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Vedic Mathematics

The art of mathematics as practiced in ancient India, which is truly marvellous and exciting, and which I feel is my duty to make accessible to the general public for the improvement of learning and knowledge.

 

Content

  1. Preface
  2. Introduction
  3. Training applet
  4. Methods
    1. Determining the decimal value of fractions whose denominators end with 9
    2. Multiplication
    3. Division
    4. Factorisation of simple quadratics
    5. Factorisation of harder quadratics
    6. Factorisation of cubics
    7. Highest common factor
    8. Simple equations
    9. Quadratic equations
    10. Cubic equations
    11. Bi-quadratic equations
  5. Index

Preface

"Vedic Mathematics" is a book written by Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja, Sankaracarya of Govardhana Matha, Puri (published by Motilal Banarsidass Publishers Private Ltd., Delhi). I came across this book by chance and at once was totally fascinated. Basically it contains methods of performing arbitrarily complicated calculations mentally, without the help of pen and paper or a calculator, in very fast and easy ways that can be comprehended even by children. Sadly the availability of pocket calculators and computers have removed the need to calculate mentally. Therefore the methods will be useful to those only who take a special interest in this matter, exercising their minds or amazing others by doing mathematical "magic".

The good Swamiji (who was a very learned sage and a brilliant personality) discovered the instructions in the old texts of the Vedas (the Books of Knowledge of ancient India written in Sanskrit) after eight years of solitude in the forest. Previously these parts of the texts had been translated in a completely different context, seeming incoherent and useless and therefore had been discarded by less mathematically inclined scholars. For example, one sutra read:
"During the reign of King Kamsa rebellions, arson, famines and insanitary conditions prevailed."
Who would guess that this line actually contains instructions about mathematical operations! It is the amazing discovery of Sri Sankaracarya that yielded the key to the truth behind the appearances.

The book, written in a lucid and peculiar style, contains a lot of praise of the vedic system and many caustic comments on the "cumbrousness" of our modern way of calculating, which are very entertaining to read (and highly recommended). However, it does not completely overthrow western mathematical science, as it sometimes emphatically promises. In fact, its calculation instructions can be explained (and proven) using modern algebra.

In India, many scholars and vedantas proficient in metaphysics claim that the mathematical abbreviations described in this book have been noted down by wise men who received these insights by intuition or divine revelation, not by following ordinary rules of deductive or inductive logic. That be as it may; I think it doesn't matter much to me or any other practical man who is interested in their application only.

In this short treatise I will therefore try to go directly to the essence of the instructions and explain them in a concise way, omitting comparisons to and condemnations of our current system. Please note that I am using the Java programming language notation for mathematical expressions.
I hope to avoid any violations of the copyright. Any flaws and mistakes are my own; corrections and comments are greatly appreciated. A Java applet is provided on each page that might be useful for training the methods and for gaining speed and proficiency.

May the light of vedic knowledge spread throughout the world :-)

Introduction

We all use abbreviations and shortcuts when we calculate mentally. Take a simple example:
19 * 20 = 380. Instead of adding up 20 nineteen times, we could get the result in several ways:

  1. 19 * 2 = 38; add a zero for multiplication by 10, equals 380.
  2. 20 * 20 = 400, subtract 20 once because 20 - 19 = 1, equals 380.

These methods are actually taught at school and are sufficiently easy and fast. They have one thing in common: they introduce "help figures" based on arithmetical rules which make the mental calculation easy.
Example 1 can be expressed as: 19 * 20 = 19 * 2 * 10 by factorization of 20 into 2 and  the help figure 10 (a * b = a * c * d where b = c * d).
Example 2 can be expressed as: 19 * 20 = ((19 + 1) * 20) - 20, using the help figure 20 to ease multiplication (a * b =  [a+c] * b - c * b).

In this way the vedic methods work, although on more complicated schemas like binomial formulae etc. The idea is that almost all calculations can be broken down into very few steps with simple operands.

An example is provided here.

Fractions whose denominators end in 9
e.g. 1/19, 1/29 etc.

Rules:

  1. Calculate the ekadhika purva, abbreviated as ep, by increasing the second-rightmost digit of the denominator by 1.
  2. Divide the first digit of the denominator by ep, getting the quotient and the remainder.
  3. Write down the quotient as the first decimal of the result. The next number to be divided is the quotient prefixed with the remainder.
  4. Divide the obtained number by ep, getting the quotient and the remainder.
  5. Write down the quotient as the next decimal of the result. The next number to be divided is the quotient prefixed with the remainder.
  6. Repeat steps 4 and 5 until the desired numbers of decimals has been reached or the decimal repeats itself.

Example 1: 1/19

  1. (1) Calculate the ekadhika purva. ep = 2, as the penultimate number of the denominator is 1.
  2. (2) The division of the first digit of the denominator yields 0 as quotient and 1 as the remainder
  3. (3) Put down 0 as first decimal. The next number to divide is 10.
    obtained decimal 0 , 0
    numbers to divide 10
  4. (4) 10 divided by 2 yields 5, remainder 0.
  5. (5) Put down 5 as the next decimal. The next number to be divided is 05 = 5.
    obtained decimal 0 , 0 5
    numbers to divide 10 5
  6. (4) 5 divided by 2 yields 2, remainder 1.
  7. (5) Put down 2 as the next decimal. The next number to be divided is 12.
    8. obtained decimal 0 , 0 5 2
    numbers to divide 10 5 12
  8. (4) 12 divided by 2 yields 6, remainder 0.
  9. (5) Put down 6 as the next decimal. The next number to be divided is 6.
    10. obtained decimal 0 , 0 5 2 6
    numbers to divide 10 5 12 6
  10. (6) repeating this process...
    obtained decimal 0 , 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1
    numbers to divide 10 5 12 6 11 15 17 18 9 14 7 13 16 8 4 2 1