Vedic Mathematics

The goal of introducing a sampler of “Vedic Mathematics” to kids (nine years of age and older) is to expose them to our rich heritage, and to ancient thoughts that continue to be relevant to modern mathematics.  Specifically, the readily available “Vedic Mathematics” resources on the Web and in books contain short-cuts for performing basic arithmetic, but they do not contain easy-to-see justifications for why the short-cuts work, when they fail, and how one can recover from apparent failures, by fixing the intermediate solutions in a systematic way. To remedy this gap, presentations accessible to kids (with some assistance from parents) have been developed. (The parents may also benefit from additional advanced material included.) The entire current set of presentations in MS Powerpoint is available by following links from this Web page, and is also hosted at:

http://www.cs.wright.edu/~tkprasad/assortedbookmarks.html

                An overview of the mathematical content of the presentations follows.
(1) Abstraction: This lecture introduces the idea of abstraction to distinguish the concept of number and its numerical representation.  It covers Egyptian numerals, Roman numerals, and Hindu-Arabic numerals to better motivate the characteristics of a desirable system of representation.
(2) Positional Number System: This lecture discusses the benefits of positional number system by explaining the invention of zero (credited to Hindu Mathematicians) and the Hindu Arabic Numerals.
(3) Arithmetic Operations I: This lecture discusses single digit multiplication of large digits in terms of multiplication of smaller digits using Vedic method. It illustrates the approach with simple examples and provides a rigorous justification for why it works for the examples shown. It then shows examples where the approach breaks down and explains how to fix the problem systematically.
(4) Arithmetic Operations II: This lecture discusses 2-digit multiplication of numbers in terms of “easier” multiplication of smaller numbers using Vedic method. It generalizes single digit approach to 2-digits.
(5) Divisibility: This lecture discusses and justifies tests for divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11.
(6) Division by 9: This lecture discusses an efficient Vedic method for dividing a number by 9.
(7) Digital Sum/Root: This lecture defines digital sum/root and explores its properties. It also introduces the Vedic Square, Checksums, and Additive Persistence of a number (a rapidly growing function).
(8) Arithmetic Operations Revisited: This lecture builds on earlier lectures to square a number, and generalizes multiplication of 3-digit numbers using “Working Base”.
(9) Prime Numbers: This lecture defines prime numbers (the fundamental building blocks of Number Theory) and perfect numbers, and explores their relationships. It explains the well-known Eratosthenes Sieve Method of generating primes. It also lists a few easy to describe but very hard to prove claims in Number Theory, to shed light on the intrinsic nature of pure mathematics (Ramanujam’s forte).
(10) Evolution of Numbers: This lecture introduces various kinds of numbers: whole numbers, integers, rationals, irrationals, reals, and complex numbers. It explains Pythagoras' Theorem and its proofs (including the one attributed to Bhaskara). It illustrates “proof by contradiction” and “non-constructive proofs”.