Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Arya-siddhanta.
For his explicit mention of the relativity of motion, he also qualifies as a major early physicist.
The Gupta Empire was an Indian empire during the classical period of the Indian subcontinent which existed from the mid 3rd century to mid 6th century CE. At its zenith, the dynasty ruled over an empire that spanned much of the northern Indian subcontinent. This period has been considered as the Golden Age of India by some historians, although this characterisation has been disputed by others. The ruling dynasty of the empire was founded by Gupta.
The high points of this period are the great cultural developments which took place primarily during the reigns of Samudragupta, Chandragupta II and Kumaragupta I. Many Hindu epics and literary sources, such as the Mahabharata and Ramayana, were canonised during this period. The Gupta period produced scholars such as Kalidasa, Aryabhata, Varahamihira and Vatsyayana, who made significant advancements in many academic fields. Science and political administration reached new heights during the Gupta era. The period, sometimes described as Pax Gupta, gave rise to achievements in architecture, sculpture, and painting that "set standards of form and taste [that] determined the whole subsequent course of art, not only in India but far beyond her borders". Strong trade ties also made the region an important cultural centre and established the region as a base that would influence nearby kingdoms and regions in India and Southeast Asia. The Puranas, earlier long poems on a variety of subjects, are also thought to have been committed to written texts around this period. Hinduism was followed by the rulers and the Brahmins flourished in the Gupta empire but the Guptas were tolerant towards people of other faiths as well.
Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments of the period include extension of the place-value system to include decimal fractions, the systematised study of algebra and advances in geometry and trigonometry.
The medieval Islamic world underwent significant developments in mathematics. Muhammad ibn Musa al-Khwārizmī played a key role in this transformation, introducing algebra as a distinct field in the 9th century. Al-Khwārizmī's approach, departing from earlier arithmetical traditions, laid the groundwork for the arithmetization of algebra, influencing mathematical thought for an extended period. Successors like Al-Karaji expanded on his work, contributing to advancements in various mathematical domains. The practicality and broad applicability of these mathematical methods facilitated the dissemination of Arabic mathematics to the West, contributing substantially to the evolution of Western mathematics.
Patna (Hindi: Paṭanā, pronounced [ˈpəʈnaː] ), historically known as Pāṭaliputra, is the capital and largest city of the Indian state of Bihar. According to the United Nations, as of 2018, Patna had a population of 2.35 million, making it the 19th largest city in India. Covering 250 square kilometres (97 sq mi) and over 2.5 million people, its urban agglomeration is the 18th largest in India. Patna also serves as the seat of Patna High Court. The Buddhist, Hindu and Jain pilgrimage centres of Vaishali, Rajgir, Nalanda, Bodh Gaya and Pawapuri are nearby and Patna City is a sacred city for Sikhs as the tenth Sikh Guru, Guru Gobind Singh was born here. The modern city of Patna is mainly on the southern bank of the river Ganges. The city also straddles the rivers Son, Gandak and Punpun. The city is approximately 35 kilometres (22 mi) in length and 16 to 18 kilometres (9.9 to 11.2 mi) wide.
One of the oldest continuously inhabited places in the world, Patna was founded in 490 BCE by the king of Magadha. Ancient Patna, known as Pataliputra, was the capital of the Magadha Empire throughout the Haryanka, Nanda, Mauryan, Shunga, Gupta, and Pala dynasties. Pataliputra was a seat of learning and fine arts. It was home to many astronomers and scholars including Aryabhata, Vātsyāyana and Chanakya. During the Maurya period (around 300 BCE) its population was about 400,000. Patna served as the seat of power, and political and cultural centre of the Indian subcontinent during the Maurya and Gupta empires. With the fall of the Gupta Empire, Patna lost its glory. The British revived it again in the 17th century as a centre of international trade. Following the partition of Bengal presidency in 1912, Patna became the capital of Bihar and Orissa Province.
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, Varāhamihira, and Madhava. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometrywas further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.
Ancient and medieval Indian mathematical works, all composed in Sanskrit, usually consisted of a section of sutras in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered so important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical document produced on the Indian subcontinent is the birch bark Bakhshali Manuscript, discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan) and is likely from the 7th century CE.
In geometry, calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is where b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term "base" denotes any side, and "height" denotes the length of a perpendicular from the vertex opposite the base onto the line containing the base. Euclid proved that the area of a triangle is half that of a parallelogram with the same base and height in his book Elements in 300 BCE. In 499 CE Aryabhata, used this illustrated method in the Aryabhatiya (section 2.6).
Although simple, this formula is only useful if the height can be readily found, which is not always the case. For example, the land surveyor of a triangular field might find it relatively easy to measure the length of each side, but relatively difficult to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. Other frequently used formulas for the area of a triangle use trigonometry, side lengths (Heron's formula), vectors, coordinates, line integrals, Pick's theorem, or other properties.
Bhāskara I (c. 600 – c. 680) was a 7th-century Indian mathematician and astronomer who was the first to write numbers in the Hindu–Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work. This commentary, Āryabhaṭīyabhāṣya, written in 629, is among the oldest known prose works in Sanskrit on mathematics and astronomy. He also wrote two astronomical works in the line of Aryabhata's school: the Mahābhāskarīya ("Great Book of Bhāskara") and the Laghubhāskarīya ("Small Book of Bhāskara").
On 7 June 1979, the Indian Space Research Organisation launched the Bhāskara I satellite, named in honour of the mathematician.
