9 - The Vedic Triangle



Vedic Mathematics is becoming increasingly popular as more and more people are introduced to the beautifully unified and easy Vedic methods. The purpose of this Newsletter is to provide information about developments in education and research and books, articles, courses, talks etc., and also to bring together those working with Vedic Mathematics. If you are working with Vedic Mathematics- teaching it or doing research- please contact us and let us include you and some description of your work in the Newsletter. Perhaps you would like to submit an article for inclusion in a later issue.
If you are learning Vedic Maths, let us know how you are getting on and what you think of this system.


This issue's article:

This is written by Barbara Salmon who has been researching pattern in number and Vedic Mathematics for some years.


The Vedic Triangle, contains the same numbers and number patterns as the Vedic Square (see Newsletter number 7), but laid out in the shape of a triangle. There are two main steps:-

    Step 1 - create a multiplication triangle;
    Step 2 - reduce the triangle to Digit Sums.

When treated as exercises, steps such as these offer a simple and interesting way to explore various aspects of number.


We can create a multiplication triangle, by multiplying digits 123456789 x 987654321 (i.e. digits 1 to 9 as ascending and descending) using the 'Vertically and Crosswise' step sequence described in the original Vedic Mathematics Treatise (Chapter 3) by Sri Bharati Krsna Tirthaji. This gives us the product of every number by every other number, in the range 1 to 9. Following this step sequence automatically gives us the products, in the shape of a triangle.

Here is a small example of that particular step sequence, using the first two digits of the multiplier 12, and multiplicand 98:-

     1             2  x         The step sequence is:-
     9             8              vertically 1 x 9 = 9;
     ------------            crosswise 1 x 8 = 8 then 9 x 2 = 18;
     9:    8:   16              vertically 2 x 8 = 16.
1   1    7      6

( If we want the product of 12 and 98 we carry the 1 of 16 and 1 of 18 one column to the left to get 1176.)

     9:    8:    16       The individual multiplication products now form the
           18                 the shape of a small triangle.

If we extend the multiplication from 12 x 98 to 123456789 x 987654321 we have a multiplication triangle as follows:-

           1            2            3             4             5              6           7          8
           9            8            7             6             5              4           3          2
                               MULTIPLICATION TRIANGLE
       9:    8:   7:    6:   5:      4:     3:   2:      1:    2:   3:      4:    5:   6:   7:    8:   9
          18:27:36:45:   54:63:72:   81:72:63:54:   45:36:27:   18
                   16:14:12:10:      8:     6:    4:   6:     8:10:12:14:16
                          24:32:40:48:56:       64:56:48:40:32:24
                                  21:18:15:12:         9:   12:15:18:21

Viewed this way 'Vertically and Crosswise' is acting more like a 'modeling' tool for Number. The pattern of the Multiplicand and Multiplier together shape the pattern of the multiplication products.

Exercise: Create your own multiplication triangle - using squared paper (or lined paper overlain at right angles), or the 'table' option if you have a text processing package on your computer. Colour in all the products of the five times table using the same colour.


To reduce a number to its Digit Sum, simply add the digits of the original number together. Repeat the process if necessary until only one digit remains. For example, 48 is 4 + 8 = 12, which is 1 + 2 = 3. Usually 'Digit Sums' are used in arithmetic to cross-check results
e.g.: 359 x 257 = 92263
Digit Sum: 8 x 5 = 40 (reducing to 4 ) so the answer is probably correct. In this article, we are using Digit Sums to see how numbers interrelate.

                                         VEDIC TRIANGLE

       9:   8:   7:   6:   5:   4:   3:   2:   1:   2:   3:   4:   5:   6:   7:   8:   9:
              9:    9:   9:  9:   9:   9:   9:   9:   9:   9:   9:   9:   9:   9:   9:
                      7:   5:  3:  1:    8:   6:   4:   6:   8:   1:   3:   5:   7
                             6:  5:  4:    3:   2:   1:   2:   3:   4:   5:   6
                                   3:   9:   6:   3:   9:   3:   6:   9:   3
                                          1:   8:   6:   4:   6:   8:   1
                                                 6:   2:   7:   2:   6
                                                       3:   9:   3

Exercise: Copy out the triangle onto squared paper and colour in the numbers which correspond to the five times table products you highlighted in the Multiplication Triangle.

Are/were you able to see the following ? :-
* There are eight vertical pairs of numbers.
* One number (7) only appears once.
* Each vertical pair of numbers appears twice e.g.. 5 and 9.
* Each pair reduces to 5 e.g.. 5 + 9 = 14, which is 1 + 4 = 5.
* 7 is the Digit Sum for 5 squared i.e.. 25 which is 2 + 5 = 7.
* If we double 7 to make a pair, we get 14, which also reduces (1 + 4) to 5.
* If we add together all the pairs (including 2 x 7) we have a total of 45, which reduces (4 + 5) to 9.
* If we add together the digits of each pair of pairs e.g.. (1 + 4) + (1 + 4) they reduce to 1.
* Similarly if we double the 7:7 pair, again we have (7+7) + (7 + 7) = 28, which also reduces to one.
* If we add together all the 'pairs of pairs' we have 5 !

The recognition of pattern in number, as illustrated by the types of exercise described above, helps us to develop our mental agility, improvise solutions and think for ourselves.

You will see by visiting the Multiplication and Addition parts of the Creative Maths website, that 'Vertically and Crosswise' offers a broad range of multiplication styles (including the ones we commonly use today). The alternative styles were improvised by myself in March this year. Once the pattern was seen, - each style took a few minutes to develop. The only thing that is changing in each example, is the sequence of multiplication steps (which in all cases can be from left to right or right to left - or even a bit of both!) and the way we record the sums. The underlying principle is the same! This demonstrates great flexibility within what is undoubtedly a unified system - the system of Vedic Mathematics.

Creative Maths website http://www.users.waitrose.com/~dabsalmon



In March of this year Dr S K Kapoor, a mathematician and author of several books on Vedic Mathematics, had an insight which apparently led to a proof of Goldbach's Conjecture. The origin of this Conjecture dates from the year 1742 when Professor Christian Goldbach wrote a letter to Leonard Euler the great Swiss mathematician in which the opinion was first expressed: that every even number above two can be expressed as the sum of two primes. So, for example, 10=3+7 (where 3 and 7 are both primes) and 18=7+11. (The number 1 is not normally considered to be prime).

In spite of its simplicity and in spite of the extensive efforts of the most distinguished mathematicians since then, the problem has so far remained unsolved and has ranked with Fermat's Last Theorem as one of the great unsolved problems of mathematics.

Dr Kapoor's book "GOLDBACH THEOREM" has just been published by M/s. Arya Book Depot, 30, Nailwala, Karol Bagh, New Delhi 110005 (INDIA), Phone : (00-91-11)5721221, (00-91-11) 5720363, FAX: (00-91-11)5767012. ISBN-81-7063-113-0. Price : Rs. 175/-. We have not yet seen this proof but hope to have a copy of the book soon.

For his excellence and service rendered in the field of Vedic mathematics, Dr Kapoor has been awarded "Shri Guru Gangeshwaranandji Veda Ratna Puraskar-1997" by Bharatiya Vidya Bhavan, Bangalore. The Veda Ratna Puraskar as these are called have come to be accepted as highest epitomes of all honours that a Vedic scholar can strive to achieve in his life time. We have more details about this book and the author which will soon be made available on the web site.

Five courses are being organised for the near future. More details can be found on the web site (click on Calendar) but here we summarise them.
1. Seminars and classes at the University of Skovde in Sweden. 4-8 September, 2000
2. Twelve week introductory course at Bhartaiya Vidya Bhavan, Manchester, England. Starting 16th September, 2000, 11am-12.
3. Ten week introductory course at Imperial College, London. Starting 9th October, 2000, 6.30-8.30.
4. Saturday School for primary school teachers at Manchester Metropolitan University, Manchester, England. 7th October, 2000.
5. Four week evening class at Manchester Metropolitan University, Manchester, England. 17th January - 14th February, 2001. VEDIC MATHS

Apparently there was a news item on Vedic Mathematics in Dubai, United Arab Emirates on 22nd May. This seems to have created a lot of interest over there.

A student at Edge Hill College in Lancashire, England intends to carry out a study comparing a class of schoolchildren who have studied Vedic Mathematics with a class who have not. Michelle Thompson, who is training to be a mathematics teacher, will teach Vedic as well as conventional methods of multiplication and division. This will be interesting!

A PhD student in the U.S. is also interested in making a study of the Vedic methods.

Some more references have come to light for those interested in following up the notes in previous newsletters.

Mathematics Teaching (UK teaching journal) Nos. 62, 63. The article in No. 62 is by Joseph Howse and the one in No. 63 is a response from R Thatcher.

There are sections on the Square in the book "The Language of Pattern" by Albarn, Smith, Steele, Walker. Thames and Hudson, London, 1974. ISBN 0500231907.

As part of recent research - a kind of A4 size brochure has appeared - dated December 1973 called 'islamathematica' by fred ros, keith albarn, antony hutt and joseph howse. (museum voor land- en volkenkunde rotterdam.) it has 53 numbered pages. It covers Islamic art/architecture in relation to number pattern, and includes a section at the end directly summarising aspects of Vedic Maths (vedische wiskunde/joseph howse). This is written in Dutch. If you are able to translate this into English for us please let us know.

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7th July 2000


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