Welcome to the Online International Journal of Vedic Mathematics.

Here we offer articles on Vedic Mathematics that are a bit technical or long for the Newsletter.

We welcome new articles. If you would like to submit an article to be published here please contact Kenneth Williams at . Articles should aim at the Vedic ideal of simplicity and directness and should be original. They should be well written and checked, and will be carefully examined by our team of Vedic Maths experts before acceptance.

Please note: occasionally embedded PDF articles fail to display. Refreshing the page can solve this. However if the problem persists, then clicking on the red PDF icon next to the article link displayed below, displays the PDF of the article directly in your browser.


62.  “Generality & Applicability of Vedic Polynomial Factorization: Quadratics (Simple & Homogenous), Cubics and GCF” by Riddhiman Jain and Shreyansh Jain, 2024.
Five different Vedic mathematics sutras are proved and discussed using modern mathematical methods.

61. "Polynomial Osculation" by K. Williams, 2023.
Two interlinked methods of factorising polynomials are shown: How Tirthaji’s osculation technique can be applied to polynomials and how it not only detects factors of polynomials but gives the quotient as well. It can therefore give two factors simultaneously. Furthermore, if the result of the osculation is not zero that result can itself be used to factorise the polynomial.

60. "Proof of Nikhilam Sutra by using modular arithmetic"
by Ranjeet Kumar, K. Senthil, S. Mitra, Amitava Roy, R. I. Bakhtsingh, 2023.
The range of application of the Nikhilam Sutra is explored and proved for general modular number systems.

59. "DNA and the Vedic Maths Sutras"
by K Williams, 2023
The paper shows parallels between the structure of the DNA and the structures within the Vedic maths Sutras outlined in the recent paper (item 57 below).

58. "Steps To Glimpse And Imbibe The Organization Format Features Of Text Of Vedic Ganita Sutras And Upsutras"
by Dr S K Kapoor, 2022
Dr Kapoor gives a comprehensive and far-reaching account of how the VM Sutras and sub-Sutras are related to Sanskrit texts and Vedic knowledge.

57. "Structures in the Vedic Mathematics Sutras"
by Kenneth Williams, 2021    short video   OR
By correlating the sixteen Vedic Mathematics Sutras with natural mental functions, observable by anyone, the seemingly obscure list of Sutras fall into simple patterns and structures with multiple applications. And they are carefully arranged by Tirthaji to take us through a sequence of analytic and synthetic processes from initiation of an inquiry to its conclusion.

56. "General Case Integer Subtraction"
by Chard Aye R. Alova, 2021
With the system of Vedic Mathematics, subtraction can be done with a much simpler, easier, and more creative method which can change the way pupils see subtraction as a
fundamental operation in Arithmetic and in the whole of Mathematics. This paper shows the firepower of the sutras All from Nine and the Last from Ten for the General Case Arithmetic Subtraction, and with the help of the sutra Transpose and Apply for the General Case Integer Subtraction.

55. "Cube and Cube Root"
by Angela Pierri, 2021
In the first part this paper presents an alternative method to calculating the Cube by a number of 2 digits, or Binomial. Possibly extendable to a number of 3 digits, too. In the second part it presents a method for extracting Cubic Root, about a base of 3 digits, in a few steps.

54. "Continued Fractions using the Sesa Sutra"
by Kenneth Williams, 2021
One of the Sūtras of Vedic mathematics is shown to have useful applications in various ways in relation to continued fractions. We see how to easily convert a given fraction to a continued fraction and vice versa, how to get the convergents and the accuracy involved in switching to any given convergent. Some applications of continued fractions are also briefly described.

53. "Highest Common Factor and Lowest Common Multiple using the Sesa Sutra"
by Kenneth Williams, 2021
This paper shows how the Sutras of Vedic mathematics can be used to obtain highest common factors and lowest common multiples of two or more numbers, using the Vedic Mathematics Sutra The Remainders By the Last Digit. Includes applications to polynomial expressions.

52. "Simple Harmonic Motion using Proportion"
by Kenneth Williams, 2020
This paper shows that there is an alternative approach to this topic, allowing easy solutions using only Proportion, and showing simple relationships between distances, velocities and accelerations. This makes the solution to such problems child’s play and like solving a puzzle by finding missing values in a 3 by 3 grid.

51. "Earth's rotation and revolution"
by Kenneth Williams, 2020    short video
This paper takes up Tirthaji's reference to the rotation of the Earth and its revolution around the Sun and explores how the Sun's position, for any observer on the Earth and for any time of day or year, may be predicted, relative to the observer's horizon.

50. "Why in many situations Vedic Mathematics is a much better choice than modern Mathematics"
by Pankaj Dhar, 2020
This paper offers a comparison of Vedic Mathematics with Modern Mathematics, comparing and contrasting their respective approaches and advantages.

49. "Arcs and Chords of Circles, Angles and Sines of Angles"
by Kenneth Williams, 2020.  short video
On a neat method for evaluating Sines, Cosines and their inverses and Tangents. And how the method throws light on an approximate general method for evaluating other functions.

48. "Expressing a Number as a Sum of Two Square Numbers"
by Kenneth Williams, 2020.  short video
On how to express a given number as a sum of two squares - special and general method. The general method is in relation to Tirthaji's chapter 31.

47. "Multiplication by Nines and its Use in Astronomy"
by Kenneth Williams, 2019.  short video
Shows how Tirthaji's method for multiplying by 9, 99, 999 etc. (and by 399 etc.) can be used to determine astronomical positions.

46. "Practical Mathematics"
by Kenneth Williams, 2019.  short video
Shows how to find the equation of a perpendicular bisector, equation of a circle or parabola through 3 given points, and how these simple and similar patterns suggest a more practical approach to mathematics.

45. "Sum and Difference of squares"
by Kenneth Williams, 2019.  short video
This paper examines, explains and discusses Tirthaji's chapter 31.

44. "Multiplication of n numbers close to a base using ‘Nikhilam’ sutra"
by Vijayakrishna J, Rajagopala Rao M, 2019.
This paper shows how any number of numbers close to a base number can be multiplied.

43. "Permutations and Combinations"
by Kenneth Williams, 2019.
This paper shows how the methods of Vedic mathematics can be used in evaluating Permutations and Combinations.

42. "Factorisation and Differentiation"
by Kenneth Williams, 2019.
This paper explains and enlarges Chapter 22 of Tirthaji's book

41. "Roots of Polynomials"
by Kenneth Williams, 2018.
This paper shows how we can easily get sums of any like powers of roots of any polynomial without the need to actually find the roots themselves.

40. "Powers of 9"
by M Rajagopala Rao, 2018.
This paper shows how the coefficients in Pascal's Triangle can be used to easily obtain powers of 9.

 39. "Statistics and Vedic Mathematics"
by Kenneth Williams, 2018.
In this paper from the 2018 Online VM Conference a chart is presented that shows the whole subject. Then how the techniques used in Statistics are related to the way the mind works and hence to the Sutras of Vedic Mathematics.

 38. "Finding Cumulative Binomial Probabilities using Only the Last Terms"
by Kenneth Williams, 2018.
This paper from the 2018 Online VM Conference shows how the columns of Pascal’s Triangle give coefficients for cumulative Binomial probabilities.

 37. "From Vedic Square to the Digital Root ‘Clockworks’ of Modulo 90 Factorization"
by Gary Croft 2018.
This paper demonstrates that the fundamental patterns underlying modulo 90 factorization at the digital root level can be traced to the Vedic Square.

 36. "Squaring Numbers Consisting of a Repeating Set of Digits"
by Mike Greenlaw, 2018.
How to square numbers like 365365 or 365365365 etc. by simple addition.

 35. "Solution of Triangles"
by Kenneth Williams, 2017.
Paper from the Delhi 2017 Conference, showing how triangles can be easily solved using the Vertically and Crosswise Sutra.

 34. "Swami Tirtha’s Crowning Gem"
by Kenneth Williams, 2016.
Paper from the 2016 Online Conference, showing how the Crowning Gem method of VM extends to powers, roots and solution of polynomial equations.

 33. "Teaching Calculus"
by Kenneth Willliams, 2015.
Paper from Bangkok Conference 2015, showing a new approach to calculus teaching.

 32. "Complement Method"
by Vitthal B.Jadhav, November 2017.
Left to Right Addition / Subtraction of Two Numbers by using 10 Point Circle (COMPLEMENT METHOD)

 31. "Bharati Krishnas Special Cases"
by Kenneth Williams, June 2017.
Kolkata conference article on VM Special Methods.

 30. "General Formula for Multiplication"
by Romith Rao, April 2017.
A 3-step method to multiply any two integers.

 29. "Cube Roots"
by Rohan Chandra, December 2016.
A variation of finding the cube roots.

 28. "Multiplying numbers nearing multiples of different bases"
by Pietro Nolasco, November 2016.
Multiplying numbers near different multiples of different bases.

 27. "Base Division"
by Nathan Annenberg, November 2015.
A Way to Better Focus the VM Base Division Algorithm

 26. "Multiplying Numbers with Different Bases"
by Sameer Mittal, June 2015
Multiplying Numbers with Different Bases

 25. "Power of Numbers"
by Sanjay Dixit, May 2015
Expansion of application of Binomial theorem to find higher powers of two, three and four digit numbers.

 24. "Vedic Mathematics"
by Dr. S. K. Kapoor, May 2015
Organization  format of Ganita Sutras runs parallel to the organization format of Sakala Rig-Ved Samhita, the source Vedic scripture.

 23. "Squaring Made Easy"
by Rohan Chandra, May 2015
An easy way to square numbers 2-digit numbers where the last digit is over 5.

 22. "Cubing 3 Digit Numbers by the Ratio Method"
by Kenneth Williams, March 2015
Extending Swami Tirtha's ratio method to cubing 3-digit numbers.

 21. "Divisibility Tests"
by Rohan Chandra, January 2015
Divisibility Tests.

 20. "Decimal Form of a Fraction - Part 1"
by Badriya Raihani, October 2014
This is a good introduction to recurring decimals.

 19. "The Psychology of Vedic Mathematics"
by James Glover, January 2014
This discusses the question as to what is the nature of the Vedic sutras of Sri Bharati Krishna Tirthaji’s system such that they could cover all of mathematics.

 18. "Calculating Powers Near a Base Number"
by Stephen Mogomotsi Modisane, November 2013
This extends Bharati Krsna’s method for squaring and cubing near a base number.

 17. "Powers and Cyclicity"
by Debasis Basak, November 2013
Finding the last digit(s) of any number to any power (cyclicity)

 16. "Observations from 'Figuring' by Shakuntala Devi"
by Sarwan Aggarwal, November 2013
Notes and discusses some connections between this book and Vedic Mathematics.

 15. "Extending the application of the Vedic Maths sutras"
by James Glover, October 2013
Extending the Application of the Vedic Maths Sutras

 14. "Two Step method"
by Vitthal B. Jadhav, October 2013
Novel Method for Squaring and Cubing of Any Number

 13. "A Different Osculation Approach to Test Divisibility of Numbers"
by Raajesh Srinivasa Rama, August 2013
This shows four easy but very effective modifications that can be applied to Tirthaji’s osculation procedure.

 12. "Nth Power of Pingala Chanda"
by Ranjani Chari, July 2013
This shows a method that was given by Pingala Chanda in 200 BC of raising a number to some power.

 11. "Novel Algorithm for 'Nth Root of Number' using Multinomial Expansion"
by Vitthal Jadhav, March 2013
This gives a general algorithm to extract the nth root of any number. It also discusses various novel approaches to expand multinomial.

 10. "Novel Methods for 'Reciprocal of Prime Number' using VM Osculator"
by Vitthal Jadhav, January 2013
This shows a number of applications of the osculator in recurring decimals.

 9. "Solving Systems of linear equations using the Paravartya rule in Vedic Mathematics"
by D.K.R. Babajee, August 2012
This develops Tirthaji's method for solving simultaneous equations, showing how it can be used to solve systems of equations.

 8. "Recursion based Derivation of Duplex Square Method"
by Vitthal Jadhav, June 2012
This proves the Vedic duplex method of squaring numbers.

 7. "A Method for Finding the Square of any Number"
by D.K.R. Babajee, June 2012
An unexpected extension of the method for squaring a number near a base which leads to a general method of squaring.

 6. "Faster Division by 10n-1 or Monodigit Number"
by Vitthal Jadhav, April 2012
Dividing by monodigit numbers like 7777, with proof and example.

 5. "Ellipse Problem Solved using Triple Code Numbers"
By Steven Vogel, March 2012
Here we see how to use triple code numbers to solve the problem of finding integral points on an ellipse where the ellipse must also be defined in terms of integers.

 4. "Triples and the Mandelbrot Set"
by Kenneth Williams, March 2012
This introduces the Mandelbrot Set and shows how it can be explained more simply than the usual way (which involves a knowledge of complex numbers). No prior knowledge of triples or the Mandelbrot set is required.

Link to PDF for "Magic of Sub-Sutra" article3. "Magic of Sub-Sutra"
by Sarwan Aggarwal, February 2012
Showing applications of the Vedic sub-sutra: "For 7, multiplier is 143"

Below are articles from previous newsletters which are also of a more technical nature.

2. AN INTERESTING VEDIC MATH APPLICATION (on compound interest) by Steven Vogel


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  • “Pupils who would normally not want to volunteer are putting their hands up to answer questions so quickly that it's hard not to look surprised!”
    Paul Phillipson-Masters, maths teacher, Scotland
  • ”… have been trying some of the methods with my students and they have started to create a real buzz around school. ”
    Garry Marshall, Maths Consultant, Manchester Enterprise Academy.
  • ” Even during the break they stayed at their desks, taking numbers randomly and doing subtractions. ”
    Daniela Panait, Romanian teacher
  • ”now we can see clearly the beauty of VM and how blessed my youngest child and my students will be to experience VM very young. ”
    Sonya Post, Teacher, USA
  • ”Near 80 multiplications. They were eager to learn and to perfect their skills. They don’t want to write multiplications any more. It was wonderful. At the end, they were not tired and asked for more. ”
    Didier Auroy, Maths teacher, France
  • ”They were amazed and would never forget the lesson for rest of their life…”
    Rajni Obhrai, maths teacher, London
  • ”She is now pushing me to learn more and more Vedic maths to keep up with her insatiable appetite for mathematics. ”
    Vera Stevens, maths teacher, Australia
  • ”But the interesting part of it is once you start doing it you start understanding the power of VM and you just get drawn towards it. ”
    Lalit Shah, Maths teacher, USA
  • ”… and am amazed at such simplicity! ”
    Claudia Brow
  • ”My entire attitude about math changed today. ”
    Barbara Kithcart, USA
  • ”All I can hear from them is "wow" and they get frenzy when they started to be able to solve math problems up to 4-digits times 4-digits multiplication in their heads. ”
    Rosendo Jr. Agulto, Philippines
  • ”For the first time students were eager to do maths and wanted more of it. ”
    Newsletter 18
  • ”I taught calculating squares and square roots to 9th grade kids on this street and they had their mouth wide open in amazement. ”
    Sunitha Ramaiah, maths teacher, India
  • ”If I'd have had VM 15 years ago it would have saved our family years of grief and stress. ”
    Sonya Post, Teacher, USA
  • ”My grandchildren will not have a fear of math because of what you taught me today. ”
    Barbara Kithcart, USA
  • ”Once you get into it you just don’t want to stop because it’s so powerful. ”
    Lalit Shah, Maths teacher, USA
  • ”The more I am learning the more awed I feel about the simplicity of the Vedic System. ”
    Mudaliar Shraddha Teacher, USA
  • ”I shared one of the addition technique with some my students and reactions on their faces were amazing. ”
    Mudaliar Shraddha Teacher, USA
  • ”Mom said it's clearly due to the VM lessons and she is now using them all the time! ”
    Hagit Genosar, Maths teacher, USA
  • ”They have started loving maths. ”
    Shylashree Ravishankar, Teacher, India
  • ”It’s much easier, simpler, faster, more interesting, more exciting, and full of fun. ”
    Tarlika Desai, Maths Teacher, USA
  • ”I feel overwhelmed by the power of this Vedic Math- not known or heard before. ”
    Tarlika Desai, Maths Teacher, USA
  • ”I feel the more I delve into it, more treasure I am getting from it. ”
    Tarlika Desai, Maths Teacher, USA
  • ”These days he is following only VM to do multiplications and he feels as if he is a magician. ”
    Madhavi Bandi, Maths teacher, UK
  • ”I'm demonstrating . . . to PhD-students . . . They all wonder why they have not been taught this method... ”
    Eric Groenendijk, Mathematician, The Netherlands
  • ”And I see my kids walk into the class with that extra bounce and energy. ”
    Mudaliar Shraddha Teacher, USA
  • ”Vedic Math has shown me new path to light of mathematics. . . . I have never enjoyed Math like this before. ”
    Tarlika Desai, Maths Teacher, USA
  • ”I am here now learning Vedic Mathematics and enjoying every moment of it. I cannot believe myself that I am taking so much interest in learning and practising mathematics. ”
    Ashok Nair, Teacher, Canada
  • ”Words really don’t begin to describe this innovative way to solve math problems”
  • ”. . . it will really set your mind to thinking about math differently. ”
  • ”The thing that surprised me most is the fact that VM is a whole system. VM by nature is fun to work out. ”
    Sonya Post, Teacher, USA
  • ”And when I introduced them to the method left to right, the reaction is something memorable. The most dumb students (branded by other teachers) now compete with toppers to answer the sums. ”
    Shylashree Ravishankar, Teacher, India
  • ”I had never seen math like this before. My whole attitude towards learning math and teaching math to my kids has changed. Vedic math is a complete math system that teaches us to find patterns, simplify complex equations and make it fun. ”
    Sangeeta Ahuja, Occupational Therapist, USA
  • ”To them, the techniques you shared with me will make math seem like a game. If I had learned 40 years ago what you taught me, I would have chosen a different field of study. ”
    Barbara H. Kithcart, Librarian Hancock Elementary School
  • ”Vedic Maths is simply incredible but incredibly simple - and I'm loving it!”
    Paul Phillipson-Masters, maths teacher, Scotland
  • ”The best part was we all had fun and after the presentation I was bombarded with please teach my son/daughter Vedic Math. ”
    Mudaliar Shraddha Teacher, USA
  • ”I was amazed how fast I could do maths…”
    Shubham Goyal, High School Senior student in USA
  • ”This time I was using the Vedic Mathematics and not only ended up finishing but I got 1st place in my entire school. ”
    Shubham Goyal, High School Senior student in USA
  • ”I have found that VM has opened my mind and expanded the possibilities when dealing with different maths problems. ”
    Damen Pitiroi, maths teacher, New Zealand
  • ”"I have also received responses like "why are they not teaching this in all schools?" or "I wish I was taught this system when I was in school"”
    Damen Pitiroi, maths teacher, New Zealand
  • ”Vedic is about working smart, not hard.”
    Mia Liley. Maths teacher, USA