Number and the Suya Indians: Lynne McTaggart Explains Why 10-3 = 13

Read Lynne McTaggart’s post on the Suya Indians of Mato Grosso, Brazil, and how their way of thinking about number is completely different from our own.

When 10-3 = 13
May 14th, 2010 by admin · 1 Comment

Recently, an American researcher from the University of California was conducting research on the Suya Indians of Mato Grosso, Brazil, attempting to determine how they count. This group of Amazonian Indians are largely famous for their music; Anthony Seeger, a Professor of Ethnomusicology at the University of California, Los Angeles, who has produced a book called Why Suya Sing, says that their singing is used to create community, establish relationships and social identity and also formulate ideas about time and space.

Singing, to a Suya, is hard and soft science.

Math lesson
This particular researcher was investigating the level of sophistication of the Suya concerning mathematics. Many scientists examining cultural differences over number systems operate on the assumption that many native cultures basically don’t have language to describe quantities of things; for instance, the Piraha people use the same word ‘hoi’ to describe ‘about one’ and ‘about two’; the only difference is a subtle alteration in inflection of pronunciation. The much-studied Munduruku in the Amazon have words for numbers only up to 5.

This has led many scientists to examine whether human beings have innate numerical skills or whether it is simply a part of cultural conditioning. Is it possible to operate entirely without numbers?

So this particular researcher asked a member of the Suya tribe what was the correct answer to the following numerical problems: If you had 10 fish and gave away three fish, how many would you have?

The Suya answered without hesitation and as though the researcher were a bit dull-witted to have even asked the question.

As anybody in the village could tell you, the answer, of course, is 13.

Minus equals plus
This was how he worked it out. In the Suya tradition, whenever you give something away to someone else, the recipient pays you back double. So if he gave three fish to his brother, he said, his brother would have to give him back two times three fish, or six. So added to his 10 original fish he would first have 16 fish.

Once he deducted the three fish he originally gave his brother, he would have a net increase of three, or 13.

So, 10-3 = 7 in Western mathematics transforms into 10 + (2×3) – 3 = 14 in Suya mathematics.

In fact, the native was dismayed at the American version of the equation. He does not view giving away as equivalent to subtraction. He finds the entire notion of it abhorrent.

“Why is it that ‘giving’ is always seen as a ‘minus’ for white people?’ another member of the tribe asked. “I know that you want me to use the minus sign instead of the plus sign, but I don’t understand why.”

This was a little shocking to Alex Bellos, the author of the recently published Alex’s Adventures in Numberland (Bloomsbury, 2010), a study of cultural differences in mathematics. He began the study of his fascinating book with the belief that numbers are a universal language – the way in which we could, say, communicate with extra terrestrials — only to find that our basic understanding of arithmetical relationships depends upon cultural context.

Relationships in the numbers
I find the story delightful for several reasons.

It reveals something very profound not simply about mathematics but about how different cultures view relationships in general, particularly how we view ourselves in relation to other things.

Our sense of mathematics very much depends upon how we define our world, and whether we view ourselves and all the things around us as individual entities separate from each other or inherently intertwined.

Many non-Western societies — pre-literate cultures such as the Aborigines, the ancient Greeks and the Egyptians, the adherents of Eastern religions such as Buddhism, Zen and Taoism, and a number of modern indigenous cultures — conceive of the universe as inseparable, connected by some universal energy ‘life force’. The beliefs of many tribal societies about this central energy force have many similarities, suggesting that an intuitive understanding of the interconnectedness of all things is fundamental to human experience.

This central belief breeds an extraordinarily different way of seeing and interacting with the world. These traditional cultures believe that we are in relationship with all of life – even with the earth itself. They hold a very different notion of time and space as one vast continuum of ‘now’ and ‘here’.

They even perceive the world out there very differently. We see the thing; they see the totality, the relationship between the things. To an indigenous native, math and the song are equivalent — all about the plus sign, the connection, in this instance, between the man with the fish and his brother.

We would do well to take a few math lessons from the Amazon.

For Lynne McTaggart’s blog and RSS feed click here!

Published in: on May 14, 2010 at 2:01 pm  Leave a Comment  

The problem of zero – Graham Flegg’s discussion in Numbers through the Ages


The following information is quoted from Graham Flegg’s wonderful account of number writing over historical time.

“The Sanskrit name for zero is sunya; the Arabic name is as-sifr. Both words have the same signification: ‘the empty’. The Arabic word sifr was transcribed into medieval Latin as cifra or zefirum. However, these two Latin words acquired quite different meanings. The word zefirum (or cefirum, as Leonardo of Pisa wrote it in the thirteenth century) retained its original meaning ‘zero’. In Italian it was changed to zefiro, zefro or zevero, which was shortened in the Venetian dialect to zero. On the other hand the word cifra acquired a more general meaning: it was used to denote any of the ten signs 0,1,2,…,9. Hence the French word chiffre, and the English word cipher. The old English expression ‘he is a mere cipher’ meaning ‘he counts for nothing’, still shows that the original meaning of cipher was ‘zero’.”

Flegg goes on to show that this ambiguity existed in French as well. Although the word cipher was eventually used for the Hindu-Arabic numerals themselves, Flegg explains that 15th century people found it puzzling that zero was “nothing,” on the one hand, but on the other hand, by placing it after another digit the value of the numeral was multiplied by ten.

Apparently even in the early 15th century, Germans continued to use the Roman numerals. It was only in the merchant and banking cultures of Italy that the Hindu-Arabic numerals, together with the place-value system and 10 distinct numerals, finally took hold.

Numbers Through the Ages, edited by Graham Flegg. Macmillan: The Open University, 1989, pp. 126-129.

Published in: on April 29, 2010 at 4:03 am  Leave a Comment  

Thoughts on Vedic Math

The following notes are a personal record of my discovery of Vedic math. I am a language teacher with an ordinary school math background, and I first encountered Vedic math on the Internet in the process of looking for an appealing approach to use in teaching math to my own children. This discovery has been one of the great surprises of my life, and I hope to briefly be able to explain why I think this was the case.

Prior to my encounter with Vedic math, I had become discouraged with conventional math instruction, due in part to the seemingly random aspects of the learning process. The problems seemed randomly chosen and arranged, and the emphasis was more on accurate arithmetic than on interesting concepts, so typical worksheets seemed laborious. I had a vague idea that mathematical patterns appear in nature, but nothing I had done in my schooling pointed to anything striking here. Discovering Vedic math was a lot like wandering in an arid valley and coming across potshards from a long-lost civilization. Whether or not people believe that the mathematical understanding of this long-lost civilization could have been meditated through contact with a piece of text, Vedic math discloses a world of complex and beautiful patterns in nature reflecting themselves in the number world, and vice versa.

There were several surprises here. The first surprise was that the Vedic math tools were empowering. The method of transportation, so to speak, made me feel like I was wearing seven-league boots or stilts, or that I had grown wings. It was fun because I was making a great leap each time I calculated. I didn’t always understand what was powering my vehicle, not having a full understanding of the underlying patterns, but it was a pleasure to apply the simple method of squaring two-digit numbers ending in 5 and the rapid way to multiply two numbers close to a base, as well as seeing the magic squares uncovered by multiplication. The one-line method itself, even if one makes mistakes in arithmetic, is very appealing because each result is a satisfying step forward.

It was also empowering to realize that there is more than one way to solve a problem. I found that there were four different ways of squaring two-digit numbers ending in five, for example, and each one is easier than the last. The doubling and halving exercise also shows that there is more than one way to multiply two numbers. I had always calculated with a pencil in hand, and it had never occurred to me to turn 14 X 3 into 2 X 21, and this technique opens up quite a few vistas. I was so thrilled with these methods that I feared there were only a few of them. When I showed my friends the patterns of nine in digit sums, they also asked “if there were more like that.” I was having so much fun that I was afraid the amazing patterns would be exhausted with the amazing digit sums of nine and its multiples. My fear turned out to be quite groundless. Recently I watched a video that showed that water that had been subject to certain positive conditions – analogous to the positive environment of a child who is loved — fell into amazingly beautiful patterns, while under other distressed conditions, this was not the case. It appears that patterns underlie everything in the universe, and Vedic math commands an understanding of this.

The third surprise was the fact that so few people had any familiarity with these methods. When I showed the multiplication of two-digit numbers near a base and the squaring of two-digit numbers ending in five to a medical doctor friend and a businessman friend who calculates rapidly in his head, I found they were completely unfamiliar with this approach. Both could calculate mentally much more rapidly than I can with conventional methods, but even their rapid mental skills were no match for what I could do with the Vedic approach. I am interested in what accounts for this situation. I think of it a lot like a valley in which a people has been living for millennia. There are mountains on every side of the valley. One might suppose people could climb the mountains and see what is on the other side, but few people do so. The mountains are not especially forbidding, but the belief system of the people restricts their freedom of movement, so to speak. One of the assumptions or beliefs in their system is that there is simply nothing much to see on the other side of the mountain. What about paths crossing the mountains? They are overgrown.

A recurrent memory helped me to understand my reasoning. I was about 8 years old and sitting at the top of the stairs in the old house we lived in, and the number 23 occurred to me. I was very fond of magic tricks at this time, and I was fond of creating number codes for time capsules which I would hide in places in the house that I thought no one would find until years later. Thinking about the number 23, it occurred to me to add the digits, as part of a code, though I suspected that if it were in any way valuable or meaningful to add digits, I would have been told about this already – presumably in school. The concept of irrational numbers also put me off. I was familiar with the term irrational mainly to refer to behavior that was considered stormy, unreasonable or inappropriate, so the idea that numbers were irrational resulted in a mild distaste for fractions that could only be turned into decimals that went on forever. What a surprise to learn that decimals translated from fractions such as 1/7 or 1/17 or 1/19 re-occur in sometimes amazing and even predictable patterns that can be worked out with simple methods.

I will list here a few additional beliefs that were shattered by my encounter with Vedic math, along with the new concept that took their places:

* You need a pencil to do a math problem – you can do complex math in your head.

* Math has to be taught – Math is intuitive;

* Math is for grownups – Math can be done by children, effectively and rapidly;

* The early steps of any learning project in math have to be repetitive and arduous – The early steps of learning math can be fun for children.

* A child cannot understand algebra. – Children reason naturally.

* Algebra and multiplication are separated by a wall. One must work extensively with arithmetic before venturing onto algebra, and the same applies to other aspects of math, that one must be “conquered” before the next concept is addressed.

* Math is about getting the right answer – Adding and subtracting/math is about understanding patterns.

* Complex math involves complex reasoning and laborious effort. – Recognizing underlying patterns can speed the process.

* Mathematics and images have nothing to do with each other – Mathematical images are beautiful and well-ordered.

* Creativity is appropriate in drawing or science, not in math – The Vedic approach enables people to experiment creatively with several approaches.

In looking at the valley surrounded by mountains, I think it is best to view the valley as our own consciousness, the mountains as the immensity of different practical tasks in the world and the paths our intuitive reasoning. On the other side, unbeknownst to most of us still, is a boundless sea of understanding that can be skillfully navigated by Vedic wisdom.

Robin Jackson
March 13, 2010
Birmingham, Alabama USA

This is also published in Kenneth Williams’ Vedic Math Newsletter, Nr. 68

Published in: on April 9, 2010 at 6:37 pm  Leave a Comment  

What is vedic mathematics?

Kenneth Williams in his recent Vedic Mathematics Newsletter, Issue 67 (, wrote an article discussing the question of what Vedic mathematics is. It is a question I have been asked a number of times, and I have never been sure how to reply. This is a helpful approach. In general, he defines an approach to mathematics as specifically vedic if it is a basically one-line method. The article follows.


The last 10 years have seen a huge increase in interest in Vedic Mathematics. The system reconstructed by Sri Bharati Krsna Tirthaji almost a century ago (and now known by the term “Vedic Mathematics”) is at last being recognised as having tremendous potential in all sorts of areas: educational, computational, scientific, psychological and so on. This new decade is sure to see this continue, and develop and expand further.

The influential and unbiased article by Dr N. M. Kansara1 in 2000 dealt with the criticisms aimed at Bharati Krsna’s use of the term Vedic Mathematics.

But though there is much positivity and much to look forward to there is one important area that needs to be clarified, and that is the answer to the question: “What is Vedic Mathematics: what is included within this term and what is not?”

Different people may have different ideas about the answer to this question. And it is not enough to simply refer to the book by Bharati Krsna as new material is being produced and termed Vedic Mathematics all the time.

How do we decide if a particular piece of mathematics is Vedic Mathematics or not? What is the defining characteristic, or characteristics, of Vedic Mathematics by which we can recognise something as part of the Vedic system?

Why is this question important? It is vital that this question is answered because there has to be some clear boundary to what is Vedic Mathematics and what is not. Otherwise people can declare all sorts of obscure mathematical results and claim them to be Vedic Mathematics.

Another reason why the question is important is that for many people Vedic Mathematics has become synonymous with tricks, short cuts and fast methods. This is unfortunate as it means it is not seen seriously by mathematicians and educationists, and it entirely misses the comprehensive and complete nature of the system of Vedic Mathematics.

Here are some possible answers to the question: what characteristic of a method, proof etc. makes it Vedic?

A method, proof etc. is Vedic if:

1) It comes under one or more of the Vedic Sutras

2) It follows a method given by Bharati Krsna

3) It is one-line

Answer 1 I must reject as I believe the Sutras describe natural ways in which the mind works 2, so any way of thinking or any method must use the Sutras. Even the current long multiplication method (that cannot be described as Vedic) uses these Sutras.

Answer 2 does not allow the possibility of methods being Vedic that are not given by Bharati Krsna and so is too restrictive.

For Answer 3, first look at the title of Bharati Krsna’s book:

“Vedic Mathematics
Sixteen Simple Formulae from the Vedas
(For One-line Answers to all Mathematical problems)”

This is the full title: first Vedic Mathematics, then an alternative title, and then the words in brackets.

This title implies that the Sutras are the basis for Vedic Mathematics, and that they give one-line answers to all mathematical problems.

This is perhaps the criterion we are looking for: a method must be one-line. But this answer is no use to us unless we can say what ‘one-line’ means.

In the area of computation we can say ‘one-line’ means that the answer can be given digit by digit with an occasional carry digit (or digits), which can be held in the mind. The term also suggests that the flow of attention is one-line: that the attention is not fragmented.

A good illustration is perhaps the obtaining of the product of two numbers by the Vertically and Crosswise method and by the usual conventional method. In the Vedic method the attention moves through the numbers being combined, obtaining the digits of the answer one after the other (from right to left or from left to right) using a simple pattern.

The conventional method is fragmented, obtaining first one row of figures and then another until finally these rows are added up. In fact the same number of products are found in both methods, but the Vedic method flows, and flows in one direction: i.e. it is one-line.

Getting an answer digit by digit with an occasional carry digit also suggests that such a calculation can be carried out mentally, and in fact Bharati Krsna writes in his Explanatory Exposition:

“…by means of what we have been describing as straight, single-line, mental arithmetic”, suggesting that ‘one-line’ is equivalent to ‘mental’.

I therefore propose that the main criterion for a method to be Vedic is that it has this one-line feature. But a technique should not be rejected outright on the basis that it is not one-line. If it is better than the conventional method, it is worthy of consideration with a view to further development.

The above illustration of multiplication happens to be a convenient one. Mathematics is not just about computation though, but about proof, problem solving, structure and so on. So how does the one-line flow of attention criterion apply more generally?

It is hoped that this article will generate some response to these questions in those interested, and if so you are requested to email with your views. Without a clear idea of the boundaries of Vedic Maths the subject may gradually disappear, with only certain techniques which are popular remaining but not being known to be part of a complete system of Vedic Mathematics.

My thanks are due to Andrew Nicholas for his helpful comments during the formulation of this article.

1 “Vedic Sources of ‘Vedic Mathematics”, Sambodhi, Vol. XXIII, 2000.

2 Please see: “The Sutras of Vedic Mathematics”, by Kenneth Williams, in the Journal of the Oriental Institute, Vol. L, Nos 1-4, Sept 2000 – June 2001, pages 145 to 156.

Kenneth Williams

Published in: on February 7, 2010 at 3:38 am  Comments (3)  

Vedic math and adding fractions

I always found the process of looking for the lowest common denominator to be a little laborious, and I always had to re-learn how to do it. The Vedic approach is simpler:

1/2 + 1/3 = ?

Step 1: Cross multiply: 1 X 3 and 1 X 2.
Step 2: Add the results: 3 + 2 = 5
Step 3: Multiply the denominators: 2 X 3 = 6
The answer is 5/6.

Published in: on February 5, 2010 at 2:53 am  Leave a Comment  

Bach and Number Symbolism

This is an interesting and hotly debated topic. One web discussion of it, by the name of “Number Symbolism in Bach Cantatas,” gives us some idea of how the issue has fared and suggests that there is an infinite amount of exploration to be done on the fascinating issue of number and number symbolism in Bach. Thomas Braatz, the writer who started the discussion, was especially interested in the question of whether there was a tradition for theological interest in number symbolism in Bach’s day. He cites a book he read called “The Occult Bach.” This appears to be impossible to find, but I believe it is written in Dutch by Dutch musicologists Kees van Houten and Marinus Kasbergen. Another book on the subject was written by Friedrich Smend (1947). Smend argued that Bach used the alphabet as code. Attempts have been made by Ruth Tatlow, among others, to debunk the work of these three scholars on the grounds that the arguments are implausible and implicate Bach in occult reasoning. However, people continue to be fascinated with the incredible ways in which Bach’s music expresses itself in number relations.

Thomas Braatz’s comments on problems finding the source mentioned above are as follows: “The book is based primarily on the work of Marinus Kasbergen and Kees van Houten’s “Bach en het getal” (1985), a book I have tried to order from some German on-line booksellers, but to no avail. Must be a popular book! The other references were to the MGG I, 1028ff and MGG 16, 1971ff (MGG=Die Musik in Geschichte und Gegenwart, the German equivalent to The New Grove Dictionary of Music) Just a few important authors to mention: Arnold Schering, 1940; Martin Jansen, 1937; Friedrich Smend, 1947-8; A. Schmitz, 1950, Walter Blankenburg who wrote one of the articles for the MGG, Henk Dieben (1954-5)”Getallenmystiek bij Bach”. This is all very new to me, but very interesting.” (Braatz from the above-mentioned 2001 discussion based on the question he asked “Was there a tradition for theological interest in number symbolism in the Lutheran Church in Bach’s day?”)

Published in: on January 18, 2010 at 9:35 pm  Leave a Comment  

Greetings and…

Welcome to my Vedic Math blog, inspired by Sacred Geometry,  Hinduism, Mental Math, and the Vedas!

Published in: on August 18, 2009 at 3:38 pm  Comments (1)