Vedic Mathematics: A Mathematical Tale from the Ancient Veda to Modern Times

Vedic mathematics is primarily a collection of methods, called the sutras, that aim at making one faster at numerical computations. These formulas are claimed to have been retrieved from the Vedas (Khare 2006); however, there is considerable disagreement on this issue (Dani 2001; Plofker 2008).

The Vedas are religious texts, originating in ancient India, composed in Sanskrit, and considered the oldest and holiest scriptures of Hinduism.

The authors, at the very outset, confess the fact that none of them is well-versed in Vedic mathematics. As luck would have it, the primary author, Giuseppe Dattoli happened to visit India for a conference during which one of his Indian friends gifted him a book, Vedic Mathematics: Sixteen Simple Mathematical Formulae from the Vedas, of a revered learned Indian monk, His Holiness Jagadguru Shankaracharya Swami Bharati Krishna Tirtha Maharaj. This book, first published in 1965, is considered to be the oldest on this subject. It has been the author’s only encounter with Vedic mathematics.

As prerequisites, the authors claim that the book calls for knowledge of the calculus and a fair disposition in the art of computing on the part of readers. However, even a first reading of the book makes readers realize that knowledge of the basics of linear congruences, matrices, and recurrence relations will make the reading of the text more enjoyable and meaningful.

In its six chapters, the book, on account of various historical snippets, attempts to trace a brief history of mathematics from the ancient Vedic sources to the current times.

Chapter 1, Mixing Up Ancient and Modern, the shortest chapter of the book, starts with an engaging discussion on how Pythagoras’ theorem was treated in the works of Euclid and that of ancient Indian Vedic texts. The authors find the Vedic treatment more effective from a practical point of view, despite its evident shortcoming in matters of rigor. Using the proof of Pythagoras’ theorem, as given in the Vedic texts, the authors go on to demonstrate how Euclid’s theorems could be derived as its consequences, contrary to the Greeks’ practice of deducing the former from the latter. The authors also bring to light that the Babylonians were well-versed in the solution of a quadratic equation, and the calculation of cube roots.

They also show how, despite being similar in principle, the search for a divisibility criterion is, in practice, unlike the process of factorization of polynomials.

Chapter 2, Divisibility Criteria, Osculator Numbers and Roots, contains discussion on the divisibility tests for primes up to 23. A comparison is drawn between the modern-day techniques using polynomials and the Vedic mathematical techniques for divisibility testing. The Vedic approach makes use of the Osculator Number (ON), the Reduced Number (RN), and the Dividend Number (DDN). Examples, wherever needed, have been supplied for better and practical understanding. A separate section, Osculator and Numbers, has been dedicated to how geometry’s osculating circle is connected to the divisibility of numbers. Techniques for the calculation of square roots and cube roots have also been discussed and sufficiently illustrated.

The discussion, in Chapter 3, Continued Fractions, Nested Radicals, Fibonacci, Pell… and Transcendental Numbers, shifts to the world of continued fractions and transcendental numbers, among others. A beautiful example of the application of continued fractions is given concerning the reform of the Julian calendar. It so happened that long back Pope Gregorio XIII commissioned a reform of the Julian calendar proposed by Julius Caesar in 45 BC. The calendar was based on some approximation of the Earth’s period of revolution around the Sun. The approximation, successively, culminated into a substantial error of 10 days, in 1582, aka the year of the reform. Repeated approximations, using continued fractions, allowed the realignment of the calendar with several astronomical events, such as equinoxes and solstices.

Chapter 4, Imaginary Numbers, Just an Unhappy Way of Saying, shows how the concept of imaginary numbers is connected to the roots of algebraic equations. The authors also attempt to highlight that the use of the phrase “imaginary numbers” is somewhat misleading and inappropriate. Imaginary numbers are not unreal, but only different from real numbers. The chapter starts with a discussion on the genesis of the famous Napier number, $e=\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n\approx 2.71828$. This number is ubiquitous and surfaces from several unexpected places (Maor 2009), very much like the number $\pi$ denoting the ratio of the circumference and the diameter of any circle. It is shown how introducing the imaginary unit as an operator acting on two numbers leads to displeasing formal incoherence. A more convenient and suitable tool, that of matrices, is introduced which can very conveniently represent all complex numbers, for example the imaginary unit $i=\left(\text{}\text{}\begin{array}{cc}0& -1\\ 1& 0\end{array}\text{}\text{}\right)$. The discussion is shown to lead to a more sophisticated but marvelously beautiful Q-trigonometric representations. This chapter also contains a discussion on the extraordinary meanings and properties of Fibonacci numbers.

In Chapter 5, Algebraic Equations: Vedic and Western Points of View, the authors take a U-turn from their standpoint of declaring the use of the word “imaginary” as unhappy for the square roots of negative numbers. It is pertinent to mention that in spite of the common understanding that the solution of quadratic equations can be traced back to the Assyrians (who are credited as the discoverers of the square complement method), the idea of imaginary numbers came to visibility way afterwards, thanks to the efforts of the Italian algebraists. And, more interestingly, these numbers came to the surface in the study of cubic equations.

Chapter 6, The Two Souls of Mathematics and the Last of Vedas, is the longest chapter of the book. It talks about the greatest Indian mathematician of the last century, an immensely talented young man with little formal training in mathematics, Srinivasa Ramanujan (Mala and Wilson 2022). For some time, Ramanujan worked as an accountant, and at that time, accountancy was based on some ancient calculation methods. It seems that Ramanujan, owing to his uncanny knack of dealing with numbers, intuition, and ability to see forms and not just formulas, had considerably refined these existing methods.

The book’s deceptive title gives it the feel of ethnomathematics that attempts to bring to light the contributions to mathematics that originated from a particular place or culture (Mala 2022). However, it is not entirely dedicated to Vedic mathematics. It is more of a history-of-mathematics book than a treatise on Vedic mathematics. It attempts to make the point that mathematical thinking is the outcome of varied experiences and geo-social factors. It demonstrated, via judiciously chosen examples, how the ancient Babylonians and the Egyptians had developed a rich corpus of mathematical items, and successfully employed the same in navigation, trade, engineering, and astronomy.

Firdous Ahmad Mala
Government Degree College Sopore, Baramulla,
Jammu and Kashmir, India
[email protected]