This page explores some ways of finding spiritual values in mathematics, including Vedic Mathematics and a foundational theory based on Vedic Mathematics.

Vedic Mathematics

Vedic Mathematics is the mathematics of the ancient Vedic tradition of India. The Vedic tradition is an all-encompassing field of knowledge that is currently being revived through Maharishi Vedic Science. Vedic Mathematics has many definitions and many aspects, ranging from the governing intelligence of the universe to procedures of practical calculation.

Maharishi’s Absolute Theory of Defence—Sovereignty in Invincibility  •  •  A 668-page book by Maharishi Mahesh Yogi, the founder of the Transcendental Meditation program. This book describes how the principles of modern science and Maharishi Vedic Science can be used to create invincibility for any individual and any nation. It contains 90 pages on Maharishi Vedic Mathematics and 33 pages on modern mathematics.  •  In this book, Maharishi describes Vedic Mathematics as the mathematics of the Veda, the mathematics of the governing intelligence of the universe, which calculates instantly, without steps, in a way which is both orderly and evolutionary, on the level of wholeness rather than parts.

Summaries of Published Sources by Maharishi Mahesh Yogi on Mathematics, including modern mathematics and Vedic mathematics  •  200K  •  Summaries of sections of publications and press conferences in which Maharishi Mahesh Yogi has addressed any subject in modern or Vedic mathematics.  •  Revised 9 November 2014.

Is Consciousness a Number? How Maharishi Vedic Mathematics Resolves Problems in the Foundations and Philosophy of Mathematics  •  100K  •  How Maharishi Vedic Mathematics resolves problems in the foundations and philosophy of mathematics. Based on my Masters thesis for a degree in Maharishi Vedic Science at Maharishi International University.  •  Revised 8 January 2005.

Dr. Narinder Puri and Dr. Michael Weinless on Vedic Mathematics  •  2.5M  •  Three articles about Drs. Puri and Weinless endorsing the use of the sūtras of Vedic Mathematics, and its connection to the development of consciousness, as described by Maharishi Vedic Science. Includes “Vedic Mathematics: The Cosmic Software For The Cosmic Computer”, a paper presented at the National Council of Teachers of Mathematics Annual Conference, Chicago, IL, 1988.

Vedic Mathematics Academy  •  •  One of several sites on the system of calculation known as Vedic Mathematics, or Sūtra-Based Computation. This system is based on 16 sūtras or aphorisms and was first published in 1965 by Swāmī Bhāratī Kṛiṣhṇa Tīrtha, Śhaṅkarāchārya of Govardhan Maṭh, Puri, in the east of India. This system uses sūtras and alternative methods of calculation to do arithmetic and other calculations in a way which is easier, more creative, and more enjoyable. This site is aimed at those interested in learning this system, for home or classroom study.

Institute for the Advancement of Vedic Mathematics  •  •  Another site for the sūtra-based system of Vedic Mathematics. This site is promotes research into and adoption of this system, through courses, seminars, workshops, lectures, and conferences.

Sūtras of Vedic Mathematics  •  The 16 sūtras and 16 upasūtras, in Sanskrit and in English translation by Swāmī Bhāratī Kṛiṣhṇa Tīrtha. An alternative system of calculation using these sūtras is outlined in Swāmī Bhāratī’s book Vedic Mathematics. The sūtras can be applied to any branch of mathematics to make it easier and more rewarding.


Numeristics is an alternative, number-based foundational theory for mathematics. Using tools from Vedic Mathematics and other sources, numeristics provides practical, intuitive procedures for developing mathematical structures from a basis in what Vedic Mathematics describes as the absolute number or zero. An extension of numeristics, equipoint analysis, provides procedures for doing calculus by operating within zero.

An Overview of Numeristics and Equipoint Analysis A brief overview of the four papers listed below, each of which is monograph length. The overview paper summarizes key findings. Definitions, proofs, and secondary findings are in the papers below.

Numeristic Mathematics, paperback edition  •  •  Also available from these Amazon marketplaces: Canada, UK, Germany, France, Spain, Italy, Japan  •  A softbound printed collection of the four numeristics documents below.

Numeristic Mathematics, Kindle edition  •  •  Also available from these Amazon marketplaces: India, Canada, UK, Germany, France, Spain, Italy, Japan, Netherlands, Brazil, Mexico, Australia  •  A Kindle collection of the four numeristics documents below.

Numeristic Mathematics, hardback edition  •  •  A hardbound printed collection of the four numeristics documents below.

Numeristics: A Number-Based Foundational Theory of Mathematics  •  600K  •  Inspired in part by the recent revival of the Vedic tradition of India, as expressed by Maharishi Mahesh Yogi in his Vedic Mathematics. This paper gives the fundamental ideas of numeristics as they apply to arithmetic and elementary algebra. Numeristics includes an alternative approach to analysis called equipoint analysis, described in the paper below.  •  Seventh edition, 11 August 2020.

Equipoint Analysis: A Numeristic Approach to Calculus  •  900K  •  Extends the concepts of numeristics to analysis. It develops a theory of analysis based on infinitesimals which are all exactly equal to zero, and infinite values that are their reciprocals. Concepts derive from Maharishi Mahesh Yogi’s Vedic Mathematics, Charles Musès’s analysis of zero and infinity, and Abraham Robinson’s non-standard analysis. This theory uses multiple levels of sensitivity to evaluate equality and defines derivatives and integrals solely in terms of elementary arithmetic operations.  •  Eighth edition, 11 August 2020.

Divergent Series: A Numeristic Approach  •  600K  •  Infinite divergent series can generate some striking results but have been controversial for centuries. The standard approaches of limits and methods of summation have drawbacks which do not account for the full range of behavior of these series. A simpler approach using numeristics is developed, which better accounts for divergent series and their sums. Numeristics is introduced in a paper above.  •  Tenth edition, 11 August 2020.

Repeating Decimals: A Numeristic Approach  •  400K  •  Derives several theorems on terminating and repeating decimals, along with illustrative examples. It is shown that every rational number has a unique decimal representation, except for a few which have two representations. The criteria by which a rational number has a terminating, pure repeating, or mixed repeating decimal are derived. Using numeristics, infinite decimals on the left side of the decimal point are explored. Numeristics is introduced in a paper above.  •  Fifth edition, 11 August 2020.

Beautiful Mathematics

There is much that is beautiful in mathematics. Here are some forumulas and figures that I particularly like.

Beautiful Arithmetic  •  100K

Beautiful Algebra  •  100K

Beautiful Trigonometry  •  200K  •  I took a particular liking to trigonometry. This document is a fairly comprehensive collection of trigonometric formulas.

Beautiful Calculus  •  100K

Gödel theorems and philosophy

Quotations by Kurt Gödel, recorded by Hao Wang in his biographies of Gödel and by Rudy Rucker in one of his books.

Gensler’s Gödel’s Theorem Simplified: Summary and revisions  •  300K  •  A summary and extension of Harry Gensler’s book on Gödel’s famous incompleteness theorem. Gödel’s theorem is widely regarded as technically difficult, but Gensler has developed an admirable framework for a simplified approach which still retains the theorem’s essence. Gensler develops a set of forumulas which go about 90% of the way towards proving the theorem. This document has a revised set which provide the other 10%.  •  Revised 27 June 2017.

Math videos

Powers of ten videos

These videos zoom in and out of ordinary objects to show the size scales of physical structures in the universe, from very small to the very large.

Powers of Ten  •  Charles and Ray Eames  •  1977  •

Cosmic Eye  •  DanApps  •  2012  •

Cosmic Voyage  •  Produced by Jeffrey Marvin, directed by Bayley Silleck  •  1997  •

Powers of Ten—Ultimate Zoom  •  Aurelius  •  2010  •


Numberphile  •  •   •  A series of short videos (5 to 15 minutes) about numbers and other mathematical topics for the general public. Marvelously clear, concise, and accessible.

Other math videos

Donald Duck in Mathmagic Land  •  Walt Disney  •  1959  •  •  Donald Duck is guided by the “true spirit of adventure” through the “wonderland of mathematics.”

My Hero Zero  •  Schoolhouse Rock  •  Bob McDorough  •  1973  •  •  The virtues of a powerful nothing.

Amateur mathematicians

Famous amateur mathematicians, an annotated list of individuals who have made significant contributions to mathematics without an advanced degree in mathematics.

Journal articles

My Google Scholar profile  •

Articles of mine that have been published in academic journals, with citation count from Google Scholar as of 2 September 2017

Circular and hyperbolic quaternions, octonions, and sedenions, Applied Mathematics and Computation 28 (1988) 47–72 67 citations
Circular and hyperbolic quaternions, octonions, and sedenions—further results, Applied Mathematics and Computation 84 (1997) 27–47 62 citations
Modular parts of a function, Applied Mathematics and Computation 36 (1990) 63–74 2 citations
Finite Fourier series, Applied Mathematics and Computation 28 (1988) 89–96 2 citations

Articles in which I received an acknowledgement

M. P. Ulmer et al., Spartan 1 X-ray observations of the Perseus Cluster: Comparison of the iron abundances and temperatures in the inner and outer regions of the cluster, The Astrophysical Journal 319 (1987) 118–125 Acknowlegement p 124
N. Kawai et al., X-ray observations of the Galactic Center by Spartan 1, The Astrophysical Journal 330 (1988) 130–141 Acknowlegement p 139
W. Snyder et al., Spartan 1 X-ray observations of the Perseus Cluster. II. The distribution of flux and hardness ratio out to a radius of 50 arcminutes, The Astrophysical Journal 365 (1990) 460–470 Acknowlegement p 469