This book is designed for teachers of students in grades 9 to 14.
It shows how Vedic Mathematics can be used in a school course but
does not cover all school topics (see contents). The book can be used
for teachers who wish to learn the Vedic system or to teach courses
on Vedic mathematics for this level. Non-teachers who have a background
knowledge of mathematics may also find it appropriate (see contents).
Topics included: calculus, series, logs and exponentials, trigonometry
(including solving trig equations, proving identities), solution of
equations (special types, quadratics, cubics, transcendental), complex
numbers, coordinate geometry, transformation geometry, Simple Harmonic
Motion, projectile motion, forces in equilibrium, work & moments,
270 + xi pages. Size: 24cm by 16cm. Paperback. 2005; Author: Kenneth
ISBN 978-1-902517-18-6. Revised edition 2010.
Price $30 (including postage).
Please note that these Manuals do not form a sequence: there is some
overlap between the three books.
Review: I have got the book
- advanced teachers manual; it is really amazing. It explains each
method so easily covering basic concepts and that too within minimal
Hats off to you.
Manual is the third of three self-contained Manuals (elementary, intermediate
and advanced) and is designed for adults with a good understanding
of basic mathematics to learn or teach the Vedic system. So teachers
could use it to learn Vedic Mathematics, or it could be used to teach
a course on Vedic Mathematics. It is suitable for teachers of children
aged about 13 to 18 years.
The eighteen 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. Some of the material here is more advanced than
would be given to the average 18 year old student but this is what
the teachers wanted on the courses and so the same is given here.
The lessons in this book however probably
contain more material than could be given in a one and a half hour
lesson. The teacher/reader may wish to omit some sections, go through
the material in a different sequence to that shown here or break up
All techniques are fully explained and proofs
and explanations are given, 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
Vedic system pupils are encouraged to be creative and use whatever
method they like.
Some topics will not be found in this text:
for example, there is no section on area and volume. 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
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
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 Vertically and
Crosswise and By One More than the One Before. 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 so on.
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).
Many 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.
TO RIGHT CALCULATIONS
Advantages OF LEFT TO RIGHT CALCULATION
WRITING LEFT TO RIGHT SUMS
CHECKING SUBTRACTION SUMS
ALL FROM 9 AND THE LAST FROM 10
FROM A BASE
OF BAR NUMBERS
MULTIPLICATION NEAR A BASE
Numbers just below the
Above the base
Above and below
With different bases
DIVISION BY NINE
A Short Cut
Dividing by 8
Dividing by 11, 12 etc.
3 RECURRING DECIMALS
DENOMINATOR ENDING IN 9
A SHORT CUT
DENOMINATORS ENDING IN 8, 7, 6
DENOMINATORS ENDING IN 1
DENOMINATORS ENDING IN 2, 3, 4
WORKING 2, 3 ETC. FIGURES AT A TIME
Triples for 45°, 30° and 60°
Variations of 3,4,5
5 GENERAL MULTIPLICATION
Explanation/ The Digit Sum Check
The Digit Sum Check
Writing Left to Right Sums
From Right to Left
setting the sums out
Using Bar Numbers
6 SOLUTION OF EQUATIONS
TRANSPOSE AND APPLY
MORE THAN ONE X TERM
6.2a GENERAL SOLUTION
ONE IN RATIO THE OTHER ONE ZERO
WHEN THE SAMUCCAYA IS THE SAME IT IS ZERO
Samuccaya as a common
Samuccaya as the Product of the Independent Terms
Samuccaya as the Sum of the Denominators
Samuccaya as a Combination or Total
6.6e other typeS
THE ULTIMATE AND TWICE THE PENULTIMATE
ONLY THE LAST TERMS
SUMMATION OF SERIES
AND SQUARE ROOTS
Squaring 2-FIGURE NUMBERS
Squaring Longer Numbers
7.4a Left to Right
7.4b Right to Left
Square Roots of Perfect Squares
Angle Between Two Lines
8.5c Circle Problems
EQUATION OF A LINE
Testing Longer Numbers
The Negative Osculator
OSCULATING WITH GROUPS OF DIGITS
Single Figure on the Flag
Short Division Digression
Decimalising the Remainder
Negative Flag Digits
Square Root of a Perfect square
General Square Roots
Changing the Divisor
11.5 ALGEBRAIC SQUARE
12 TRIPLE TRIGONOMETRY
12.1 COMPOUND ANGLES
12.2 INVERSE FUNCTIONS
12.3 THE GENERAL TRIPLE
A SPECIAL TYPE
13 COMBINED OPERATIONS
SUMS OF PRODUCTS
ADDITION AND DIVISION
MEAN AND MEAN DEVIATION
DIVIDING SUMS OF PRODUCTS
14 SOLUTION OF POLYNOMIAL
x > 1
x < –1
0 < x < 1
0 < x < 1 and x2 Coefficient > 1
–1 < x < 0
HIGHER ORDER EQUATIONS
A CUBIC WITH 0 < x < 1
15 CALCULUS METHODS
Integration by ‘Parts’
BINOMIAL AND MACLAURIN THEOREMS
Derivatives of a Product
Derivative of A Quotient
Differential Equations – 1
Differential Equations – 2
16 APPLIED MATHEMATICS
SIMPLE HARMONIC MOTION
FORCES IN EQUILIBRIUM
WORK AND MOMENT
17 TRIGONOMETRIC FUNCTIONS
17.2 SERIES EXPANSIONS
17.3 INVERSE TRIGONOMETRIC FUNCTIONS
EVALUATING TRIGONOMETRIC FUNCTIONS
18 TRIGONOMETRIC AND
SUTRAS AND SUB-SUTRAS
INDEX OF THE VEDIC FORMULAE INDEX
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 Advanced Manual is the
third of three self-contained Manuals 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 Vedic methods.