Vedic mathematics

Summary: Vedic mathematics involves challenging mental calculations and was transmitted orally.

Vedic mathematics is a system of mathematics associated with India’s upper Indus Valley prior to 1000 b.c.e. Originally transmitted orally, the Vedic mathematics known in the twenty-first century was abstracted from ancient Sanskrit texts, known as “Vedas.” Sri Bharati Krsna Tirthaji rediscovered the Vedas in the early 1900s, but his scholarly results were not published until 1965.

The Vedas covered all areas of knowledge, with the mathematics created to support this knowledge. Since recording mechanisms were not available, Vedic mathematics involves creative mental calculations, often at very challenging levels. Through Arab and Islamic writers in the 770s c.e., some Vedic mathematics was transmitted and became part of European mathematics, including elements such as the Arabic numerals, the multiplication sign, and a symbol for zero. However, the mental aspects of Vedic mathematics were not known until 1965, and these “secrets” have provided scholars, mathematicians, and students interesting explorations into multiple areas, including basic arithmetic computations, factoring, exponents, algebra in the form of linear through cubic equations, elementary number theory, analytic geometry involving the conic sections, the Pythagorean theorem, and differential calculus.

The Sutras

Sixteen formulas (or Sutras, which means “thread”) form the foundation of Vedic mathematics, along with fourteen “sub-Sutra” corollaries. Expressed as word phrases, each formula acts as a “thread” woven throughout the Vedic mathematics system, assuming the role of a unifying element.

For example, Sutra #2 states: “All from 9 and the Last from 10.” Sutra #3 states: “Vertically and Crosswise.” The combined importance of both Sutras is best explained within the context of mental multiplication, such as finding the “sum” 88×98. Both numbers are close to the “base” 100, involving “deficiencies of 12 and 2,” respectively. The desired product is obtained using these deficiencies (Sutra #2), then represented either mentally or symbolically (by Sutra #3):

88 -- 12
98 -- 2
86/24

In these operations, the deficiencies 12 and 2 are placed to the right of the original numbers, 88 and 98. The 86 is found by subtracting a deficiency from the other number in the product (98-12=86=88-2), while the 24 is the product of the deficiencies. Finally, the desired result is found: 88×98=8624, as the 86 actually represented 8600. Though this process involves a sense of magic, it is much easier than the modern computational algorithm commonly used in the twenty-first century.

It is not only important to investigate why this Sutra-based technique works, but also determine possible constraints or exceptions. For example, applying the Sutra to the product 25×57, the process becomes:

25 -- 75
57 -- 43
-18/3225

Because the desired product can be obtained via -1800+3225=1425, the power and the limitations of the Sutra become more evident, especially the emphasis on the numbers 9 and 10. The technique is not useful in this example because the large internal products and need for a negative quantity become obtrusive. However, the method does work, and it can be proven true algebraically. Suppose the desired product is a×b, where a and b are whole numbers less than 100. Using the respective deficiencies (100-a and 100-b), the Sutra’s process leads to the algebraic identity ab=100[b-(100-a)]+(100-a)(100-b). Thus, the numbers a and b could be any numerical values—positive, negative, fractions, irrational, or even complex numbers.

Finding Decimals

As another example, Sutra #1 states: “By One More than the One Before.” This Sutra is used in the construction of the number system, as each whole number is one greater than its predecessor (akin to the Peano postulates or axioms formulated in the nineteenth century). However, the Sutra’s power is its application in other situations as well. Suppose the problem was to find the repeating decimal equivalent to the “vulgar” fraction 1/19, usually obtained by laboriously dividing 19 into 1. The Sutra suggests a focus on “one more than the number before” the 9, or the number 2, which is one more than the 1 which appears before the 9. The 2 (called Ekadhika for “one more”) becomes the new divisor in lieu of the troublesome 19. The “strange” decimal resulting from this division of 2 into 1 is

0.105126311151718914713168421… .

To explain this strange expression, start with a 0 and a decimal point. Then, 1 divided by 2 is 0 remainder 1, represented by placing a 0 in the decimal expression, preceded by a subscripted 1 as the remainder. The process is repeated, where 10 (or the visual of the subscripted 1 and adjacent 0) is divided by 2, resulting in 5 with remainder 0. Thus, the 5 in the decimal expression now is not preceded by a subscripted number. Next, 5 divided by 2 results in 2 remainder 1, which are represented as before with the remainder becoming the preceding subscript. And, in subsequent divisions, 12 divided by 2 is 6 remainder 0, 6 divided by 2 is 3 remainder 0, 3 divided by 2 is 1 remainder 1, and so on. Finally, to get the final value of the decimal expression for 1/19, the subscripted values are removed: 1/19=0.052631578947368421, as they are needed only as “mental” reminders of the division process by 2. The mathematical explanation underlying this process is quite complex, but can be found in Chapter 26 of Tirthaji’s Vedic Mathematics.

These two examples illustrate the enjoyment of investigating Vedic mathematics. On one level, the 16 Sutra and their corollaries provide efficient mental algorithms that become very powerful and efficient in special instances. On a second level, the careful examination of the Sutra and its application provides a rich opportunity to understand the role of generalization and algebraic identities.

Bibliography

Bathia, Dhaval. Vedic Mathematics Made Easy. Mumbai, India: Jaico Publishing, 2006.

Howse, Joseph. Maths or Magic? Simple Vedic Arithmetic Methods. London: Watkins Publishing, 1976.

Tirthaji, B. K. Vedic Mathematics. Delhi: Motilal Banarsidass, 1965.

Williams, Kenneth, and Mark Gaskell. The Cosmic Calculator: A Vedic Mathematics Course for Schools. (Books 1, 2, and 3). Delhi: Motilal Banarsidass, 2002.