This Manual is the first of three (elementary,
intermediate and advanced) Manuals which are designed for adults with
a basic understanding of mathematics to learn or teach the Vedic system.
So teachers could use it to learn Vedic Mathematics, though it is
not suitable as a text for children (for that the Cosmic Calculator
Course is recommended). Or it could be used to teach a course on Vedic
The sixteen lessons of this course are based on
a series of one week summer courses given at Oxford University by
the author to Swedish mathematics teachers between 1990 and 1995.
Those courses were quite intensive consisting of eighteen, one and
a half hour, lessons.
All techniques are fully explained and proofs
are given where appropriate, the relevant Sutras are indicated throughout
(these are listed at the end of this Manual) and, for convenience,
answers are given after each exercise. Cross-references are given
showing what alternative topics may be continued with at certain points.
It should also be noted that the Vedic system
encourages mental work so we always encourage students to work mentally
as long as it is comfortable. In the Cosmic Calculator Course pupils
are given a short mental test at the start of most or all lessons,
which makes a good start to the lesson, revises previous work and
introduces some of the ideas needed in the current lesson. In the
Cosmic Calculator course there are also many games that help to establish
and promote confidence in the ideas used here.
Some topics will be found to be missing in this
text: for example, there is no section on area, only a brief mention.
This is because the actual methods are the same as currently taught
so that the only difference would be to give the relevant Sutra(s).
Vedic Mathematics is an ancient system of mathematics which was rediscovered
early last century by Sri Bharati Krsna Tirthaji (henceforth
referred to as Bharati Krsna).
The Sanskrit word “veda” means “knowledge”. The Vedas are ancient
writings whose date is disputed but which date from at least several
centuries BC. According to Indian tradition the content of the Vedas
was known long before writing was invented and was freely available
to everyone. It was passed on by word of mouth. The writings called
the Vedas consist of a huge number of documents (there are said to
be millions of such documents in India, many of which have not yet
been translated) and these have recently been shown to be highly structured,
both within themselves and in relation to each other (see Reference
2). Subjects covered in the Vedas include Grammar, Astronomy, Architecture,
Psychology, Philosophy, Archery etc., etc.
A hundred years ago Sanskrit scholars were translating the
Vedic documents and were surprised at the depth and breadth of knowledge
contained in them. But some documents headed “Ganita Sutras”, which
means mathematics, could not be interpreted by them in terms of mathematics.
One verse, for example, said “in the reign of King Kamse famine, pestilence
and unsanitary conditions prevailed”. This is not mathematics they
said, but nonsense.
Bharati Krsna was born in 1884 and died in 1960. He was a brilliant
student, obtaining the highest honours in all the subjects he studied,
including Sanskrit, Philosophy, English, Mathematics, History and
Science. When he heard what the European scholars were saying about
the parts of the Vedas which were supposed to contain mathematics
he resolved to study the documents and find their meaning. Between
1911 and 1918 he was able to reconstruct the ancient system of mathematics
which we now call Vedic Mathematics.
He wrote sixteen books expounding this system, but unfortunately
these have been lost and when the loss was confirmed in 1958 Bharati
Krsna wrote a single introductory book entitled “Vedic Mathematics”.
This is currently available and is a best-seller (see Reference 1).
There are many special aspects and features of Vedic Mathematics
which are better discussed as we go along rather than now because
you will need to see the system in action to appreciate it fully.
But the main points for now are:
1) The system rediscovered by Bharati Krsna is based on sixteen
formulae (or Sutras) and some sub-formulae (sub-Sutras). These Sutras
are given in word form: for example By One More than the One Before
and Vertically and Crosswise. In this text they are indicated
by italics. These Sutras can be related to natural mental functions
such as completing a whole, noticing analogies, generalisation and
2) Not only does the system give many striking general and
special methods, previously unknown to modern mathematics, but it
is far more coherent and integrated as a system.
3) Vedic Mathematics is a system of mental mathematics (though
it can also be written down).
of the Vedic methods are new, simple and striking. They are also beautifully
interrelated so that division, for example, can be seen as an easy
reversal of the simple multiplication method (similarly with squaring
and square roots). This is in complete contrast to the modern system.
Because the Vedic methods are so different to the conventional methods,
and also to gain familiarity with the Vedic system, it is best to
practice the techniques as you go along.
COMPLETING THE WHOLE
The Ten Point Circle
Multiples of Ten
Deficiency from Ten
Deficiency and Completion Together
Completing the Whole
Columns of Figures
By Addition and By Subtraction
Subtracting Numbers Near a Base
DOUBLING AND HALVING
Multiplying by 4, 8
Dividing by 4, 8
Extending your Tables
Multiplying by 5, 50, 25
Dividing by 5, 50, 25
Dividing by 5
Dividing by 50, 25
The Nine Point Circle
Casting out Nines
Digit Sum Puzzles
More Digit Sum Puzzles
The Digit Sum Check
The Vedic square
Patterns from the Vedic Square
LEFT TO RIGHT
Addition: Left to Right
Multiplication: Left to Right
Doubling and Halving
Subtraction: Left to Right
Checking Subtraction Sums
FROM 9 AND THE LAST FROM 10
All From 9 and the Last from 10
One Less Again
Numbers just Over Ten
Multiplication Table Patterns
Numbers Close to 100
Numbers Over 100
Russian Peasant Multiplication
Numbers Above the Base
Another Application of Proportionately
Multiplying Numbers near Different Bases
Squaring Numbers near a Base
Digit Sum Check for Division
The First by the First and the Last by the Last
The First by the First
The Last by the Last
Divisibility by 4
Divisibility by 11
Remainder after Division by 11
Another Digit Sum Check
Removing Bar Numbers
All from 9 and the Last from 10
Creating Bar Numbers
Using Bar Numbers
10 SPECIAL MULTIPLICATION
Multiplication by 11
By One More than the One Before
Multiplication by Nines
The First by the First and the Last by the Last
Using the Average
11 GENERAL MULTIPLICATION
Multiplying 3-Figure Numbers
that end in 5
Squaring Numbers Near 50
Digit Sums of squares
Square Roots of Perfect Squares
3 and 4 Figure Numbers
Unification of Operations
A Short Cut
Division by 8 etc.
Division by 99, 98 etc.
Divisor Below a Base Number
Divisor Above a Base Number
16 THE CROWNING
on the Flag
Short Division Digression
Negative Flag Digits
Decimalising the Remainder
INDEX OF THE VEDIC FORMULAE
Vedic Mathematics was reconstructed
from ancient Vedic texts early last century by Sri Bharati Krsna Tirthaji
(1884-1960). It is a complete system of mathematics which has many
surprising properties and applies at all levels and areas of mathematics,
pure and applied.
It has a remarkable coherence
and simplicity that make it easy to do and easy to understand. Through
its amazingly easy methods complex
problems can often be solved in one line.
The system is based on sixteen
word-formulae (Sutras) that relate to the way in which we use our
The benefits of using Vedic
Mathematics include more enjoyment of maths, increased flexibility,
creativity and confidence, improved memory, greater mental agility
and so on.
This Elementary Manual is
the first of three designed for teachers who wish to teach the Vedic
system, either to a class or to other adults/teachers. It is also
suitable for anyone who would like to teach themselves the basic Vedic