By Lt Gen SRR AIYENGAR PVSM AVSM VSM

“Vedic Mathematics is a treasure trove of techniques that simplify complex calculations and enhance mental agility." - Shakuntala Devi

"Vedic Mathematics exemplifies the harmony between mathematical precision and spiritual wisdom." - Harish Johari

Mathematics in the subcontinent boasts a rich and extensive history that spans over 5000 years, flourishing for centuries even prior to the advancements in Europe. Its influence extended far and wide, reaching China, Southeast Asia, the Middle East, and Europe. Indian mathematicians have left a lasting impact on various fields, introducing fundamental concepts such as zero and significantly shaping geometry, arithmetic, binary mathematics, negative numbers, algebra, trigonometry, and calculus, among others. Notably, India is attributed with the development of the decimal place value system, a contribution that has been embraced worldwide.

Known as Ganita, mathematics has been an integral component of Indian culture since ancient times. The early Indians developed a multitude of mathematical concepts while addressing real-life challenges that emerged in their daily existence. For instance, within the realm of the Vedangas, Jyotisha focused on employing mathematical principles, measurements, and approximation techniques to predict the movements of celestial bodies in the heavens. This necessity persisted from ancient times through the post-Vedic era, forming an unbroken tradition that yielded numerous invaluable contributions to the realm of mathematics.

Within the vibrant tapestry of ancient Indian knowledge systems, a captivating journey unfolds—a journey that intertwines seamlessly with the realm of mathematics. Encompassing millennia of exploration and inquiry, Indian scholars embarked on a profound tryst with mathematical concepts, enriching the intellectual landscape and contributing lasting insights to the world. The subsequent paragraphs briefly explore some of the distinctive features of Indian mathematics.

Indian Numeral System -A Lasting Legacy

The Indian numeral system stands as a remarkable and foundational contribution to both mathematics and ancient knowledge systems. With origins dating back to antiquity, this decimal system brought about a revolutionary transformation in the representation and calculation of numbers, thereby shaping the contemporary understanding of mathematics across the globe.

At the core of the Indian numeral system lies the pivotal concept of "zero." In contrast to earlier numeral systems that lacked or only partially grasped the notion of zero, Indian mathematicians introduced a symbol to denote zero, representing a vacant place value. This seemingly elementary addition carried profound consequences, enabling more efficient calculations and laying the essential groundwork for the emergence of sophisticated mathematical concepts.

The foundation of the Indian decimal system is rooted in a place value notation, wherein a digit's value hinges on its position within the numeral. The value of each digit is magnified tenfold in relation to its rightward neighbour, yielding an uncomplicated and versatile means of expressing numbers. The strategic use of zero as a placeholder amid significant digits facilitates the smooth representation of vast numerical quantities.

Empowered by this numeral system and its associated place value doctrine, intricate arithmetic operations, spanning addition, subtraction, multiplication, and division, became attainable. Indian mathematicians ingeniously devised algorithms and methodologies to tackle practical challenges, profoundly aiding diverse fields such as astronomy, trade, and engineering.

The impact of the Indian numeral system transcended geographical borders. Through trade and the exchange of cultures, its influence extended to various corners of the world, encompassing China, the Middle East, and Europe. Arab scholars played a pivotal role in disseminating this knowledge to the Western sphere, eventually labelling it as "Arabic numerals."

The introduction of the Indian numeral system marked a pivotal juncture in the annals of mathematical history, leaving an indelible imprint on science, technology, and commerce. Its enduring legacy stands manifest in its continued adoption as the universal standard numeral system, rendering it an integral part of our day-to-day lives and a testament to the inventive prowess of ancient Indian mathematicians.

Trigonometry

Trigonometry occupies a significant and distinguished position within the achievements of ancient Indian mathematicians, effectively showcasing their profound comprehension of mathematical concepts and their practical applications. Driven by the imperative to unravel real-world predicaments, these mathematicians contributed notably to trigonometry, solidifying it as an integral facet of their mathematical legacy. Trigonometry, a pivotal branch of mathematics focused on deciphering the relationships between the angles and sides of triangles, boasts deep-seated origins within ancient Indian knowledge systems. Remarkably, Indian mathematicians and astronomers augmented the understanding and practical utility of trigonometry through their considerable contributions.

The earliest references to trigonometry within Indian texts are discernible in the "Sulba Sutras," a component of the ancient Vedic texts recognized as the "Shrauta Sutras." Originating approximately in the 800s BCE, these texts proffer geometric guidelines for erecting sacrificial altars, encompassing an understanding of right-angled triangles and their attributes.

Nonetheless, trigonometry encountered substantial expansion during subsequent eras, particularly within the classical epoch of Indian mathematics (roughly spanning the 5th to 12th centuries CE). Eminent Indian mathematicians such as Aryabhata, Bhaskara I, and Bhaskara II, engendered remarkable progress within the realm of trigonometry through their scholarly endeavours.

Aryabhata, a venerable mathematician and astronomer from the 5th century CE, introduced the notion of trigonometric functions and defined sine (jya), cosine (kojya), and versine (utkrama-jya) within his seminal opus, the "Aryabhatiya." His contributions also encompassed furnishing tables of values for these trigonometric functions.

In the 7th century CE, Bhaskara I scrutinized the lengths of shadows cast by objects at various times of the day, entailing the application of trigonometric principles. Subsequently, Bhaskara II, also identified as Bhaskaracharya, propelled trigonometry forward during the 12th century CE. His magnum opus "Lilavati" encompasses a rich repository of trigonometric outcomes and predicaments. Among the concepts he explored were the properties of chords (jya), half-chords (ardha-jya), and versines (utkrama-jya).

In their quest for precise astronomical calculations, Indian astronomers extensively harnessed trigonometry. This mathematical framework was instrumental in predicting planetary positions, eclipses, and the trajectories of celestial entities.

These historical contributions in the domain of trigonometry laid a sturdy foundation for subsequent advancements, both within India and beyond its borders. Throughout the medieval period, Arab scholars, particularly during the Islamic Golden Age, facilitated the translation and widespread propagation of Indian mathematical works, including trigonometry, to diverse regions. Ultimately, this knowledge traversed continents, significantly influencing the European Renaissance and catalysing the progression of trigonometry in Western mathematics.

Algebra

Algebra occupies a significant and venerable position within the legacy of ancient Indian mathematicians, underscoring their advanced grasp of mathematical concepts and their pragmatic applications. These mathematicians, pioneers in their field, made seminal strides in algebra, a cornerstone that profoundly shaped their mathematical heritage.

Algebra, the branch of mathematics concerned with symbols and the rules governing their manipulation to solve equations and analyse mathematical structures, boasts a rich lineage within ancient Indian knowledge systems. Indian mathematicians left an indelible mark on algebra, paving the path for its evolution and widespread utilization across diverse domains.

The origins of algebraic thought in ancient India trace back to the Vedic era, spanning approximately from 1500 to 500 BCE. Notably, Vedic texts harbour early inklings of algebraic notions, especially evident in contexts pertaining to the resolution of linear equations and predicaments intertwined with ritualistic practices.

However, it was during the classical epoch of Indian mathematics, encompassing the period from the 5th to the 12th centuries CE, that algebra flourished into an autonomous discipline. Distinguished Indian mathematicians such as Brahmagupta and Bhaskara II played pivotal roles in nurturing and expanding the frontiers of algebra through their scholarly contributions.

Brahmagupta, a luminary mathematician and astronomer hailing from the 7th century CE, is acclaimed for formalizing algebraic notation. He introduced rules governing arithmetic operations involving positive and negative numbers, in addition to devising methodologies for solving linear and quadratic equations. Brahmagupta's opus, particularly his treatise "Brahmasphutasiddhanta," emerged as a cornerstone in the advancement of algebraic ideas.

Continuing this tradition of algebraic innovation, Bhaskara II, renowned as Bhaskaracharya, propelled algebraic insights into the 12th century CE. Within his works "Lilavati" and "Bijaganita," he adeptly navigated through an array of algebraic techniques, encompassing the resolution of indeterminate equations, quadratic equations, and cubic equations. Bhaskara II also engineered methodologies to tackle problems associated with planetary positioning.

Significantly, the application of algebraic techniques found another vital domain within ancient Indian astronomy. Algebraic equations and methodologies were harnessed by Indian astronomers to orchestrate meticulous computations concerning planetary trajectories, eclipses, and various celestial occurrences.

Moreover, the notion of "Bija," meaning "seed" in ancient Indian mathematical parlance, resonates with modern-day algebraic variables. This conceptual framework empowered mathematicians to grapple with unknown quantities and artfully encapsulate relationships between them symbolically.

The indelible contributions of ancient Indian mathematicians to the realm of algebra laid a bedrock for subsequent advancements. In the medieval era, Indian algebraic treatises traversed borders, finding resonance within the Islamic realm, profoundly influencing Islamic mathematics. This transmission eventually found its way to Europe during the Islamic Golden Age, exerting a momentous impact on the trajectory of algebra's development in Western mathematical thought.

Computation of Square and Cube roots

Ancient Indian knowledge systems propelled substantial strides in comprehending and calculating square and cube roots. Indian mathematicians devised intricate methodologies to approximate square and cube roots, serving as bedrock for their pragmatic integration across diverse practical domains.

Among the earliest documented techniques for approximating square roots within ancient India, the "Bakhshali method" emerges prominently. Preserved within the "Bakhshali Manuscript," dating back approximately to the 3rd and 4th centuries CE, this method embraces a recursive algorithm for estimating square roots, bearing resemblance to the modern "Babylonian method" or "Heron's method."

The "Brahmagupta's Formula," conceived by the 7th-century luminary mathematician Brahmagupta, extended a pathway for computing square roots of perfect squares. This formula posits that the square root of a perfect square equates to half the number's value multiplied by the square root of the nearest preceding perfect square.

Extending their expertise beyond square roots, Indian mathematicians embarked upon the exploration of cube root methods. The "Chakravala method," devised by Bhaskara II, also known as Bhaskaracharya, an eminent mathematician from the 12th century, manifested as an algorithmic strategy for resolving indeterminate quadratic equations. While not intrinsically linked to cube roots, this method underscored the advanced algebraic techniques harnessed by Indian mathematicians during that era.

These mathematicians carved a pioneering niche within the realm of indeterminate equations, delving into the quest for integers satisfying specific algebraic expressions. While not serving as direct avenues for calculating cube roots, these inquiries contributed to a broader grasp of algebraic manipulations, subsequently paving the way for more intricate root-finding methodologies.

The acumen surrounding square and cube roots in ancient India yielded pragmatic applications spanning astronomy, architectural pursuits, and commercial endeavours. Precise computations of these roots assumed paramount importance in exact measurements, construction undertakings, and economic transactions.

The triumphs of ancient Indian mathematicians within the realm of square and cube roots radiated their influence, transcending borders through trade and cultural interactions. As a result, the methodologies and insights incubated within ancient India reverberated on the global stage, leaving an indelible imprint on the evolution of mathematics worldwide.

Ancient Indian Mathematicians and their Salient contributions

Few of the well-known mathematicians of ancient times in the BCE era are given below. This does not include many famous mathematicians during the CE era: 

• Vedic Texts. 3000BCE or earlier-Number systems, Decimal system, concept of Infinity.
• Sulba-sutras. 800-600 BCE. Earlier texts of Geometry, Exact procedures for the construction and transformation of squares, rectangles etc.
• Pingala-300 BCE. Binary sequences.
• Jaina Mathematical Works. 200BCE to 300CE. Concept of logarithms, Decimal system, Approximation of π.
• Bauddha Mathematical Works. About 500 BCE to500 CE. Multi- valued logic. Discussion of Indeterminate and Infinite numbers.
• Aryabhata. 476-550 CE. Algorithm for square root, cube root, quadratic equations.
• Bhaskara1. 600-680 CE. Expanded Aryabhata’s work.
• Other reputed mathematicians include Aryabhata, Bhaskara I &II, Brahmagupta, Virahanka, Sriharacharya, Madhava of Sangamagrama (founder of Kerala School of Mathematics) and many more. For more details, please refer to the Table 8.2 at pages 195-196 of “Introduction to Indian Knowledge System-Concepts and Applications” by Profs B. Mahadevan, Vinayak Rajat Bhat & Nagendra Pavana R.N.

These are just a few examples of the rich mathematical heritage of ancient Indian knowledge systems. Ancient Indians developed several concepts of mathematics primarily because they needed to solve a lot of real-life problems that they were facing even during the Vedic period. There has been an uninterrupted tradition of mathematical thinking and it has widely spread across the length and breadth of India.