Indian Mathematics Research Articles

From Concrete to Abstract in Indian Mathematics

Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the historical conditions under which it originated and evolved has not been studied much. The focus has been more on achievements than on how they developed. One also tends to read the ancient texts with the present concepts and methods in mind. The absence of writing over a long stretch of Indian history too gets overlooked in such readings. The purpose of this article is to explore the journey of mathematics by examining what the ancient texts on arithmetic, geometry and algebra tell us about the nature of mathematics in their times. These investigations reveal that over a period of a thousand or more years Indian mathematics transitioned from concrete and context-bound phase to context-free, abstract phase accompanied by several conceptual leaps. Invention of writing in the 3rd century BCE greatly facilitated this transition.

Contribution of Sanskrit in Mathematics and Astronomy

यथा शिखा मयूराणाां नागाणाां मणयो यथा। तथा वेदाङ्गशास्त्राणां ज्योतिषं मूर्धनिस्थितम् ।। “Like the combs of the peacocks and the crest-jewel of the serpents, the science of, Jyotiṣa is situated at the head of all the sciences forming the auxillaries of the vedās (Vedāṇgas)” says Lagadha in Vedāṅga Jyotiṣa . Introduction Sanskrit’s contributions to the field of Mathematics and Astronomy are well known. Both Mathematics and Astronomy has played a vital role in all aspects of life in the Indian society. ‘Jyotiṣam’ an auxiliary of Veda(s), is a branch of study that includes Mathematics, Astronomy and Astrology. Mathematics forms the base for the study of Celestial bodies. The study of mathematics was always systematic in the Indian tradition. This paper is an attempt to summarize briefly the notable contributions of Indian Mathematicians / Astronomers.

Design and Performance Evaluation of Brent Kung Adder based 8-Bit Vedic Multiplier

Multiplier is an essential functional block of a microprocessor because multiplication is needed to be performed repeatedly in almost all scientific calculations. Therefore, design of fast and low power binary multiplier is very important particularly for Digital Signal Processors. Vedic mathematics has improved the performance of multiplier. Vedic mathematics, a system of ancient Indian mathematics, which has a unique technique of solutions based on only 16 sutras. The novel point is the efficient use of Vedic algorithm (sutras) that reduces the number of computational steps considerably compared with any conventional method. This paper presents design and Performance Evaluation of Brent Kung Adder based 8-Bit Vedic Multiplier. Urdhva Tiryagbhyam sutra has been used for multiplication purpose. The partial product addition in Vedic multiplier is realized using Brent Kung Adder. Simulation results shows that described Brent Kung Adder based 8-Bit Vedic Multiplier is efficiently decreases the Delay, power consumption and Area than other multipliers.

AN INVESTIGATION ON CONTRIBUTION OF MATHEMATICIANS IN DEVELOPMENT OF MATHEMATICS

Mathematics is an international subject, and it has no geographical boundary for its development. Many countries have contributed for the development of the discipline of mathematics. The development of mathematical concepts started dates back to more than thousand years. For the development of the discipline mathematics, many countries mathematicians have contributed such as the Babylonian, Egyptian, Greek, Chinese and Indians etc. The concept of mathematics have come to us only for the need of calculations in daily life and commercial activities, measurement of land around for crop and to predict astronomical events happening around us by observing day and night. Therefore, mathematics is a science, because it is based on observation and recorded systematically in logical order. The contribution of Indian mathematicians in the field of mathematics is far more than western mathematicians. Hence, this study is suggesting that, there is a need to introduce among us a good number of Indian Mathematician contributions to develop positive attitude among the learners and to appreciate the mathematics subject and contribution of Indian mathematicians in the field of mathematics. Ancient Indian mathematicians have contributed immensely to the field of mathematics. The invention of zero is attributed to Indians and this contribution outweighs all other made by any other nation since it is the basis of the decimal number system, without which no advancement in mathematics would have been possible. The number system used today was invented by Indians and it is still called Indo-Arabic numerals because Indians invented them and the Arab merchants took them to the western world. It is also noticed that there is a distinct and inequitable neglect off the contributions of the sub-continent. Many of the developments of Indian mathematics remain almost completely ignored, or worse or attributed to the scholars of other nationalities, often European. However a few historians (mainly European) are reluctant to acknowledge the contributions of Indian mathematicians. Some of the great mathematicians are Pythagoras, Fibonacci, Euclid, Archimedes, Rene Descartes, and Aryabhata, who have contributed significant inputs in the field of mathematics in ancient times.

KaTaPaYa Saankhya Sutra Extended as an Encoding Scheme for Secure Communication in IoT Devices

Modern computers linked to IoT devices constantly produce vast amounts of data and information. Information security is crucial in securing this ever-generating vast volume of data. Modern encryption algorithms significantly enlarge files. Therefore, more bandwidth is needed for safe data transfer. Transforming a message into codes is the act of encoding. The study of ancient Vedic encoding techniques is applied to the contemporary period in the current work. Most of the Vedic literature is in Sanskrit through Poetry rather than narration. This Sanskrit poetry of the Vedic and its contemporary periods not only laud the Gods but also includes numerical codes. The Katapayadi technique of writing numbers in Sanskrit words was used in ancient India. Numerous ancient Indian mathematicians employed this and other similar systems. This tries to reduce the file size while simultaneously encrypting the content using this antique encryption method via Sanskrit as the medium.In this paper "Katapayadi Sankhya Sutra" (KTPY Rule) was studied and implemented. The "Katapayadi Sankhya Sutra - Extended" (KTPY-E Rule) is the improvement suggested for this KTPY Rule. An effective encryption and decryption approach was proposed using this KTPY-E Rule. In the suggested KTPY-E technique, English text is converted to the Unicode code KTPY encoded form, and then it is stored in its Binary Format indexed by its initial character set.

Evolution and Significance of Sine Function in Ancient Indian Mathematics

This research article delves into the historical and mathematical intricacies of the construction of sine functions, particularly focusing on ancient Indian mathematical concepts and their evolution. It begins by highlighting the significance of sine (jya) as a fundamental trigonometric function in Indian mathematics, alongside its synonyms in Sanskrit. The historical journey of the term “sine” is traced back to ancient Indian mathematical texts, which were later translated into Arabic and then into Latin, ultimately leading to the adoption of the term “sine” in modern mathematics. The article elaborates on the definitions and interpretations of sine (ज्या), cosine (कोटिज्या), versed sine (उत्क्रमज्या), and co-versed sine (कोट्युत्क्रमज्या) in Indian mathematical traditions. It explains how these functions were conceptualized and utilized in astronomical calculations, emphasizing their geometric and trigonometric significance. Furthermore, the article discusses the division of a circle into quadrants (Vṛttapādas) and the concept of Trijyā (radius), highlighting the mathematical sophistication and terminology used by ancient Indian mathematicians. Several mathematical formulas related to the construction and calculation of sine functions are presented, including formulas for finding sine values at specific angles such as 45 degrees, 18 degrees and 72 degrees. These formulas showcase the mathematical ingenuity and problem-solving skills of ancient Indian mathematicians. Overall, this research article provides a comprehensive overview of the historical development and mathematical foundations of sine functions in ancient Indian mathematics, shedding light on their enduring legacy and contributions to modern trigonometry.

Translating Indian Algebra in the Seventeenth Century: ʿAtāʾollāh Roshdi’s Persian Version of Bhāskara II’s Bījagaṇita

Abstract One of the most important works in the field of medieval Indian mathematics corresponding to algebra is Bījagaṇita (Seed Calculation), a Sanskrit treatise by the famous mathematician and astronomer Bhāskara II (or Bhāskarācārya, c. 1114–aft. 1183). This paper is a general overview of the Tarjoma-ye Bij ganet (The Translation of the Bījagaṇita, 1634), a Persian text composed by the mathematician ʿAtāʾollāh Roshdi (or Rashidi), who served at the court of the Mughal emperor Shāh Jahān (r. 1628–58). The translation is divided into an introduction, subdivided in six chapters (bāb), and five books (maqāla). After discussing both Bhāskara II and Roshdi, I compare the latter’s translation with the original Sanskrit text on the basis of available English translations. I will also demonstrate how an intermediate language, which played an important role in the connection of South Asian scientific sources, facilitated this translation.

A Comparative Study of Geometric Principles in the Sulba Sutras and the Pythagorean Theorem: Historical Context and Mathematical Applications

This research delves into the geometric principles detailed in the ancient Sulba Sutras and their parallels with the Pythagorean Theorem, traditionally attributed to the Greek mathematician Pythagoras. While the Pythagorean theorem is widely regarded as a cornerstone of Greek geometry, recent studies suggest that its formulation appeared centuries earlier in Indian mathematics, as evidenced in Baudhāyana’s Sulba Sutra (circa 800 BCE). These texts offer extensive insights into the geometric constructions used in Vedic rituals, including the use of Pythagorean triples and transformations between geometric shapes. The Sulba Sutras present a rich mathematical framework, emphasizing practical applications in altar construction, land measurements, and urban planning. This study aims to bridge the research gap by comparing the geometric developments in the Sulba Sutras with Greek formulations, particularly focusing on their mathematical methods, cultural contexts, and historical significance. The findings underscore the advanced state of Indian mathematics and its profound influence on subsequent mathematical traditions, offering a nuanced understanding of the parallel evolution of geometric knowledge across cultures.

Permutations and Oblong Numbers in the Theravāda-vinaya: A New Intersection of Buddhism and Indian Mathematics

Within the context of Indian religions, Jainism has long been recognized for its extensive use of permutations and combinations. However, the application of these principles within Buddhist scriptures has received relatively little scholarly attention. This paper introduces a new example of the specific application of permutations and combinations in Buddhist scriptures. In this paper, we focus on the first saṅghādisesa rule in the Theravāda-vinaya, which lists a series of element sets and arranges these elements according to a certain pattern known as “ten-roots” (mūla), and we discover that these arrangements form a regular numerical sequence, called “oblong numbers”. Moreover, similar patterns with different quantities are also found in the fourth Pārājika and the fifth saṅghādisesa rules. This indicates that the compilers of the Theravāda-vinaya did not use this mathematical knowledge without basis. Interestingly, we also found the use of this sequence in the Bakhshālī manuscript. Therefore, in this article, after summarizing and verifying the arrangement rules of the Theravāda-vinaya, we discuss whether the oblong numbers were influenced by Greek mathematics.

МАТЕМАТИКАНЫ ЖАНА АЗЫРКЫ ЗАМАНДЫ ӨНҮКТҮРҮҮ БОЮНЧА КЫРГЫЗ ЫКМАСЫ

The educational process consists of two main types of activities: the teaching activity of the teacher in educating students in knowledge, skills, and life experience, and the learning activity of the student in mastering the content taught by the teacher. Defining the learning process, teaching guides and determines learning. Therefore, the learning process is often explained as the process of educating the student. As a result, the learning process is perceived as a mechanism for the "transmission" of teaching content from teacher to student, with the teacher directly shaping the student. It formalizes the algorithm for solving problems − the problem-solving algorithm using the method of Diophantus al-Khwarizmi. This article presented a solution to text problems in algebra for grade 7 using the Kyrgyz method and a system of linear equations. It should be noted that this type of thinking was successfully used by many ancient Egyptian, Babylonian, Chinese and Indian mathematicians. The generalization of the described method can be successfully used to solve various problems.

A Review of Brahmagupta's Contributions

The life, accomplishments, and legacy of prominent Indian mathematician and astronomer Brahmagupta from the 7th century are examined in this review. Brahmagupta, who is well-known for his contributions to algebra, arithmetic, and astronomy, provided ground-breaking ideas in his "Brāhmasphuṭasiddhānta,"a foundational work that included the rules for mathematical operations involving zero and negative integers. His efforts established the groundwork for contemporary mathematics, inspiring other researchers in mediaeval Europe as well as the Islamic world. In order to emphasize Brahmagupta's methods, his contribution to the advancement of mathematical thought, and the ongoing influence of his work on current mathematical practices, this review synthesizes historical accounts and contemporary interpretations.

Extension of Vedic mathematics sutras for square calculation

Abstract Square is one of the basic arithmetic operations. Referring to the earlier contributions provided by ancient Indian mathematicians concerning square calculation, various techniques have been suggested by Arybhata, Bhaskara I, Sridhara, Mahavira and Bhaskara II to quote a few. Besides, P. Swami Bharati Krishna Tirthji discussed various techniques for square calculating based on various sutras. In this paper, we summarise the works done by ancient Indian mathematicians and discuss the applications of calculating squares using different sutras. In addition, we propose a new method of calculating square based on the drawbacks identified and that can be applied for the square calculation of any number. Our method splits the number and hence makes the calculation simpler. Moreover, for bigger values, nested calculations can also be applied.

Evolution and Significance of Sine Functions in Ancient Indian Mathematics

Evolution and Significance of Sine Functions in Ancient Indian Mathematics

Extending the Aryabhatan algorithm for the extraction of higher order roots

In schools, children are taught algorithm for extracting the square and cube roots of a number. However, they are not informed that Aryabhata (b. 476 CE) was the first Indian mathematician to present a systematic algorithm for the extraction of the square and cube root of a number. This algorithm has been succinctly presented by Aryabhata in two successive verses in the Ganitapada of his Aryabhatiyam composed in 499 CE. A careful study of the algorithm given by Aryabhata shows that it can be easily extended to find higher order roots. It also clearly reveals that the decimal place value system should have been in vogue in India for quite a long time before him, because this algorithm is heavily based on a thorough understanding of the decimal place value system. In our paper, we will discuss some of these aspects as well as outline how his algorithm can be extended to find the higher order roots.

A Biography of Sridharacharya, an Ancient Indian Mathematician

A Biography of Sridharacharya, an Ancient Indian Mathematician

МАТЕМАТИЧНА ПОЕЗIЯ ТРАГIЧНОЇ НЕСКIНЧЕННОСТI

The article, very briefly, tells about the life path and scientific achievements of the Indian mathematician Srinivasa Ramanujan Iyengar, who made a significant contribution to number theory, mathematical analysis and other areas of mathematics. With no formal training in mathematics, he taught himself the discipline, relying on limited resources and his own intuition. He formulated many theorems, made assumptions that were later proved by other mathematicians, and developed new research methods. His works had a significant impact on the development of mathematics in the 20th century.

Aryabhata-I (476-550 CE) : The great indian mathematician and astronomer of early India

Aryabhata-I (476-550 CE) : The great indian mathematician and astronomer of early India

A STUDY ON MATHEMATICIANS OF ANCIENT INDIA AND THEIR CONTRIBUTIONS

Indian mathematics has its roots in Vedic literature. It is imperative to know about the ancient, medieval and modern time Indian mathematicians and their contribution in the field of Mathematics. Indian mathematicians made significant early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed here. Ancient Indian mathematicians have contributed immensely to the field of mathematics. The invention of zero is attributed to Indians and this contribution outweighs all other made by any other nation since it is the basis of the decimal number system, without which no advancement in mathematics would have been possible. The number system used today was invented by Indians and it is still called Indo-Arabic numerals because Indians invented them and the Arab merchants took them to the western world. It is also noticed that there is a distinct and inequitable neglect off the contributions of the sub-continent. Many of the developments of Indian mathematics remain almost completely ignored, or worse or attributed to the scholars of other nationalities, often European. However a few historians (mainly European) are reluctant to acknowledge the contributions of Indian mathematicians. They believe Indians borrowed the knowledge of mathematics from Greeks. Even after all those criticisms it is a great pleasure for us to have been born in such a country where so many great mathematicians like Aryabhata, Brahmagupta, Srinivasa Ramanujan, P.C. Mahalanobis , Calyampudi Radhakrishna Rao, D.R. Kaprekar, Satyendra Bose ,Bhaskara I, Mahavira, Bhaskara II and many others born.

Ancient Indian mathematics needs an honorific place in modern mathematics celebration

Ancient Indian mathematics needs an honorific place in modern mathematics celebration

CONTRIBUTION OF ANCIENT INDIAN MATHEMATICIANS

The history of any ancient civilization is often surrounded by mystery and controversy. Still, Western scholars have not given sufficient credit to the work of some ancient Indian mathematicians, notably Brahmagupta, Bodhayana and Bhaskara II, despite the fact that their entire works were translated into English by the nineteenth-century British Sanskrit scholar Henry Thomas Colebrooke. In the initial part was done by The purpose of this article is to lay out the facts and leave the reader free to interpret and evaluate them according to his own understanding of the subject. Sincere efforts have been made to bring the contribution of some ancient Indian mathematicians to the public. Keywords:- Right angled triangle, Decimal system, Alphanumeric system, Shulva Sutra.