"Aryabhata: The Father of Indian Mathematics and His Enduring Legacy". Contribution in mathematics

 

    

Aryabhata

 

The father of Indian mathematics, Aryabhata, made a groundbreaking contribution to mathematics. Abstract The father of Indian mathematics, Aryabhata, was a well-known mathematician and astronomer from India. His excellent works, legacy, and inventions in various domains have earned him international recognition. During the ancient era of Indian mathematics and astronomy, he was the first mathematician and astronomer. Among his works are the now-lost Aryasiddhanta and the Aryabhatiya (c. 499), the principal surviving text from Aryabhata's classical works, which is composed of 118 lines that describe Hindu mathematics up to that point.He composed at least these two treatises in Kusumapura, which is close to Patalipurta (Patna), the Gupta dynasty's capital at the time. It covers topics including algebra, arithmetic, quadratic equations, and plane and spherical trigonometry. Aryabhata's explicit description of the relativity of motion makes him a famous pioneer physicist as well. The foundational concepts of trigonometry—cosine, sine, inverse sine, and versed sine—are also found in Aryabhata's work. He was also the first mathematician to arrive at a calculation using four decimal places of precision from 0 to 180 degrees using sine and versine tables. Patliputra served as the hub for network channel communications, allowing access to education from around the globe.

"We owe a lot to the ancient Indians teaching us how to count," is a good quote to begin the opening section of this survey work. This would not have been conceivable without the majority of contemporary scientific advances. One of the oldest civilizations in the world, according to Albert Einstein, is Indian civilization. Many people were unaware of this civilization because they were reluctant to acknowledge it appropriately for hundreds of years. It plays a significant and direct role in the development of mathematics in the modern period.On the surface, it is meant to highlight how context affects how the phrase "Indian Mathematics" is used. This phrase could potentially refer to mathematics that started in India and spread to Bangladesh, Nepal, Pakistan, and Sri Lanka. The Brahmans and other intellectuals were appointed as priests and formed a religion of learning. The trading group and the agricultural group were referred to as the Vaishyas, while another cult of soldiers and nobles was designated as the Kshatriyas. Additionally, the Shudra was used to adorn the craftspeople and workers. Thus, this was the basis for the division of Hindus into four varnas, or "classes." The caste system is unrelated to the terminology. In ancient India, religion served as the impetus for a wide range of movements.

Aryabhata's Foundational Works Aryabhata is supposed to have attended Nalanda University. He went on to become the department head. Math, astronomy, physics, medicine, biology, and other topics are among the subjects he has conducted research on at Nalanda. Nalanda provided him with his essential information source. His well-known works were based on earlier findings made by Greeks, Mesopotamians, and Nalanda University. In the Indian mathematical literature that has persisted to the present day, Aryabhatiya, a Compendium Commentary on Mathematics and Astronomy, was used as a source. The Aryabhatiya's mathematical portion details his thorough understanding of algebra, spherical trigonometry, plane trigonometry, and arithmetic. A table on sines, sums-of-power series, quadratic equations, and continuing fractions are also included.

Only the treatise known as Aryabhatiya is used to examine the empirical working characteristics of Aryabhata. It is referred to as Ashmakatantra, or the treatise from the Ashmaka, by his follower Bhaskara I. Because the text materials contain 108 verses, it is also occasionally referred to as Arya-shatas-aShTa. It is mentioned in the succinct way that is characteristic of Sutra literature. It is divided into the following four Padas, or Chapters. 1. Gitikapada, who composed thirteen pieces: Its different time units—kalpa, yuga, and manvantra—describe a different cosmology than that which has been previously articulated in works like Lagadha's Vedanga Jyotisha (c. 1st century BCE). A table on sines, or jya, written in single is also available.It is known that a mahayuga's planetary revolutions lasted 4.32 million years. 2. Ganitapada, which has 33 compositions, examines gnomon, or easy shadows, indeterminate equations, quadratic equations, simultaneous equations, and mensuration, or kṣetra vyavahara, arithmetic, and geometric progressions. 3. Kalakriyapada, who composed twenty-five pieces: It uses a seven-day week naming pattern for the days of a week, computations pertaining to the intercalary month, and a variety of time units and a mechanism for figuring out the locations of planets for a given day.

With twenty-five compositions, Kalakriyapada uses a seven-day week naming pattern for the days of a week, as well as different time units and a system for figuring out the placements of planets for a given day and computations pertaining to the intercalary month. 4. Golapada, which has 50 compositions, explores the ecliptic, node, celestial equator, geometric and trigonometric fields of the celestial sphere, earth's size and shape, day-and-night factors, the emergence of zodiacal signs on the horizon, and other properties of objects. Additionally, some editions include a colophon citation as an addendum at the conclusion that describes the works' attributes, etc.

In his work Aryabhatiya, he introduced some improvements in astronomy and mathematics in the form of commentary versions that have influenced the field's advancement for decades. He is well-versed in his explanation of relativity in motion. Taking note of Aryabhata's contemporary Varahamihira and later mathematicians Brahmagupta and Bhaskara I allows one to understand the "Arya-siddhanta," the lost works on astronomical computations. This ancient text seems to be based on the earlier "Surya Siddhanta," a Sanskrit synopsis of Greek and Mesopotamian astronomical and mathematical systems that applies by displaying the mid-night day counting rather than the dawn in Aryabhatiya.Al-nanf or Al-ntf is a third text that might have stood up in the Arabic translation. Although Aryabhata is credited with translating it, the Sanskrit title of this work was not yet known. Abu Rayhan al-Biruni, the Persian Scholar and Analyst of India, may have referenced it as early as the ninth century.

The Algebraic Works of Aryabhata Aryabhata was also given the title of "Father of Algebra" in addition to Al-Khwarizmi and Diophantus due to his remarkable comprehension and elucidation of planetary systems. His investigation of the astronomical problem of estimating the planets' periods led to the creation of these algebraic components. When 𝑎, 𝑏, and 𝑐 are integers, it describes integer solutions to equations of the type 𝑏𝑦 = 𝑎𝑥 + 𝑐 and 𝑏𝑦 = 𝑎𝑥 − 𝑐. To address issues of this nature, Aryabhata employs the "Kuttaka" algorithm. Kuttaka is believed to mean "to pulverise" and refers to the process of decomposing an issue into smaller ones, with the coefficients getting smaller and smaller with each stage.The method uses the Euclidean algorithm to determine the greatest common divisor of 𝑎 and 𝑏, and is linked to continuing fractions. In Aryabhatiya, Aryabhata presented beautiful findings for summation of series of squares and cubes, such as the following: 1^2 + 2 ^2 + ⋯ + 𝑛^2 =𝑛(𝑛+1)(2𝑛+1)/6

,1^3 + 2^3 + ⋯ + 𝑛^3 = (1 + 2 + 3 + ⋯ + 𝑛)^2

It is discovered that the formulas provided by Aryabhata for the areas of a triangle and a circle are accurate. For the sum of an arithmetic progression 𝑆𝑛, he presented a mathematical algorithm. Considering that the sum is obviously in the knowledge domain, he also developed another one for the number of terms 𝑛. He uses the mathematical link between the beginning term 𝑎 and the common difference 𝑑 of alternate terms to calculate the sum of an arithmetic progression.

The Works of Aryabhata on Indeterminate Equations There are two or more unknowns to solve for in second-degree or higher degree indeterminate equations. Diophantine systems of equations are polynomial expression equations with highly sought-after rational or integer solutions. Although these are somewhat similar, the phrase usually suggests that one needs integer solutions. There are more unknowns than equations in an indeterminate equation system. These unknowns are also susceptible to other restrictions, such as being rationals, integers, or positive integers, among others.Indeterminate equations are known to be non-uniquely solvable. It may even have an endless number of solutions in some situations. Finding integer solutions to Diophantine equations in the typical format of 𝑎𝑥 + 𝑏𝑦 = 𝑐 has been a challenge of great interest to Indian mathematicians from ancient times. The Chinese Remainder Theorem is a common name for the answer to this problem, which was also studied in classical Chinese mathematics.Diophantine equations can be famously challenging to solve in the real world. They were covered in great detail in the Sulba Sutras, an early Vedic literature. Its more classical elements may have been created as early as 800 BCE. The kuṭṭaka, which Bhaskara developed in 621 CE, is Aryabhata's technique for resolving these kinds of confusions. It uses a recursive procedure to write the original factors in smaller ones. In Indian mathematics, this process is transformed into the accepted technique for resolving first-order diophantine equations. Intriguingly, the entire field of algebra was formerly known as kuṭṭaka-gaṇita or kuṭṭaka. Even now, the Kuttaka method is regarded as the accepted approach for solving these kinds of equations.

      

Aryabhata

Aryabhata’s Legacy and Invention Aryabhata demonstrated the most exceptional imaginative approach. His work was widely applicable in Indian astronomy. It influenced adjacent cultures and civilizations through translational labor. Arabic translations during the Islamic Golden Age (c. 820 CE) were significant. Al-Khwarizmi references some of his results from the 10th century. According to Al-Biruni, followers of Aryabhata held the belief that the Earth rotated on its axis.Trigonometry was born as a result of his definitions of sine, cosine, versine, and inverse sine. Additionally, he was the first to give sine and versine (1−cos x) tables to up to four decimal places, with 3.75° intervals between 0° and 90°. The words jya and kojya, which were introduced by Aryabhata, are really mistranslated as "sine" and "cosine" in modern usage. As previously stated, Gerard of Cremona misinterpreted them when translating an Arabic geometry treatise into Latin after they were rendered as jiba and kojiba in Arabic. Since "fold in a garment" is the meaning of the Arabic term jaib, he thought that jiba was the same as L. sinus (c. 1150).The astronomical computation techniques of Aryabhata also had a significant influence. They were used extensively in the Islamic world in addition to the trigonometric tables, and they were used to calculate numerous Arabic astronomical tables (zijes). The most precise ephemeris used in Europe for centuries was the astronomical tables in the 11th-century work of the Arabic-Spanish astronomer Al Zarqali, which were translated into Latin as the Tables of Toledo in the 12th century.India has long used the calendric computations developed by Aryabhata and his adherents for practical purposes related to establishing the Panchangam, or Hindu calendar.They served as the foundation for the Jalali calendar in the Islamic world, which was created in 1073 CE by a group of astronomers that included Omar Khayyam. The compositions of this calendar, which were altered in 1925, are now the national calendars used in Iran and Afghanistan. Like the Aryabhata and earlier Siddhanta calendars, the Jalali calendar's dates are truly determined by the solar transition. An ephemeris is necessary for this kind of calendar in order to calculate dates. The Jalali calendar had fewer seasonal flaws than the Gregorian calendar, despite the fact that dates were challenging to calculate.Both the lunar crater and India's first satellite, Aryabhata, bear his name. Additionally, the Indian 2-rupee currency's reverse featured the Aryabhata satellite. Near Nainital, India, the Aryabhata Research Institute of Observational Sciences (ARIES) is a research institute for atmospheric sciences, astronomy, and astrophysics. In 2009, ISRO scientists identified a kind of bacterium called Bacillus aryabhata in the stratosphere. The interschool Aryabhata Mathematics Competition was also named after him.