Nilesh's Weblog

From the Vedic Perspective

What is an ancient Indian technique that is being taught in schools in UK but not in India? What is it that beautifully corelates all the mathematics operators if they were never different. What is it that allows you to solve quadratic equations using differential calculus yet still do it mentally? You guessed it. It is Vedic Mathematics. This long forgotten technique for mathematical calculations is gaining popularity by the exponent.

The Vedic system of mathematics was rediscovered from some ancient Sanskrit texts in the last century between 1911 and 1918 by Sri Bharati Krsna Tirthaji. The system is based on 16 sutras or axioms. These formulae describe the way one's mind naturally works towards a solution. There are also 13 upasutras or corollaries which add to the basic formulae.

Let's take a simple example. The sutra Nikhilam Navtashcharam Dashatah or All from 9 and the last from 10 describes how we mentally subtract say 793 from 1000.

    1000 -    7     9      3
              |     |      |
           (9-7) (9-9) (10-3)
              |     |      |
         =    2     0      7

Here we subtracted all digits of 793 by 9 except the last one. The simplest of all calculations.

Another example: This one is based on the sutra Yavadunam or By the Deficiency (from the base). This is multiplying two numbers near(above or below)to a base (10/100/1000)

          88 x 92 = 8096

We proceed as follows:
1. 88 is 12 below the base 100
2. 92 is 8 below the base 100
3. 92-12 = 88-8 = 80. 
   This is the first part of the answer.
4. Now multiply 12 by 8. 12x8=96
   This is the second part of the answer.
5. The answer is 8096.


     88    12  (difference from 100)
  x  92     8  (difference from 100)
    ----------
           96

     88    12   (either 92-12 or 88-8)
         /
  x  92     8
    ----------
     80    96

Isn't that very interesting? This calculation hardly takes a few seconds if you do it mentally. It takes about the same time entering this in a calculator!

The corollary Aanurupyena or Proportionately can be applied with the above sutra. If we multiply the numbers 103 and 107, 103 x 107 = 11021. If the numbers were 203 and 207, the first part of the answer should be doubled. 203 x 207 = 42021. <2(203-(-7))><3x7>. This is because our base has also doubled(2x100)

One very useful example I would like to give using the sutra Ekadhikena Purvena or by one more than the one before - Converting a fraction to recurring decimal form. These fractions have the denominator as a prime factor other than 2 or 5 and ending in 9. Lets take 1/19 as an example. 19 is a number which has prime factors other than 2 and 5.

1/19. 
 1. Take the digit 1 from denominator.
 2. Add one to it. i.e. 1+1=2. 
 3. '2' is the key digit for Ekadhikena.
 4. Dividing 1 by 2, we get answer as 0 and remainder 1.
 5. 1/19 = 0.0
 6. Take remainder 1 and last digit 0. Divide 10 by 2
 7. 1/19 = 0.05
 8. Next 5 divided by 2 is 2 remainder 1
 9. 1/19 = 0.052
10. Take remainder 1 and last digit 2. Divide 12 by 2
11. 1/19 = 0.0526
12. And so on till you get 1 again with no remainder
    1/19 = 0.052631578947368421

A last example: Using Urdhva-tiryaghyam or Vertically and Cross-wise for multiplication of numbers

       275
     x 513
    -------
    141075
	
1. vertically, last digit,
   (5x3) = 15 => write 5 carry 1
   
2. crosswise, last two digits,
   (3x7)+(1x5)=26 + carry 1 = 27 => 7 carry 2
   
3. vertically and crosswise(all three digits), 
   (2x3)+(5x5)+(7x1)=38 + carry 2 = 40 => 0 carry 4
   
4. crosswise, first two digits,
   (2x1)+(7x5)=37 + carry 4 = 41 => 1 carry 4
   
5. vertically, first digit,
   (2x5) = 10 + carry 4 = 14 => 14
   
Therefore, 275x513 = 141075

The theory behind this is:
    ax² +bx +c
 x  dx² +ex +f
-------------------------------------------
adx4 +(ae+bd)x³ +(af+be+cd)x² +(bf+ce)x +cf

Here x is 10.

These are only a fraction of what Vedic Maths 'teaches'. Also these methods do not mean there are no other ways of solving a problem. It only emphasizes on the simplicity of the sutras of vedic mathematics. I hope I have aroused your interest in Vedic Maths. Search on the web for more sutras and their application to maths. Try searching for the sutra to solve quadratic equations.