ISSUE No. 16

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.

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This issue's article is written by Blidi S. Stemn, Assistant Professor at Northeastern University, School of Education, 50 Huntington Ave, Boston, MA 02115, USA. It is from an article in this month's teaching children mathematics journal. You will see it is about what we have previously called the Vedic Square. The figures referred to in the article are not given here but they can be reconstructed from the descriptions in the text.

The Vedic Matrix/Square is a nine by nine square array of numbers formed by taking a multiplication table (up to nine times nine) and replacing each number by its digit sum. The digit sum is found by adding the digits in a number and adding again if necessary:
42 becomes 6 and
56 becomes 11 which becomes 2.
So the first row consists of 1,2,3,4,5,6,7,8,9 and the second row is
2,4,6,8,1,3,5,7,9 and so on.

## VEDIC MATRIX

I did this activity with a group of 13 year olds in a school in Massachusetts, United States. To generate the Vedic Matrix, I asked the students, each student was given a 9 by 9 multiplication matrix and asked to complete the table. If any of the products in cell was more than 9, students were to repeatedly add the digits until the sum was less than or equal to 9 and the result recorded in the corresponding cell on a separate matrix. For example, 8 x 7 = 56; 56 is greater than 9 so add 5 and 6 to get 11. Since 11 is greater than 9 add the digits, i.e., 1+1 = 2 so 2 is recorded in the cell. After generating the matrix, the students were challenged to find as many patterns as possible.

Discussion of the Patterns Identified
· In the 3rd row or column, 3 + 6 = 9 and in the 6th row 6 + 3 = 9.
· In any vertical or horizontal set of numbers, the sum of the first and last numbers is 9 (ignoring the last column and row). For example, in the second row they noticed that 2 +7 = 9; 4 +5 = 9; 6 + 3 = 9; and 8 + 1 = 9. These pairs of numbers can be written as ordered pairs: (1, 8), (2, 7), (3, 6), and (4, 5).
With the exception of the 9th row and column, the sum of the numbers in each column or row is 45 and that when you add the digits of the sum the result is 9 (e.g., 4 + 5 = 9). The sum of the numbers in the 9th row or column is 81 and the sum of the digit is 9.
· If you add the first and the last numbers in each row or column you get the following sequence of numbers: 10, 11, 12, 13, 14, 15, 16, 17, 18. When you add the digits you get 1, 2, 3, 4, 5, 6, 7, 8, and 9.
· The first four numbers generated in the 7th column and row, again ignoring the last number in the column and row, are odd numbers (i.e., 7, 5, 3, 1) while the next four numbers were even (i.e., 8, 6, 4).
· The numbers in the 1st row or column are the reverse of the numbers in the 8th row or column (without taking into account the 9th row and column). This is true for rows/columns 2 and 7; 3 and 6; 4 and 5.

The students were guided to arrive at the following relationships:
· The pair of numbers (1, 8), (2, 7), (3, 6), and (4, 5) from the matrix have some relationship with the nine times table. For example,
(1, 8):    1 + 8 = 9;    18 = 9 x 2, or 81 = 9 x 9
(2, 7)     2 + 7 = 9     27 = 9 x 3, or 72 = 9 x 8
(3, 6)     3 + 6 = 9     36 = 9 x 4, or 63 = 9 x 7
(4, 5)     4 + 5 = 9     45 = 9 x 5, or 54 = 9 x 6

There are twelve 3s and twelve 6s, twenty-one 9s, six 1s and six 8s, six 2s and six 7s, and six 4s and six 5s. Now let us examine some calculations using the above data.
(1, 8): (1 x 6) + (8 x 6) = 6 + 48 = 54 and 5 + 4 = 9
(2, 7): (2 x 6) + (7 x 6) = 12 + 42 = 54 and 5 + 4 = 9
(3, 7): (3 x 12) + (6 x 12) = 36 + 72 = 108 = 2(54) and 1 + 8 = 9
(4, 5): (4 x 6) + (5 x 6) = 24 + 30 = 54 and 5 + 4 = 9
For the 9s: 9 x 21 = 189 = 1+ 8 + 9 = 18; 1 + 8 = 9

One of the fascinating things about this activity is that opportunities exist for making numerous connections among different mathematics concepts at multiple grade levels.

Many students identified a variety of number patterns and their relationships. For example, some found that in the 3rd row, 3 + 6 = 9 and in the 6th row 6 + 3 = 9. Similarly, others noticed that in any vertical or horizontal set of numbers, the sum of the first and last numbers is 9 (ignoring the last column and row). For example, in the second row they noticed that 2 +7 = 9; 4 +5 = 9; 6 + 3 = 9; and 8 + 1 = 9. These pairs of numbers can be written as ordered pairs: (1, 8), (2, 7), (3, 6), and (4, 5).
Some students pointed out that the sum of the numbers in each column or row is 45 and that when you add the digits of the sum the result is 9 (e.g., 4 + 5 = 9). Others noted that the first four numbers generated in the 7th column and row, again ignoring the last number in the column and row, were odd numbers while the next four numbers were even. Figure 3 contains some samples of students' conclusions or observations.
One fundamental characteristic of the Vedic Matrix in terms of digit sums (Figure 2) is that if you do not count the 9, the 1 times row and the 8 times row are the reverse of each other. This is the same with the 2 times row and the 7 times row, 3 times row and the 6 times row, and the 4 times row and the 5 times row. Also, the appearance of the number 9 in many different forms in Vedic Matrix indicates a strong relationship between the matrix (Figure 2) and the nine times row. In the nine times row, the sum of the digits of each product is 9, explaining why the 9s column and row have all 9s. Another important connection is that the above mentioned pair of numbers from the matrix have some relationship with the nine times table. For example,
(1, 8):   1 + 8 = 9;   18 = 9 x 2, or 81 = 9 x 9
(2, 7)    2 + 7 = 9    27 = 9 x 3, or 72 = 9 x 8
(3, 6)    3 + 6 = 9    36 = 9 x 4, or 63 = 9 x 7
(4, 5)    4 + 5 = 9    45 = 9 x 5, or 54 = 9 x 6

Part II: Generating Shapes
An implied premise in the use of the Vedic Matrix mentioned earlier is that when connected, numbers form symmetrical shapes (Nelson, et al, 1993; Shan & Bailey, 1991). To investigate this premise, we asked our students to work in pairs. One student was responsible for connecting all the 1s, 2s, 3s, and 4s with a straight line while the other partner connected all the 5s, 6s, 7s, and 8s. To connect each number, the students placed a tracing paper on the final matrix (Figure 2) and marked off each number using a dot. Next, they connected all the dots with a straight line making sure that all the points are connected. We did a whole class demonstration on an overhead. After the demonstration, the students completed connecting the rest of the numbers and compared their shapes with their partners. Figure 4 shows the shapes of 1 and 8 when connected with a straight line.

Discussion of the Shapes
Before they could complete drawing all the shapes, many students noticed some important connections between each pair of numbers. For example, they found that the shapes of one and eight were reflections of each other. Similar observations were made about (2, 7), (3, 6), and (4, 5) as shown in Figure 5. Some of the students conjectured that two shapes are a reflection of each other provided that the sum of the numbers they represent equals 9. For instance, 1 and 8 are reflections of each other since 1 + 8 = 9. These observations and conjectures about reflective symmetry indicate that for each horizontal set of numbers, there is an identical vertical set of numbers and in each pair of numbers, one is the reverse of the other. In addition to the above observations, some of the students identified different geometric figures they found in their shapes such as triangles, rectangles, octagons, etc. We have included samples of students' responses as figure 6.

The more I look at the Vedic Matrix, the more patterns I find.

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## NEWS

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TALKS AND BOOKS IN BIRMINGHAM

There will be two Vedic Mathematics talks at the Natural Health and Ecology Show, 2nd, 3rd June (one talk on each day) at Ryton Organic Gardens near Birmingham, England. There will be a stand for Vedic Mathematics at the show and books will be available.

CONFERENCE PRESENTATION

Andrew Nicholas gave a talk on Vedic Mathematics at the 9th SEAL (Society for Effective Affective Learning) Conference at The King's School, Canterbury, Kent, UK on 30 March. The talk was very popular with people being turned away as there was not enough space for everyone. Members of SEAL are generally people who are looking for new educational forms and they enjoyed the presentation enormously.. Word went round the conference that this mathematics was something special. One lady from Russia, who could not attend the talk, wants to invite Andrew to give a presentation there.

After the conference Andrew went to China where he researching for his new book, a novel about ancient lost knowledge.

WORKSHOPS IN INDIA

As interest grows in Vedic Mathematics we get more inquiries from people wanting to attend courses or contact others. What is really needed is a country by country list of those active in VM so that we can put inquirers in contact with those near them. We would therefore like to make a list of people in each interested country so that we can put people in touch. Please send your email address and country to us if you would like to be put on a list and also include any other information you like (your area or city would be useful). Then when we get an inquiry from your country we will pass this information on to the inquirer.

TIMES OF INDIA

An editorial in The Times of India, under the title "UGC Seeks the Philosopher's Stone", on 7th April describes a recent proposal of the University Grants Commission to introduce courses on Vedic Mathematics, astrology and vastushastra (position and orientation of land and buildings) in science curricula. You can see the article at www.timesofindia.com

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CORRESPONDENCE

EMAIL: I have been contacted by a teacher in Florida as per your recommendation and we are discussing some ways we can make vedic Math more popular in our respective areas and possibly across the country. I am thinking about forming a study group on Vedic Math. I hope I can get the approval to teach a course in VM. I intend to do a workshop first and then push for a course at my present university or elsewhere. VM is so fascinating!!!!

EMAIL: Thank you for your assistance. Your explanation cleared everything up for me! You have been most helpful. I have one other question that concerns the educational requirements one has to have to teach Vedic Mathematics in an official school environment. Is there a distinct series of courses to take in college? Are Master and Doctorate degrees offered in this subject? Has Vedic instruction taken a firm foothold in the States? My interest is rapidly growing as each chapter of Vedic Mathematics unfolds. If instruction is available in the U.S., I would probably consider a career teaching these amazing methods to others. I am a first-year college student who has at present selected a career as a Research Scientist/Botanist. Vedic mathematics has already begun to change my life--I might switch careers because of it! :)

REPLY: In spite of the obvious merits of the Vedic system it is taking a long time for it to be properly appreciated, though things have picked up a lot in the last couple of years. I think there have been a small number of VM doctorates in India and I am not aware of any standard college courses in Vedic Maths: we are not quite there yet. There have been many courses in many countries but nothing regular as far as I know.
The United States seems to be lagging in appreciating VM. Vedic Mathematics is part of a course in Edinboro University of PA, USA. You may want to contact Dr Blidi Stemn at Northeastern University, details below, who is keen to promote Vedic Maths.

EMAIL: Are you aware of any workshops or talks on Vedic math in U.S this summer or in the next academic year ? I will appreciate getting any info. on this.Thanks.

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If you would like to send us details about your work or submit an article for inclusion please let us know on

Articles in previous issues of this Newsletter can be copied from the web site - www.vedicmaths.org:
Issue 1: An Introduction
Issue 2: "So What's so Special about Vedic Mathematics?"
Issue 3: Sri Bharati Krsna Tirthaji: More than a Mathematical Genius
Issue 4: The Vedic Numerical Code
Issue 5: "Mathematics of the Millennium"- Seminar in Singapore
Issue 6: The Sutras of Vedic Mathematics
Issue 7: The Vedic Square
Issue 8: The Nine Point Circle
Issue 9: The Vedic Triangle
Issue 10: Proof of Goldbach's Conjecture
Issue 11: Is Knowledge Essentially Simple?
Issue 12: Left to Right or Right to Left?
Issue 13: The Vinculum and other Devices
Issue 14: 1,2,3,4: Pythagoras and the Cosmology of Number
Issue 15: A Descriptive Preparatory Note on the Astounding Wonders of Ancient Indian Vedic Mathematics

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Editor: Kenneth Williams

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9th May 2001