The Evolution Of Indian Mathematics: A Journey Through Time – OpEd

The development of mathematics in the Indian subcontinent is a remarkable blend of the familiar and the unique. The Indian decimal place-value numeral system, which is the foundation of the modern decimal number system, is one of the most significant contributions to mathematics.

In addition, the techniques of arithmetic and algebra that emerged in India laid the groundwork for many mathematical advancements. However, unlike modern mathematical texts, classical Indian mathematical treatises were composed in verse form, prioritizing memorization and intuition over structured formal proofs. Indian mathematics, or “ganita” (Sanskrit for “computation”), required not only a strong memory and quick arithmetic skills but also logical reasoning and numerical intuition to develop new techniques and approximations.

This article explores the history of mathematics in the Indian subcontinent, tracing its journey from ancient times to the pre-colonial period. By the 19th century, Indian mathematics had merged with Western mathematical traditions, leading to further developments in the field.

Ancient Foundations

Vedic Number Systems and Geometry

Sanskrit, the classical language of India, played a crucial role in preserving mathematical knowledge. For centuries, it maintained an oral tradition, and even after writing was introduced, traditional materials like palm leaves and birch bark did not survive well in South Asia’s climate. The earliest mathematical references are found in the Vedas, ancient religious texts that were passed down through oral transmission. The Yajurveda, for instance, contains names for powers of ten extending up to 10¹², far beyond what was common in other ancient civilizations.

A particularly interesting example from the Satapatha Brahmana (circa 1000 BCE) involves the division of 720 bricks into groups, excluding all divisors that are multiples of numbers relatively prime to 60. This approach is reminiscent of the base-60 system used in Mesopotamian mathematics, suggesting possible influences from Babylonian numerical methods.

The Brahmins, the priestly class responsible for performing ritual sacrifices, contributed significantly to early mathematical thought. Their knowledge is most evident in the Sulbasutras (“Cord-Rules”), written around the 1st millennium BCE. These texts provide instructions for constructing brick fire altars, employing geometric techniques to transform one shape into another of equal area. The Sulbasutras also contain early references to the Pythagorean theorem, estimates of π (pi), and approximations for √2, highlighting an advanced understanding of geometry. The association of mathematics with religious rituals likely fueled its development, with geometric principles being sanctified within the framework of spiritual practice.

The Post-Vedic Era

With the rise of Buddhism and Jainism after 500 BCE, mathematical thought evolved to align with new religious philosophies. Rather than focusing on altar construction, mathematics found applications in cosmology and philosophy. Jain scholars, in particular, explored large numbers and even conceptualized infinity within their models of the universe. Mathematics also played a crucial role in astronomical calculations, which were necessary for maintaining religious calendars. Some of these methods, including sexagesimal calculations and linear approximations of seasonal variations, may have been influenced by Mesopotamian sources introduced through Achaemenid Persia.

Beyond religious applications, mathematics was used in commerce and administration. Although direct evidence is scarce, a Buddhist text from the 1st century BCE, attributed to Vasumitra, mentions merchants’ “counting pits.” These pits contained tokens representing units, hundreds, and thousands, demonstrating an early understanding of place-value notation. Vasumitra likened this system to the impermanence of reality, illustrating how the same clay token could represent different values depending on its position.

Indian Numerals and the Decimal Place-Value System

The development of the Indian numeral system is one of the most significant milestones in mathematical history. Early Indian numerals, such as the Brahmi numerals, were not place-valued and instead used distinct symbols for multiples of different powers of ten. Inscriptions on monuments and deed plates reveal that these systems coexisted for centuries before evolving into a purely place-value system.

By the early 1st millennium CE, decimal place-value notation was widely used among Indian mathematicians. A Sanskrit adaptation of a Greek astrological text from the 3rd century CE provides evidence of this transition. In this work, numbers were represented using symbolic words—such as “moon” for 1, “eye” for 2, and “Veda” for 4—forming compounds like “moon-Veda-eye-moon” to denote 1,241. This suggests that the reader intuitively understood decimal place-value representation. The eventual standardization of the Indian numeral system played a crucial role in its dissemination, eventually influencing Arabic numerals and, later, the modern global numeric system.

Conclusion

Indian mathematics represents an extraordinary intellectual tradition that combined computational skill with conceptual innovation. From the Vedic period’s numerical insights and geometric constructions to the Jain and Buddhist mathematical philosophies, the Indian subcontinent has contributed immensely to the global history of mathematics. The decimal place-value system, in particular, revolutionized numerical computation and laid the foundation for modern arithmetic. While Indian mathematical texts often prioritized mnemonic and intuitive learning methods over formal proofs, they nonetheless fostered a deep understanding of mathematical principles, allowing for remarkable advances in both theoretical and applied mathematics.

References

1. Kim Plofker, Mathematics in India (Princeton University Press, 2009)
2. David Pingree, The History of Mathematical Astronomy in India (Springer, 1981)
3. George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Princeton University Press, 2011)
4. A.K. Bag, Mathematics in Ancient and Medieval India (Motilal Banarsidass, 1979)
5. Bibhutibhushan Datta & Avadhesh Narayan Singh, History of Hindu Mathematics (Asia Publishing House, 1935)

Authors:

• Dr. Baljinder Kour, Assistant Professor, Department of Mathematics, Akal University, Talwandi Sabo, Bathinda, Punjab, India.
• Prof. (Dr.) Ashish Arora, Department of Mathematics, IK Gujral Punjab Technical University, Kapurthala, Punjab, India.