtract if greater. The result in each case is ‘sphuta-bhogyakhanda’ the correct tabular difference. In modern notation, the formula is denoted as sphuta-bhogyakhanda =  ( D r + D r-1) ⁄2 ± t| D r –  D r-1 |⁄2, where ± is introduced according to   D r  < D r+1  or  D r  > D r+1  and f(a) is given by, f(a) = f r + t × sphuta-bhogyakhanda, is known as Stirling’s interpolation formula for second-order differences. 14. Algebra Brahmagupta gave the solution of general linear equations in chapter 18 of his book and wrote, The difference between rupas, when inverted and divided by the difference of the coefficients 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.” Algebraically, the above statement means that, for an equation of type bx+c = dx+e, the solution is given by x = e − c / b − d. He also gave two solutions for the quadratic equation ax² +bx = c, and wrote,   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.” By this method, the solution is given by, x = ±(√(4ac+b²) – b)⁄2a.  Whatever is the square-root of the rupas multiplied by the square and increased by the square of half the unknown, diminished that by half the unknown and divide the remainder by its square. The result is the unknown.” By this method, the solution is given by, x = ±(√(ac+b²⁄4) – b²⁄2)⁄a. Related Posts Antiphon’s Contribution in Mathematics 7 Examples of Median in Daily Life Carl Friedrich Gauss’ Contributions in Mathematics 22 Examples of Mathematics in Everyday Life Hypatia’s Contribution in Mathematics 8 Daily Life Examples Of Axioms Add comment cancel reply. Essay on Brahmagupta Students are often asked to write an essay on Brahmagupta in their schools and colleges. And if you’re also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic. Let’s take a look… 100 Words Essay on Brahmagupta Brahmagupta’s life. Brahmagupta was a great Indian mathematician and astronomer. He was born in 598 AD in Bhinmal, a town in the Jalore district of Rajasthan, India. Contributions to Mathematics Brahmagupta is famous for his work “Brahmasphutasiddhanta”. In this book, he gave rules to compute with zero and negative numbers, which was revolutionary. Astronomical Discoveries In the field of astronomy, Brahmagupta corrected the existing astronomical theories. His accurate calculations for solar and lunar eclipses are noteworthy. Brahmagupta’s contributions have greatly influenced both Indian and Islamic mathematical traditions. His work still inspires mathematicians today. 250 Words Essay on Brahmagupta Brahmagupta: the mathematical genius, life and works. Little is known about Brahmagupta’s personal life. Born in 598 CE in Bhinmal, a city in the state of Rajasthan, India, he spent most of his life in Ujjain, one of the intellectual hubs of ancient India. Brahmagupta’s most influential works are the ‘Brahmasphutasiddhanta’ and ‘Khandakhadyaka’. These texts provide valuable insights into the mathematical and astronomical theories of his time. Brahmagupta’s contributions to mathematics are diverse and groundbreaking. He was the first to provide rules to compute with zero, thus laying the foundation for algebra. His work on the solutions of linear and quadratic equations and rules for the sum of squares are noteworthy. In astronomy, Brahmagupta criticized the prevailing notion of a flat Earth and proposed a belief in a spherical Earth. He also made significant contributions to the field of trigonometry. Brahmagupta’s work has had a profound influence on subsequent generations of mathematicians and astronomers, both in India and the Islamic world. His pioneering work on zero and its operations laid the foundation for algebra, a cornerstone of modern mathematics. In conclusion, Brahmagupta’s contributions to the fields of mathematics and astronomy are immeasurable. His work continues to inspire and influence the scientific community, affirming his place as one of the great intellects of the medieval world. 500 Words Essay on Brahmagupta Introduction. Brahmagupta’s most notable work in mathematics is perhaps his accurate definition of zero. He was the first to understand and articulate that when a number is divided by zero, it results in infinity, and when zero is subtracted from itself, it remains zero. His work on zero serves as a foundation for the mathematical concept of infinity. In his seminal work, Brahmasphutasiddhanta, Brahmagupta presented solutions to linear and quadratic equations, and his methods are still taught in schools today. He also introduced the concept of negative numbers and demonstrated rules for arithmetic involving these numbers. Contributions to Astronomy Brahmagupta also developed methods for calculating the positions of various celestial bodies over time, a branch of astronomy known as celestial mechanics. His work in this field laid the groundwork for the later development of gravitational theory. Brahmagupta’s work was translated into Arabic in the 8th century, which allowed it to reach a wider audience and greatly influence the mathematical and scientific thought in the Middle East. His work later found its way to Europe, influencing the Renaissance’s scientific revolution. Brahmagupta’s profound insights into mathematics and astronomy have withstood the test of time, making him one of history’s most influential scientists. His understanding of zero, his solutions to equations, his astronomical calculations, and his pioneering work in celestial mechanics all attest to his genius. His legacy is a testament to the advanced state of science in ancient India and continues to inspire mathematicians and astronomers worldwide. That’s it! I hope the essay helped you. If you’re looking for more, here are essays on other interesting topics: Happy studying! Leave a Reply Cancel reply Your email address will not be published. Required fields are marked * Brahmagupta Reference work entry First Online: 01 January 2016 Cite this reference work entry R. C. Gupta   41 Accesses “Brahmagupta holds a remarkable place in the history of Eastern civilization” (Sachau, 1971 ). Bhāskara II described Brahmagupta as Gaṇakacakracūdāmani , jewel among the circle of mathematicians. Brahmagupta was born in AD 598 according to his own statement: “… when 550 years of the Śaka era had elapsed, Brahmagupta, son of Jisṇu, at the age of 30, composed the Brāhmasphuṭasiddhānta for the pleasure of good mathematicians and astronomers.” Thus he was 30 years old in Śaka 550 or AD 628 when he wrote the Brāhmasphuṭasiddhānta . That he was still active in old age is clear from the title epoch of AD 665 used in another of his works called Khaṇḍa-khādyaka . Pṛthūdaka Svāmin, an ancient commentator on Brahmagupta, calls him Bhillamālācārya, which shows that he came from Bhillamāla. This place has been identified with the modern village Bhinmal near Mount Abu close to the Rajasthan-Gujarat border. We have no knowledge of Brahmagupta’s teachers, or of his education, but we know he studied the... This is a preview of subscription content, log in via an institution to check access. Access this chapter Subscribe and save. Get 10 units per month Download Article/Chapter or eBook 1 Unit = 1 Article or 1 Chapter Cancel anytime Available as PDF Read on any device Instant download Own it forever Available as EPUB and PDF Durable hardcover edition Dispatched in 3 to 5 business days Free shipping worldwide - see info Tax calculation will be finalised at checkout Purchases are for personal use only Institutional subscriptions Primary Sources Chatterjee, B. (Ed. & Trans.). (1970). Khaṇḍakhādyaka with the commentary of Bhaṭṭotpala . Calcutta, India: World Press. Google Scholar   Sengupta, P. C. (Ed. & Trans.). (1934). The Khaṇḍakhādyaka of Brahmagupta . Calcutta, India: Calcutta University. Sharma, R. S., et al. (Eds.). (1966). The Brāhma-sphuṭa-siddhānta (with Hindi translation) (Vol. 4). New Delhi, India: Indian Institute of Astronomical and Sanskrit Research. Secondary Sources Colebrooke, H. T. (1817). Algebra with arithmetic and mensuration from the Sanskrit of Brahmegupta and Bhāscara . London: Murray. Gupta, R. C. (1969). Second order interpolation in Indian mathematics. Indian Journal of History of Science, 4 , 86–98. Gupta, R. C. (1972). Brahamagupta’s rule for the volume of frustum-like solids. Mathematics Education, 6 (4B), 117–120. Gupta, R. C. (1974). Brahmagupta’s formulas for the area and diagonals of a cyclic quadrilateral. Mathematics Education, 8 (2B), 33–36. Ikeyama, S. (2003). Brahmasphutasiddhanta (Ch. 21) of Brahmagupta with commentary of Prthudaka, critically edited with english translation and notes. Indian Journal of History of Science, 38 (1), S1–S74. no. 2: 75-S152; no. 3: not available. (This article is a critical edition, with an English translation and commentary, of a Sanskrit astronomical text. It is divided into three parts and treated as a supplement to issues 1, 2, and 3). Kak, S. (1989). The Brahmagupta algorithm for square-rooting. Gaṇita Bhāratī, 11 , 27–29. Kusuba, T. (1981). Brahmagupta’s Sūtras on tri- and quadrilaterals. Historia Scientarum, 21 , 43–55. Pottage, J. The mensuration of quadrilaterals and the generation of pythagorean triads etc. Archive for History of Exact Sciences, 12 , 299–354. Sachau, E. (1971). Alberuni’s India . New York: Norton. Sarton, G. (1947). Introduction to the history of science . Baltimore: Williams and Wilkins. Venkutschaliyenger, K. (1988). The development of mathematics in ancient India: The role of Brahmagupta. In B. V. Subbarayappa & S. R. N. Murthy (Eds.), Scientific heritage of India (pp. 36–47). Bangalore, India: Mythic Society. Download references You can also search for this author in PubMed   Google Scholar Editor information Editors and affiliations. Hampshire College, Amherst, MA, USA Helaine Selin Rights and permissions Reprints and permissions Copyright information © 2016 Springer Science+Business Media Dordrecht About this entry Cite this entry. Gupta, R.C. (2016). Brahmagupta. In: Selin, H. 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He made advances in astronomy and most importantly in number systems including algorithms for square roots and the solution of quadratic equations. Brahmagupta Brahmagupta (c. 598 - 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 Brahmagupta (born 598—died c. 665, possibly Bhillamala [modern Bhinmal], Rajasthan, India) was one of the most accomplished of the ancient Indian astronomers. He also had a profound and direct influence on Islamic and Byzantine astronomy. Brahmagupta was an orthodox Hindu, and his religious views, particularly the Hindu yuga system of ... Brahmagupta Biography: Life, Discoveries, and Contributions of the Brahmagupta was an ancient Indian mathematician and astronomer who lived from 598 to 668 CE. He resided in Bhillamāla in Gurjaradesa, which is now Bhinmal in Rajasthan, India. Born to Jishnugupta and a follower of Hinduism, Brahmagupta spent most of his life in this region. Brahmagupta is considered one of the most influential mathematicians ... Brahmagupta Quick Guide to Brahmagupta Brahmagupta: • was the director of the astronomical observatory of Ujjain, the center of Ancient Indian mathematical astronomy. • wrote four books about astronomy and mathematics, the most famous of which is Brahma-sphuta-siddhanta ( Brahma's Correct System of Astronomy, or The Opening of the Universe.) Brahmagupta Brahmagupta was an ancient Indian mathematician and astronomer. He made progress in number systems and astronomy. Brahmagupta Brahmagupta. Brahmagupta (c. 598-c. 670) was one of the most significant mathematicians of ancient India. He introduced extremely influential concepts to basic mathematics, including the use of zero in mathematical calculations and the use of mathematics and algebra in describing and predicting astronomical events. Brahmagupta Brahmagupta The field of mathematics is incomplete without the generous contribution of an Indian mathematician named, Brahmagupta. Besides being a great mathematician he was an even brilliant astronomer who wrote several books on these subjects. The doctrine of Barhama entitled the Brāhmasphuṭasiddhānta, is one of his early works on mathematics and astronomy. His major contribution to ... Brahmagupta Brahmagupta. (598-665?). One of the most accomplished of the ancient Indian astronomers was Brahmagupta. He also had a profound and direct influence on Islamic and Byzantine astronomy. Brahmagupta was born in 598 an orthodox Hindu, and his religious views, particularly the Hindu yuga system of measuring the ages of humankind, influenced his work. Brahmagupta Brahmagupta. Brahmagupta (listen (help·info)) (born c. 598 CE, died c. 668 CE) was an mathematician and astronomer from South Asia. He was the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma ", dated 628). Brahmagupta was the first to give rules to compute ... Brahmagupta (ca. 598-ca. 665) -- from Eric Weisstein's World of Brahmagupta (ca. 598-ca. 665) Hindu astronomer and mathematician who applied algebraic methods to astronomical problems. Brahmagupta's treatise Brâhma-sphuta-siddhânta (628; where siddhânta means "concepts"), is based on a positional number system, and is the oldest known work where the zero ( cipher) appears in arithmetical operations. BRAHMAGUPTA: MATHEMATICIAN AND ASTRONOMER Biography. Brahmagupta (598-668 CE) The great 7th Century Indian mathematician and astronomer Brahmagupta wrote some important works on both mathematics and astronomy. He was from the state of Rajasthan of northwest India (he is often referred to as Bhillamalacarya, the teacher from Bhillamala), and later became the head of the astronomical ... Ancient India's Brahmagupta: Pioneering the Foundations of Mathematics In this blog post, we will delve into the life and accomplishments of Brahmagupta, highlighting his pioneering insights into gravity. Brahmagupta: A Brief Biography Born around 598 CE in the ancient city of Ujjain, Brahmagupta belonged to a lineage of esteemed mathematicians and astronomers. BrahmaGupta, Mathematician Par Excellence Brahmagupta (Sanskrit: ब्रह्मगुप्त), the son of Jisnugupta, was an Indian mathematician and astronomer who lived between 597-668 AD and wrote two important works on mathematics and astronomy: Brāhmasphuṭa Siddhānta in 628 AD (Correctly Established Doctrine of Brahma, also called The Opening of the Universe) which is ... Brahmagupta Brahmagupta's "Brahmasphutasiddhanta" Brahmagupta composed his most famous book, the Brahmasphutasiddhanta meaning "the corrected treatise of Brahma," at the age of 30 in 628 AD. Brahmagupta Biography, Life & Interesting Facts Revealed Brahmagupta is believed to have lived and worked in Bhinmal in present-day Rajasthan, India, for a few years. The city was a center of learning for mathematics and astronomy, and he flourished as an astronomer in the intellectual atmosphere of the city. At the age of 30, he composed the theoretical treatise 'Brāhmasphuṭasiddhānta ... Brahmagupta: Biography, Facts Brahmagupta was indeed an Indian mathematician and astronomer. He decided to write the Brhmasphuasiddhnta, "fully established theory of Brahma," released in 628, a theoretical dissertation, and also the Khandakhadyaka, "edible bite," published in 665, a more functional tract. Brahmagupta was the first one to offer recommendations for working with zeroes. Brahmagupta's works were ... Brahmagupta Brahmagupta was born in AD 598 according to his own statement: "… when 550 years of the Śaka era had elapsed, Brahmagupta, son of Jisṇu, at the age of 30, composed the Brāhmasphuṭasiddhānta for the pleasure of good mathematicians and astronomers.". Thus he was 30 years old in Śaka 550 or AD 628 when he wrote the ... Brahmagupta's Contributions in Mathematics Brahmagupta's formula for the cyclic quadrilaterals is regarded as his most famous discovery in geometry. Given the sides of a cyclic quadrilateral, he provided an approximate and exact formula for the area of the cyclic quadrilateral. Essay on Brahmagupta Students are often asked to write an essay on Brahmagupta in their schools and colleges. And if you're also looking for the same, we have created 100-word, 250-word, and 500-word essays on the topic. Brahmagupta Brahmagupta. "Brahmagupta holds a remarkable place in the history of Eastern civilization" (Sachau, 1971). Bhāskara II described Brahmagupta as Gaṇakacakracūdāmani, jewel among the circle of mathematicians. Brahmagupta was born in AD 598 according to his own statement: "… when 550 years of the Śaka era had elapsed, Brahmagupta ... Brahmagupta Timeline of Mathematics. c. 300 BCE: Indian mathematician Pingala writes about zero, binary numbers, Fibonacci numbers, and Pascal's triangle. c. 260 BCE: Archimedes proves that π is between 3.1429 and 3.1408. c. 235 BCE: Eratosthenes uses a sieve algorithm to quickly find prime numbers. c. 200 BCE: The "Suàn shù shū" (Book on Numbers ... assignment essay homework paper phd report review speech thesis