Issue130 - Structures in the Vedic Mathematics Sutras

Vedic Mathematics Newsletter No. 130

A warm welcome to our new subscribers.

This issue’s article is by Kenneth Williams, newsletter editor, and is titled “Structures in the Vedic Mathematics Sutras”.
As indicated here the Vedic Mathematics Sutras can be established as describing a set of natural mental processes applicable over the entire field of mathematics, generated in a natural way and in constant use by us in our daily lives. It means in fact that we are operating mathematically much of the time without knowing it.





Featuring: Latest Research, Reports, Events, Lilavati Poetry Competition
June 12 – 13 2021

From around the globe, enthusiastic students and teachers joined in with the two-day 7th Online International Vedic Mathematics Conference during the weekend 12th – 13th  June. The conference was organised and run by the Institute for the Advancement of Vedic Mathematics (IAVM), a UK based charity operating across all continents. Over 300 enthusiasts attended.

The first day consisted of presentations of research papers demonstrating the latest developments and insights in Vedic Mathematics, together with reports from various experts on working with students during the pandemic restrictions.

One highlight came from Niveditha and Nanditha (aged 11 and 15) who have learnt Vedic Maths from their father Devaraj in Kottayam, Kerala. During lockdown, these intrepid girls realised that many kids had time on their hands at home and so they have been conducting online workshops, called Maths Made Easy, to hundreds of children right across India. Local exam boards have praised their work and this helps increase grass-roots.

Another top story was a report from Ike Prudente in the Philippines where, due to his unceasing efforts, many thousands of teachers and students are getting involved with learning the wonderful methods and quick solutions afforded by this inspirational system from India.

Day two of the conference was devoted to providing free workshops for teachers and students. This involved three breakout rooms on Zoom, each receiving four workshops given by experienced teachers and trainers. These included Vedic methods found in Counting Shapes, Exploring Geometric Designs. Kolam Designs, Digital Applications of VM, Fun Maths with Lego, Strategies in Vedic Maths, and Puzzles from Bhaskhara’s Lilavati.

A new feature for the conference was the Lilavati Poetry Competition, organised by Gowri Ramachandran. Participants were previously invited to compose a mathematical poem or riddle. A total of 61 entries were received in four categories under age groups 6 -10, 11-13, 14 - 16 and open category. These were judged and winners were announced at the conference. The winners in each category are as follows.

Category – 8-10                                                                          Category – 11-14


School / location



School / location


Tanav Patkar

Natick, MA - USA

1st Place

Arjun Patro

Holliston, MA - USA

1st Place

Krishav Thakur

Tokyo Bay Int’l School, Japan

2nd Place

Sanjana Roy

Vaels  Int’l School, India

2nd Place

Adhya Ghatole

Tokyo Bay Int’l School, Japan

3rd Place

Sabarish K

Tokyo Bay  Int’l  School, Japan

3rd Place

Category – 14 - 16                                                                             Category – Open


School / location



School / location


Mrunmayee Yawale


First Place

Sharvari Oak


First Place

Shriya Balaji


Second Place

Chard Aye Alova


Second Place

Aadhya Purohit

Vibgyor high school, India

Third Place

Colonel Ashok Naik


Third Place

Overall, this was a HAVE FUN WITH MATHS event, opening hearts and minds to the beauty and utility of this important discipline.


The Global Forum for Education and Learning (GFEL) selected Swati Dave for its ‘Excellence in Education’ award. Swati collected the award at their conference:

Nominees were rated on the following 5 parameters:

  1. Overall Reach
  2. Industry Impact
  3. Spirit of Innovation
  4. Future Readiness
  5. Market Demand

Swati writes:
“Honored and humbled to be recognized for my work in Education at the GFEL conference, and super proud to bring Vedic Mathematics to the mainstream space.

This is a recognition of our dynamic IAVM team, (James, Marianne, Gowri, Ramya, Prajakti and Raghavendra) who work tirelessly to bring paradigm shift in math learning and math teaching across the globe.

Thankful to all from bottom of our heart who have supported, encouraged and inspired us on this path.”



Dr Yogesh Chandna has started a petition at "Department of Education: Introduction of Vedic Mathematics in our School Curriculum". You can read more and sign the petition here:



The next Teacher Training Course starts 30th August 2021.

For details of this and the several other courses offered by the VM Academy please see:






The book “Vedic Mathematics” by Sri Bharati Krishna Tirthaji was published in 1965. The sub-title is

“Sixteen Simple Mathematical Formulae from the Vedas
(For One-Line Answers to All Mathematics Problems)”

and this suggests the collection of formulae/Sutras described in the book somehow provides a basis for the entire field of mathematics. These 16 Sutras are listed in the table below and have no obvious structure. Many of them are cryptic-sounding and abstruse; some have many applications and others very few. We see that Tirthaji was a holy man, a Shankaracharya, highly educated and highly respected, but we are given these formulae without explanation or derivation. Amazingly unified and coherent though Vedic mathematics is, and though it has provided remarkable breakthroughs in some areas of mathematics, we do need to have a viable explanation for these Sutras if we are to consider them as a basis for mathematics.







1. By One More than the One Before


9. Calculus

2. All from 9 and the Last from 10

10. By the Deficiency

3. Vertically and Crosswise

11. Specific and General

4. Transpose and Apply

12. The Remainders by the Last Digit


5. If the Samuccaya is the Same it is Zero


13. The Ultimate and Twice the Penultimate

6. If One is in Ratio the Other is Zero

14. By One Less than the One Before

7. By Addition and by Subtraction

15. The Product of the Sums

8. By the Completion or Non- Completion

16. All the Multipliers

Mental Functions

We know that mathematics is a vast and expanding field of knowledge, so how can 16 formulae cover it all? An explanation can be found in the way mathematics is constructed, because it is entirely a mental construct: constructed by people through their mental faculties. Therefore the mental functions and abilities which we all possess define the way we construct mathematics. If we examine our mental functions we find that there are not many specific functions we use. We can extend an idea, reverse it, generalise it and so on. We develop these functions without being specifically taught them, and we all develop the same ones. It seems reasonable therefore to look at the structures inherent in the mind to help establish foundations for mathematics.

Mental Functions and VM Sutras

It further seems reasonable to inquire if there is any correlation between natural mental processes and the Vedic Mathematics Sutras. For example the 3 functions mentioned above: extending, reversing, generalising, connect respectively with Sutras 1, 4 and 11. Looking more closely we find that not only do the 16 Sutras all correlate with specific mental functions, but they also fall into groups as described below.



In the table, the 16 Sutras are divided into 4 Tetrads (the word Tetrad simply means a group of four). In this article I would like to outline a framework that shows how the 16 Sutras can be structured in 2 groups of 8 (an analytic group and a synthetic group), and also in 4 groups of 4 Sutras, each of which describes a 3-step cycle.

Analytic and Synthetic Groups

We find that Sutras 1 to 8, in the first column (Tetrads 1 and 2), correspond collectively to processes of a general analytic nature. That is, they deal with proliferation and fragmentation, mathematically mainly involved with computational procedures. And Sutras 9 to 16 (Tetrads 3 and 4) correspond collectively to processes of a general synthetic nature dealing with generalising and consolidating, emphasising topics like algebra and calculus.


These 2 groups further divide to give 4 groups of 4. T1 (Tetrad 1) describes the basic process of generating results or ideas or data on the question or enquiry presented. T2 describes the analysis and comparison of the results from T1. In T3 we begin to synthesise the results of T2 by drawing conclusions from those results. And T4 involves the further consolidation of the results of T3 into greater wholes.

Overall Structure

The Tetrads are each composed of 4 internal mental processes, each one consisting of a 3-step cycle defined by the first 3 Sutras in that Tetrad.

In this way we find that the 16 Sutras form a unified structure consisting of 4 main processes (the Tetrads) each of which is composed of 4 internal processes. The individual Sutras precisely describe the operation of each Tetrad.



Any mathematical inquiry would start with the first Sutra but how far the inquiry gets through the Sutras and Tetrads following will depend on the nature of the inquiry. One inquiry may be ended immediately after it is started if the answer requires no thought, another may end after one or more cycles of the first Tetrad, another may end after the second Tetrad or after the third or fourth.

There is not space here to go into more detail about the operation of the individual Sutras within the Tetrads or how the 4 Tetrads work together. An article is under preparation with full explanations and examples, which also looks into the many implications raised, such as in mathematics education, problem solving, Sutra combinations, how Tirthaji may have derived the Sutras from the Vedas etc. Other outstanding questions about the Vedic system are also answered with reference to this framework.

The Sutras at first seem collectively unstructured, and individually enigmatic and obscure. The arrangement outlined here however gives structure and a unity to the set of formulae. Even Sutras that are particularly enigmatic now make sense in a new way and this could lead to further applications of those Sutras, and others too.



The fact that the 16 Sutras can be linked in this way to well-known and well-used mental processes gives welcome meaning to an otherwise random-looking collection of aphorisms.

As indicated here the Vedic Mathematics Sutras can be established as describing a set of natural mental processes applicable over the entire field of mathematics, generated in a natural way and in constant use by us in our daily lives. It means in fact that we are operating mathematically much of the time without knowing it.

‘Mathematics’ refers to the laws that govern our creation – natural laws. The word ‘Veda’ means ‘knowledge’ and so ‘Vedic mathematics’ refers to a holistic coherent approach to the study of universal law. There are therefore wider implications relating to the individual and the world that are suggested by this view of mathematics and its origins. The structures outlined here offer a more personal view of how mathematics works which we all utilise and to which we can all relate.

Relevant previous newsletter articles:

No. 95: Supersutras

No. 103: Lines of Thought

No. 108: The Sub-Sutras

No. 121: All Methods are Vedic

End of article.


Your comments about this Newsletter are invited.

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Editor: Kenneth Williams

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26th June 2021


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