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Evergreen Journey of Mathematics: From Stone Age to AI

 Evergreen Journey of Mathematics: From Stone Age to AI

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Chapter 3 

Ancient Indian Mathematics


Indian mathematics has a remarkable contributions that revolutionized numerical notations and concepts. The development of numeral systems, zero, decimal notation, and the concept of infinity stands as a testament to the brilliance of ancient Indian mathematicians. Let's embark on a journey through time to explore these fascinating advancements.

The mathematicians of ancient India made profound discoveries and laid the foundation for mathematical knowledge that continues to influence modern mathematics. In this article, we will explore the key achievements and contributions of Indian mathematics during this era.

Famous Indian Mathematicians

  1. Numerical Notation and Place Value System: One of the remarkable achievements of Indian mathematicians was the development of a decimal place value system. This system was revolutionary and laid the foundation for modern numerical notation. Key features include:
  • Decimal System: Indian mathematicians introduced the concept of zero, which became a fundamental concept in arithmetic and mathematics. This innovation facilitated efficient calculations and the representation of large numbers.
  • Place Value System: The use of positional notation allowed numbers to be represented based on their place value, with each position representing a different power of ten. This system greatly simplified arithmetic operations and expanded the range of numbers that could be represented.
  1. Arithmetic: Indian mathematicians made significant contributions to arithmetic, developing sophisticated methods and algorithms. Some noteworthy achievements include:
  • Calculation Techniques: Indian mathematicians devised various calculation techniques, including efficient methods for addition, subtraction, multiplication, and division. They also explored advanced concepts like square roots and cube roots.
  • Algebraic Concepts: Indian mathematicians developed algebraic techniques for solving linear and quadratic equations. They introduced symbolic notation for unknowns and developed systematic approaches to solving equations.
  • Numerical Series: Mathematicians in ancient India were particularly interested in numerical series and patterns. They explored sequences such as arithmetic progressions, geometric progressions, and factorial series. These investigations formed the basis for later work on infinite series and mathematical induction.
  • Indian numerals' chronology
  1. Geometry: Geometry also held a prominent place in Indian mathematics. Indian mathematicians made significant contributions to geometric principles and calculations. Some notable achievements include:
  • Measurement of Angles: Ancient Indian mathematicians introduced precise methods for measuring angles using circular arcs and trigonometric principles. They developed trigonometric tables and relationships between angles, chords, and arc lengths.
  • Geometrical Constructions: Indian mathematicians developed geometric construction techniques, such as constructing regular polygons, tangents, and perpendiculars. These methods were essential in architectural design and town planning.
  • Geometric Proofs: Indian mathematicians emphasized geometric proofs and the application of deductive reasoning. They provided rigorous justifications for theorems and geometric properties.

  • Indian work on Geometry
  1. Astronomy and Trigonometry: Indian mathematics intertwined with astronomy, leading to advancements in trigonometry. Key achievements include:
  • Trigonometric Functions: Indian mathematicians developed a rich set of trigonometric functions, including sine, cosine, tangent, and their inverse functions. These functions were crucial for astronomical calculations, such as determining planetary positions and eclipses.
  • Trigonometric Tables: Indian mathematicians constructed extensive tables of trigonometric values, facilitating precise calculations. These tables played a vital role in navigational techniques and celestial observations.
  • Spherical Trigonometry: Indian mathematicians also explored spherical trigonometry, which deals with calculations on the surface of a sphere. They developed formulas and techniques for solving problems related to celestial spheres.
  1. Numeral Systems: The foundation of numerical notations can be traced back to ancient India, where various numeral systems emerged. The most significant among them is the decimal system, which originated in India around the 5th century AD. Key features include:
  • Place Value System: Indian mathematicians recognized the significance of position in numerical representation. Each digit's value depended on its position, multiplying by a power of ten. This positional notation allowed for efficient representation of large numbers and precise calculations.
  • Base-10 System: The decimal system, also known as the base-10 system, utilizes ten digits (0 to 9) to represent all numbers. Each digit's value is determined by its position, from the units place to the highest power of ten. This system's elegance and simplicity have made it the most widely used numeral system globally.
  • Achariya Pingala and Fibonacci Pattern
  1. Zero: The introduction of zero as a distinct numeral was a groundbreaking achievement in Indian mathematics. The concept of zero revolutionized numerical notations and calculations. Key points include:
  • Placeholder for Nothingness: Indian mathematicians realized the need for a symbol to represent the absence of a value or an empty place. The introduction of zero as a placeholder transformed arithmetic and opened the door to more complex calculations.
  • Indian Origin and Diffusion: The concept of zero emerged in ancient India and subsequently spread to other civilizations, including the Arab world, where it gained further recognition and acceptance. The inclusion of zero in numeral systems transformed mathematics and played a pivotal role in the development of algebra and calculus.

  • Bhaskara II
  1. Decimal Notation: Decimal notation, characterized by the use of a decimal point, further enhanced the power of numerical representation. Indian mathematicians made significant advancements in decimal notation. Key features include:
  • Fractions and Decimal Fractions: Ancient Indian mathematicians developed a systematic method for expressing fractions in decimal form. They introduced decimal fractions, where a whole number is followed by a decimal point and decimal digits. This notation enabled precise representation of fractional values.
  • Versatility and Precision: Decimal notation expanded the range of numbers that could be represented accurately. It facilitated calculations involving fractions and allowed for more precise measurements in various fields, including astronomy, architecture, and commerce.
  1. Concept of Infinity: The concept of infinity fascinated ancient Indian mathematicians, who explored its nature and properties. Key aspects include:
  • Infinite Series: Indian mathematicians investigated infinite series and their convergence. They made significant progress in understanding the behavior of series involving numbers or variables that stretch infinitely.
  • Conceptual Exploration: Ancient Indian mathematicians pondered the idea of infinity as an abstract concept beyond the realm of finite numbers. They contemplated infinite sets, infinite processes, and the infinite nature of the universe.
  • Philosophical Connections: The concept of infinity in Indian mathematics had philosophical implications, intertwining with the Indian philosophical tradition. It reflected a deep exploration of the nature of reality and the infinite possibilities that existence holds.

3.1. Vedic Mathematics

Vedic Mathematics is a system of computation that originated in ancient India and is believed to have been rediscovered in the early 20th century by Sri Bharati Krishna Tirthaji Maharaja, a scholar and mathematician. The term "Vedic" comes from the word "Veda," which means knowledge. This system is based on the Vedas, the ancient sacred texts of India, and presents a unique and efficient approach to performing mathematical operations. Vedic Mathematics encompasses various techniques that simplify complex calculations and enhance mathematical problem-solving skills.

Extract from ancient scripture

The Foundation of Vedic Mathematics

Vedic Mathematics is an invaluable treasure of ancient Indian knowledge that has the potential to revolutionize the way we approach and solve mathematical problems. Its remarkable speed and accuracy make it an attractive option for students, professionals, and enthusiasts alike. By combining the wisdom of the past with the needs of the present, Vedic Mathematics continues to hold its relevance in modern times as a powerful tool for computation and problem-solving. Embracing this ancient system can empower individuals with mathematical prowess and unlock their potential in various fields.

Vedic Mathematics is built upon 16 sutras (aphorisms) and 13 sub-sutras (corollaries) that act as guiding principles for solving different types of mathematical problems. These sutras cover various aspects of arithmetic, algebra, geometry, and trigonometry, providing a comprehensive framework for computation.

3.2. The Sutras of Vedic Mathematics

  1. Ekadhikena Purvena: "By one more than the previous one" - This sutra simplifies multiplication of numbers that are one less than bases of 10 (e.g., 9, 99, 999).
  2. Nikhilam Navatashcaramam Dashatah: "All from 9 and the last from 10" - A quick method for subtraction involving complements to 10.
  3. Urdhva-Tiryagbhyam: "Vertically and crosswise" - An efficient technique for multiplication.
  4. Paravartya Yojayet: "Transpose and adjust" - This sutra is used for division problems.
  5. Shunyam Saamyasamuccaye: "When the sum is the same, that sum is zero" - A shortcut for finding the sum of fractions.
  6. Anurupyena: "Proportionality" - A method for solving proportions.
  7. Sankalana-Vyavakalanabhyam: "By addition and by subtraction" - Useful for solving linear equations.
  8. Puranapuranabyham: "By the completion or non-completion" - A technique for finding solutions to quadratic equations.
  9. Chalana-Kalanabyham: "Differences and Similarities" - A method for finding the square of a number.
  10. Yaavadunam: "Whatever the extent of its deficiency" - A technique for solving quadratic and higher order equations.
  11. Vestanam: "Specific and General" - A way to solve simultaneous linear equations.
  12. Yavadunam Tavadunikritya Varganca Yojayet: "By one more than the one before, and whatever is the deficit, and by the square of the deficit" - Used to find squares of numbers.
  13. Antyayor Dashake: "The last digit is ten" - A shortcut for multiplication.
  14. Antyayoreva: "Only the last terms" - Used for solving certain equations.
  15. Yavadunancha: "Whatever the extent of its excess" - A method for solving equations.
  16. Vyashtisamanstih: "Part and Whole" - A technique for solving mathematical problems involving ratios.

 3.3. Ancient Indian Mathematicians

The remarkable journey of Indian mathematics begins with Aryabhata, a mathematician and astronomer who lived in the 5th century AD. 

Aryabhata (476 AD - 529 AD):

Aryabhata, born in Kusumapura (present-day Patna, India), was a brilliant mathematician and astronomer. His contributions encompassed various branches of mathematics and astronomy.

Aryabhata's most notable contribution was the development of the decimal place value system. Recognizing the significance of zero as a placeholder, he revolutionized numerical notation. His work on place value allowed numbers to be represented efficiently and facilitated complex calculations. Aryabhata's breakthrough paved the way for the advanced arithmetic techniques used today. Some of his remarkable works include:

  • Aryabhatiya: Aryabhata's most famous work is the "Aryabhatiya," a treatise that revolutionized mathematics and astronomy. It is divided into three sections: Ganita (mathematics), Kala-kriya (time calculations), and Gola (spherical astronomy). The Aryabhatiya introduced the decimal place value system, including the concept of zero as a placeholder.

  • Aryabhata
  • Trigonometry: Aryabhata developed trigonometric functions and tables, along with methods for calculating the values of sine and cosine. He used trigonometry extensively in his astronomical calculations, including determining planetary positions and eclipses.

Brahmagupta (598 AD - 668 AD): 

Brahmagupta, born in Bhinmal (present-day Rajasthan, India), was a renowned mathematician and astronomer. His contributions to mathematics, particularly algebra and number theory, were groundbreaking.


Brahmagupta introduced the concept of negative numbers and developed rules for their operations. He laid the foundation for modern algebraic notation by using abbreviations for unknown variables. His work on quadratic equations and solutions greatly influenced the development of algebraic methods. Some of his revolutionary works are:

  • Brahmasphutasiddhanta: Brahmagupta's main work, "Brahmasphutasiddhanta," is divided into four sections: Ganita (mathematics), Grahaganita (planetary mathematics), Rekhaganita (geometry), and Golaganita (spherical astronomy). In this treatise, he discussed various topics, including arithmetic operations, quadratic equations, and geometric principles.
  • Negative Numbers: Brahmagupta introduced the concept of negative numbers and formulated rules for their arithmetic operations. His work on negative numbers laid the foundation for the development of modern algebra.
  • Quadratic Equations: Brahmagupta provided methods for solving quadratic equations and developed techniques for determining both positive and negative solutions. He made significant advancements in algebraic methods and provided systematic approaches to solving equations.

Pingala (3rd century BC):

Pingala, a mathematician and author, lived during the 3rd century BC. Believed to have hailed from the ancient city of Pratishthana (present-day Maharashtra, India), Pingala made significant contributions to the study of combinatorics and poetic meters.

His legendary work includes:

  • Chhandahshastra: Pingala's work "Chhandahshastra" explored the mathematical patterns found in Sanskrit poetry. In this treatise, he developed a binary system of counting known as "Meru Prastara." This system involved a recursive process of constructing patterns by combining short and long syllables. Pingala's work on binary numbers had a profound impact on the development of binary arithmetic and its applications in computer science.
  • Combinatorics: Pingala made significant contributions to combinatorial mathematics. He studied various poetic meters and developed rules for the arrangement of syllables. His work laid the foundation for the study of permutations and combinations.

Pingala's Meru Prastara and Pascal's Triangle

These three extraordinary mathematicians not only made significant individual contributions but also influenced subsequent generations of scholars. Their discoveries and theories formed the bedrock of Indian mathematics and enriched the global mathematical landscape.

The stories of Aryabhata, Brahmagupta, and Pingala spread far and wide, inspiring scholars and mathematicians across the world. Their innovative ideas transcended time and continue to shape modern mathematics.

Their legacies live on in the hearts and minds of those who seek to unravel the mysteries of numbers and their applications. They remind us that knowledge knows no boundaries and that the pursuit of understanding is a noble endeavor.

The contributions of Aryabhata, Brahmagupta, and Pingala in ancient Indian mathematics were instrumental in shaping the field. They introduced innovative ideas, developed new techniques, and laid the groundwork for future mathematical advancements. Their works continue to be studied and appreciated, serving as a testament to their intellect and the enduring legacy of ancient Indian mathematics.


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