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The Brāhma-sphuṭa-siddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS) is a main work of Brahmagupta, written c. 628.^[1] This text of mathematical astronomy contains significant mathematical content, including the first good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta theorem.

The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the quadratic formula (the solution of the quadratic equation).^[2]^[3]

Positive and negative numbers[edit]

Brāhmasphuṭasiddhānta is one of the first books to provide concrete ideas on positive numbers, negative numbers, and zero.^[4] For example, it notes that the sum of a positive number and a negative number is their difference or, if they are equal, zero; that subtracting a negative number is equivalent to adding a positive number; that the product of two negative numbers is positive. Some of the notions of fractions differ from the modern rational number system. For example, Brahmagupta allows division by zero resulting in a fraction with a $0$ in the denominator, and defines $0/0 = 0$ . In modern mathematics, division by zero is undefined for any field.^[5]

Influence[edit]

Ashadhara, the son of Rihluka, wrote Graha-jnana with tables based on Brahma-sphuta-siddhanta in 1132. This work is also known by the names Graha-ganita, Brahma-tulyanayana, Bhaumadi-panchagraha-nayana, Kshanika-grahanayana, or simply Ashadhara. Harihara wrote an extended version of the Graha-jnana around 1575 CE.^[6]

References[edit]

^ "Brahmagupta | Indian astronomer". Encyclopedia Britannica.
^ Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).
^ Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).
^ Henry Thomas Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London 1817, p. 339 (online)
^ Kaplan, Robert (1999). The Nothing that is: A Natural History of Zero. New York: Oxford University Press. pp. 68–75. ISBN 0-19-514237-3.
^ David Pingree, ed. (1970). Census of the Exact Sciences in Sanskrit Series A. Vol. 1. American Philosophical Society. p. 54.

External links[edit]

Brahmasphutasiddhanta at GRETIL (mathematical chapters: 12, 18-20, 21.17-23)
O'Connor, John J.; Robertson, Edmund F., "Brahmagupta", MacTutor History of Mathematics Archive, University of St Andrews

[1] "Brahmagupta | Indian astronomer". Encyclopedia Britannica.

[Bradley-2] Bradley, Michael. The Birth of Mathematics: Ancient Times to 1300, p. 86 (Infobase Publishing 2006).

[3] Mackenzie, Dana. The Universe in Zero Words: The Story of Mathematics as Told through Equations, p. 61 (Princeton University Press, 2012).

[4] Henry Thomas Colebrooke. Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London 1817, p. 339 (online)

[Kaplan-5] Kaplan, Robert (1999). The Nothing that is: A Natural History of Zero. New York: Oxford University Press. pp. 68–75. ISBN 0-19-514237-3.

[6] David Pingree, ed. (1970). Census of the Exact Sciences in Sanskrit Series A. Vol. 1. American Philosophical Society. p. 54.

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@@ Line 1: / Line 1: @@
 {{DISPLAYTITLE:''Brāhmasphuṭasiddhānta''}}
-The '''''Brāhmasphuṭasiddhānta''''' ("Correctly Established [[siddhanta|Doctrine]] of [[Brahma]]", abbreviated BSS)
+The '''''Brāhma-sphuṭa-siddhānta''''' ("Correctly Established [[siddhanta|Doctrine]] of [[Brahma]]", abbreviated BSS)
+is a main work of [[Brahmagupta]], written c. 628.<ref>{{cite web|title=Brahmagupta {{!}} Indian astronomer|url=https://www.britannica.com/biography/Brahmagupta|website=Encyclopedia Britannica|language=en}}</ref> This text of [[Theoretical astronomy|mathematical astronomy]] contains significant [[Mathematics|mathematical]] content, including the first good understanding of the role of [[zero]], rules for manipulating both [[negative and positive numbers]], a method for computing [[square root]]s, methods of solving [[linear equation|linear]] and [[quadratic equation]]s, and rules for summing [[series (mathematics)|series]], [[Brahmagupta's identity]], and [[Brahmagupta theorem]].
-is the main work of [[Brahmagupta]], written c. 628.{{cn|date=September 2014}}
-The text is notable for its mathematical content, as it  contains ideas including a good understanding of role of [[0 (number)|zero]], rules for manipulating both [[negative and positive numbers]], a method for computing [[square root]]s, methods of solving [[linear equation|linear]] and [[quadratic equation]]s, and rules for summing [[series (mathematics)|series]], [[Brahmagupta's identity]],  and  [[Brahmagupta’s theorem]].
-The book was written completely in verse and does not contain any kind of mathematical notation.  Nevertheless, it contained the first clear description of the [[quadratic formula]] (the solution of the quadratic equation).<ref name=Bradley>Bradley, Michael.  ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006).</ref><ref>Mackenzie, Dana.  ''The Universe in Zero Words: The Story of Mathematics as Told through Equations'', p. 61 (Princeton University Press, 2012).</ref>
+The book was written completely in verse and does not contain any kind of mathematical notation. Nevertheless, it contained the first clear description of the [[quadratic formula]] (the solution of the quadratic equation).<ref name=Bradley>Bradley, Michael.  ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006).</ref><ref>Mackenzie, Dana.  ''[https://books.google.com/books?id=2QYSAAAAQBAJ&q=Brahmagupta The Universe in Zero Words: The Story of Mathematics as Told through Equations]'', p. 61 (Princeton University Press, 2012).</ref>
-==Brahmasphuta-siddhantas rules for numbers==
+== Positive and negative numbers==
-''Brhmasphuta-siddhanta'' is one of the first mathematical books to provide concrete ideas on [[positive number]]s, [[negative number]]s, and zero. He wrote the following rules:<ref>[[Henry Thomas Colebrooke]]. ''Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara''. London 1817 ([https://archive.org/details/algebrawitharith00brahuoft online])</ref>
+''Brāhmasphuṭasiddhānta'' is one of the first books to provide concrete ideas on [[positive number]]s, [[negative number]]s, and zero.<ref>[[Henry Thomas Colebrooke]]. ''Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara'', London 1817, p. 339 ([https://archive.org/details/algebrawitharith00brahuoft online])</ref> For example, it notes that the sum of a positive number and a negative number is their difference or, if they are equal, zero; that subtracting a negative number is equivalent to adding a positive number; that the product of two negative numbers is positive. Some of the notions of fractions differ from the modern [[rational number]] system. For example, Brahmagupta allows [[division by zero]] resulting in a fraction with a {{math|0}} in the denominator, and defines {{math|1=0/0 = 0}}. In modern mathematics, division by zero is undefined for any [[field (mathematics)|field]].<ref name="Kaplan">{{cite book
+  | last = Kaplan | first = Robert | title = The Nothing that is: A Natural History of Zero
+  | url = https://archive.org/details/nothingthatisnat00kapl | url-access = registration | publisher = Oxford University Press | year = 1999 | location = New York
+  | pages = [https://archive.org/details/nothingthatisnat00kapl/page/68 68–75] | isbn = 0-19-514237-3}}
+</ref>
+== Influence ==
-* The [[addition|sum]] of two positive quantities is positive
-* The sum of two negative quantities is negative
-* The sum of zero and a negative number is negative
-* The sum of zero and a positive number is positive
-* The sum of zero and zero is zero
-* The sum of a positive and a negative is their difference; or, if they are equal, zero
-* In [[subtraction]], the less is to be taken from the greater, positive from positive
-* In subtraction, the less is to be taken from the greater, negative from negative
-* When the greater however, is subtracted from the less, the difference is [[additive inverse|reversed]]
-* When positive is to be subtracted from negative, and negative from positive, they must be added together
-* The [[multiplication|product]] of a negative quantity and a positive quantity is negative
-* The product of two negative quantities is positive
-* The product of two positive quantities is positive
-* Positive [[division (mathematics)|divided]] by positive or negative by negative is positive
-* Positive divided by negative is negative. Negative divided by positive is negative
-*A positive or negative number when [[Division by zero|divided by zero]] is a [[fraction (mathematics)|fraction]] with the zero as denominator
-* Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator
-* [[indeterminate form|Zero divided by zero]] is zero
+Ashadhara, the son of Rihluka, wrote ''Graha-jnana'' with tables based on ''Brahma-sphuta-siddhanta'' in 1132. This work is also known by the names ''Graha-ganita'', ''Brahma-tulyanayana'', ''Bhaumadi-panchagraha-nayana'', ''Kshanika-grahanayana'', or simply ''Ashadhara''. Harihara wrote an extended version of the ''Graha-jnana'' around 1575 CE.<ref>{{cite book |editor=David Pingree |editor-link=David Pingree |title=Census of the Exact Sciences in Sanskrit Series A |volume=1 |publisher=American Philosophical Society |year=1970 |page=54 |url=https://archive.org/details/PingreeCESS/Pingree_CESS_A1_1970/ }}</ref>
-The last of these rules is notable as the earliest attempt to define division by zero, even though it is not compatible with modern number theory (division by zero is undefined for a [[field (mathematics)|field]]).<ref name="Kaplan">{{cite book
-  | last = Kaplan | first = Robert | title = The nothing that is: A natural history of zero
-  | publisher = Oxford University Press | year = 1999 | location = New York
-  | pages = 68–75 | isbn = 0-19-514237-3}}
-</ref>
 ==References==
 {{reflist}}
-== External links ==
+==External links==
 {{wikisourcelang|sa|ब्रह्मस्फुटसिद्धान्त}}
 * [http://gretil.sub.uni-goettingen.de/gretil/1_sanskr/6_sastra/8_jyot/brsphutu.htm  ''Brahmasphutasiddhanta''] at [[GRETIL]] (mathematical chapters: 12, 18-20, 21.17-23)
@@ Line 43: / Line 25: @@
 {{Indian mathematics}}
+{{Indian astronomy}}
+{{Use dmy dates|date=December 2019}}
 {{DEFAULTSORT:Brahmasphutasiddhanta}}
 [[Category:Indian mathematics]]
 [[Category:Mathematics manuscripts]]
-[[Category:Sanskrit texts]]
+[[Category:7th-century Sanskrit literature]]
 [[Category:7th-century manuscripts]]
 [[Category:History of algebra]]
+[[Category:Treatises]]
+[[Category:Indian astronomy texts]]
+[[Category:Brahmagupta]]

v t e Indian astronomy
Astronomers	Apastamba Aryabhata I Aryabhata II Vainu Bappu Bhāskara I Bhāskara II Baudhayana Brahmagupta Sandip Chakrabarti Radhagobinda Chandra Subrahmanyan Chandrasekhar Amil Kumar Das P. Devadas Gautama Siddha M. K. Das Gupta Jyesthadeva Kamalakara Katyayana Lagadha Lalla Madhava of Sangamagrama Prasanta Chandra Mahalanobis Mahavira Manava Jayant Narlikar Parameshvara Achyuta Pisharati Venkatraman Radhakrishnan J. J. Rawal Megh Nad Saha Jagannatha Samrat Bapudeva Sastri Jai Singh II of Amber Nilakantha Somayaji Sripati Mahendra Suri Govind Swarup Utpala Varahamihira Shankara Variyar Vasistha Yajnavalkya Pathani Samanta
Works	Aryabhatiya Brahmanas Aitareya Brahmana Shatapatha Brahmana Bṛhat Saṃhitā Ganitagannadi Jyotirmimamsa Shulba Sutras Panchangam Khandakhadyaka Mahādevī Makarandasarini Siddhantas Brāhmasphuṭasiddhānta Maha-Siddhanta Pancha-Siddhantika Paulisa Siddhanta Paitamaha Siddhanta Rājamṛgāṅka (astronomy book) Romaka Siddhanta Surya Siddhanta Vasishtha Siddhanta Tantrasangraha Treatise on Astrology of the Kaiyuan Era Vākyakaraṇa Vedanga Jyotisha Vedas Atharvaveda Rigveda Yavanajataka Zij
Instruments	Armillary sphere Astrolabe Celestial globe Compass Cross-staff Dioptra Equatorial ring Equatorial sundial Globe Gnomon Mural instrument Quadrant Sextant Sundial Triquetrum Water clock Yantraraja
Concepts	Axial tilt Azimuth Cardinal direction Celestial equator Celestial sphere Celestial spheres Center of mass Circular orbit Deferent and epicycle Earth's rotation Eccentricity Ecliptic Elliptic orbit Equant Geocentrism Geoheliocentrism Globe Gravity Heliocentrism Hindu units of time Jyotiṣa Kali ahargaṇa Kalpa Karana Manvantara Moonlight Nakshatra Nityayoga Parallax Precession Ritu Spherical Earth Sunlight Tithi Vāra Yuga Zodiacal sign
Centres	Jantar Mantar Jaipur New Delhi Ujjain Varanasi Kerala school of astronomy and mathematics
Other regions	Babylonian astronomy Ancient Greek astronomy Hellenistic astronomy Islamic astronomy Chinese astronomy European astronomy
Months of the Vedic calendar	Chaitra Vaisakha Jyeshta Aashaadha Shraavana Bhadrapada Ashvin Kartik Margashirsha Pausha Maagha Phalguna