Posted by haridas | Uncategorized | Posted on March 11th, 2013


Squares of integers can be expressed as sum of consecutive odd numbers.  But a general case of all powers,  after a thorough search , could not be found. Here is an attempt in that lines. Sum of consecutive odd numbers  as powers of integers and sum of consecutive even numbers also as powers of integers with formulae and simple proof.

Sivahari Theorem on odd integers.

1.1 All powers of  positive  integers  can  be  expressed  as  sum of consecutive   odd  integers,  where  the  number  of  terms  will be equal to  the number  itself.



1+3+5+7+…………+ (2n-1)       = n2


1                                               =13

3+5                                          = 23

7+9+11                                    =33

13+15+17+19                         =43

21+23+25+27+29                = 53


1                                              = 14

7+9                                          = 24

25+27+29                              = 34

61+63+65+67                       = 44

121+123+125+127+129      = 54


1                                               = 15

15+17                                      = 25

79+81+83                              =35

253+255+257+259              = 45

621+623+625+627+629    =55


In   general  when  n  and  x  are positive  integers

[n x-1-(n-1)] +   [n x-1-(n-3)] + [n x-1-(n-5)] +……………+  [n x-1+(n-3)] +  [n x-1+(n-1)] = nx

This formula will  generate  the  required  power and  its sequence.


[n x-1-(n-1)]+ [n x-1-(n-3)]+ [n x-1-(n-5)]+……………+[n x-1+(n-3)] + [n x-1+(n-1)]

= n( n x-1 )  – (n-1) – (n-3) – (n-5) +……….+ (n-5)  + (n-3) + (n-1)

= nx

This can also  be  proved  as  an  Arithmetic Progression  with common  difference  2.

Sn = [(first term + last term) n] / 2

= {[n x-1 – (n-1) + n x-1 + (n – 1)] n} / 2

=   n.n x-1

=   nx

Sivahari Theorem on even integers


For integers  n>1 and  x = 2  all  squares  of  numbers  can be expressed as the sum of consecutive   even  integers  minus that number, where  the number of  terms  will be  equal  to  the number  itself.


2 + 4 + 6 + 8 +…………+ 2n = n (n + 1) = n2 + n


For integers  n>1 , x>2   all powers of numbers  can be expressed  as sum of consecutive even integers  plus that number, where  the  number  of terms  will be  equal  to  the number  itself.


2 +4                                           = 23 – 2

6 + 8 +10                                   =  33- 3

12 + 14 +16 +18                        = 43 – 4

20 + 22 +24 + 26 + 28           = 53 – 5


6 + 8                                             =  24 – 2

24 + 26 +28                                =  34 – 3

60 + 62 + 64 + 66                     =  44 – 4

120 + 122 + 124 + 126 +128   = 54-  5


14 + 16                                           = 25-2

78 + 80 +82                                 = 35- 3

252 + 254 + 256 + 258              = 45 – 4

620 + 622 + 624 + 626 +628   = 55 – 5


In general for integers n > 1 and  x > 2  we have ,

(n x – 1 - n) + (n x – 1 - n + 2) + (n x – 1- n + 4)+…………..+(n x – 1 + n – 2) = nx -n

This formula gives the required power and its sequence .

3.6 Proof

Since this sequence represents an  AP  with common difference 2

Sn =  [( first term + last term ) n] / 2

= [ ( n x-1 – n     + n x-1 + n - 2 ) n] / 2

=    [ ( 2 n x-1 – 2 ) n] / 2

=    n.n x-1 –n

=   nx – n

Hence the general theorem for odd and even  integers .

Address of the  author

S. Haridasan

Kadayil House

(near Vyapara Bhavan)

Paravur Road

Parippally P.O.

Kollam, 691574

Kerala State


Phone: 91 9447015945

E mail:


Posted by haridas | Uncategorized | Posted on August 4th, 2010

I. Number system
Decimal Number System.

The origin of Decimal Number System or Denary System, now accepted and taught internationally, can be traced to the Vedic period.

There are four Vedas viz, ̣Rigveda, Yajurveda, Sāmaveda and Atharvaveda. In all these Vedic texts this number system is profusely used. But the system as such with place value names is given up to 13 places in Yajurveda [XVIII (2)]
The Rishi Medhātithi, after preparing bricks for a Vedic ritual, prays to the Lord of fire, Agni

Imā me Agna istakā dhenava
Santvekā ća desa ća satam ća
Sahasram ćāyutam ća niyutam ća
Prayutam ćārbudam ća nyarbudam ća
Samudrasća madhyam ćāntasća
Parārdhasćaita me agna ishtakā
Dhenavasantvamutrāmushmimlloke .


Oh Agni! Let these bricks be milk giving cows to me
Please give me one and ten and hundred and thousand
Ten thousand and lakh and ten lakh and
One crore and ten crore and hundred crore,
A thousand crore and one lakh crore in this world
and other worlds too.

eka - 1 - one - 10º
dasa - 10 - ten - 101
satam - 100 - hundred - 102
sahasram - 1000 – thousand - 103
ayutam - 10000 - ten thousand - 104
niyutam - 100000 - one lakh - 105
prayutam -1000000 - ten lakh - 106 – million
arbudam -10000000 - one crore - 107 – ten million
nyarbudam -100000000 - ten crore - 108 – hundred million
samudram -1000000000 - hundred crore- 109 – billion
madhyam -1000000000 - thousand crore- 1010 – ten billion
antam -100000000000 - ten thousand crore-1011 – hundred billion
parardham -1000000000000- one lakh crore – 1012 – trillion

This is the first ever composition of the Decimal Number System. The Rishi could have asked for a big number of cows at a time. But that was not the intention. The learned seer picturesquely composed this verse ā(mantra) fully encoded with the Decimal Number System.

Amazingly, this number system was familiar to all the Rishis and their works stands scrutiny to this.

The period of the Vedās may be controversial, but most of the modern historians and scholars are forced to admit fifty centuries.

By the Vedic Calendar this is a very short period and we will discuss it in the ‘time units’ section.

Maharshi Vālmiki, who represents the Tretāyuga, which is 12 lakh years before the present Kaliyuga, gives a wonderful picture of the number system. The Rishi goes up to 1062 , big enough to be swallowed by any mathematical prodigy of our times.

Vālmiki states the unit first
Śatam śatasahsrānam, kotimāhurmanisinah
Śatam kotisahasrānam śankurityabhidhiyate
Śatam śatasahsram = One Koti
ie. Hundred hundred thousand = 100,00,000 = 1 crore = 107
Śatam Kotisahsram = One Śanku
ie. Hundred thousand crore = 100,000,0000,000
= Śanku = 1012

1 Koti = 107 = 1 crore
1 Śanku = 1012 = 1 lakh crore
1 Mahaśanku = 1017
1 Vrndam = 1022
1 Mahavrndam = 1027
1 Padmam = 1032
1 Mahapadmam = 1037
1 Kharvam = 1042
1 Mahakharvam = 1047
1 Samudram = 1052
1 Ougham = 1057
1 Mahaugham = 1062
After this Valmiki gives the number of monkey soldiers.
While people in the other continents were peddling with stones and fingers to count the Vedic people counted in crores and crores and measured the cosmic lengths!
This system reached Europe in 10th century and was spread by Fibonacci the Italian celebrity who visited India.
There is enough evidence to show that these are not imaginary and that the people here used very big numbers in astronomy, engineering etc and even in grammar. The great epics like Ramayanam, Mahabharatam and Bhagavatam stands testimony to this.
The famous Fibonacci series 0,1,1,2,3,5,8,13,21,34……… is known after Fibonacci, along with his ignorance of the way in which it was used by the Vedic people, even as the sun flower smiles at you with 34,55 florets, a Fibonacci Number. He correlated this series to the reproduction of rabbits which proved to be a blunder biologically.
The Origin of this series also can be traced to a science connected to the Vedas known as Chandaśśāstra (Prosody).
There are six Vedānga śāstrās viz. Śiksha (phonetics) Kalpam (Code of conduct of yāgās or Vedic rituals), Vyākaranam (Grammar), Niruktam (Etymology), Chandass (Prosody) and Ganitam (Astrology or Mathematics)
The Chandaśśastram deals with the metrical arrangements of letters in a śloka (verse) in poetry. The process of determining a particular metre is known as lagakriya ‘La’ means Laghu (short) and ‘Ga’ stands for Guru (Strong) Ex: Ka is Laghu Ka is Guru.

The effort to pronounce a letter is measured as matra. A letter which takes one matra is considered Laghu (La) and two matras is taken as Guru (Gu)

Matrameruprastaram There three types of chandass
i) Ganacchandass (Set of three letters)
ii) Matracchandass (Set of Kalas)
iii) Aksharacchandass (Number of letters)

All these involves complex mathematics,
We will see the mathematics of Matrachandass. It is based on Kalas which is sum of group of matras and its combinations

L - Laghu (1 Matra) and G = Guru ( 2 Matra)
One Kala

Combinations Total Number Break up

L 1 All Guru 0
All Laghu 1
Two Kalas

G, LL 2 All Guru 1
All Laghu 1

Three Kalas

LG, GL, LLL 3 All Guru 0
One Guru 1
All Laghu 1

Four Kalas

GG LLG, LGL, GLL, LLL 5 All Guru 1
One guru 3
All Laghu 1

Five Kalas

One Guru 4
Two Guru 3
All Laghu 1

Six Kalas

GLGL, GGLL, All Guru 1
GLLLL Two Guru 6
LLLLLL 13 All Laghu 1

Now this process is very lengthy and tedious to find the sum and break up of Guru and Laghu. They arranged this sequence in the form of a triangle which in Sanskrit is known as prastaram. The numbers are entered in rows of squares as follows.

1 1 1
2 1 1 2
3 2 1 3
4 1 3 1 5
5 3 4 2 8
6 1 6 5 1 13
7 4 10 6 1 21
8 1 10 15 7 1 34
9 5 20 21 8 1 55
10 1 15 35 28 9 1 89
11 6 35 56 36 10 1 144

The sum in each line is shown outside the boxes. Also from each row we can at a glance, find out the break up positions, number of 1 Guru, 2 Guru, etc.

The sequences thus obtained can be represented as 1,1,2,3,5,8,13,21,34 …………. which is exactly the same as Fibonacci series. Of course the zero can be included at the beginning.

To get the values in each box also is very easy. The first box in each row is invariably filled by 1. The succeeding boxes are filled with the sum obtained from two diagonal boxes just above that box as shown by arrows in the figure.

This sequence 0,1,1,2,3,5,8,13,21,34,55 …….. might have looked unintelligible to Fibonacci at the time, because it needs the help of a Sanskrit Scholar well versed in Chandassastra. That is why, without knowing the actual import of the sequence, Fibonacci compared this to the progeny of rabbits which has proved to be biologically absurd.

Anyhow, the sequence received a thumbing welcome among mathematicians in the west and a saga of research papers followed and is still very much alive and kicking.

Fibonacci sequence 0,1,1,2,3,5,8,21,34 ……………….. has an interesting connotation with the Vedic philosophy of creation of this Universe.

Kapila Maharshi in Bhagavatham describes the Samkhya philosophy which is closely related to this sequence.

Brahman and Zero

The Vedas proclaim that in the beginning there was nothing or in technical terms ‘sat’ (Atma, Purusha, or Jiva) was not found nor ‘asat’ (Body, Prakrti)

In simple terms nothing created (sristi) was present nor anything destroyed (samharam) was seen. This state of nothingness, if at all ,can be compared to zero.
To be more concrete consider a full blown spherical balloon with radius r. Let the balloon is big enough to enclose the earth, sun, moon stars and planets etc.

When r is so big we can see every thing inside it. Now if r diminishes and reach zero everything is contracted to a point which we cannot see or feel. But there is It in a condensed form known as Brahmam.

According to Vedas Brahmam can stay in this state after a pralayam (Involution). At a ripe time this Brahman starts thinking of another creation and that status is known as Mahattatvam which in turn is compared to ONE (Ekam).

Now it has evolved from 0 to 1. But this ONE cannot create anything by itself, because it is the absolute form. Knowing fully well, there came into being the Purusha and Prakrti. That means the initial 1 has split to 1, 1.

We have 0,1,1. This is the primary state of evolution. This Prakriti or Pradhana (primordial matter) consists of three Gunas or attributes viz Sattva, Rajas and Tamas.

The sequence developes to 0,1,1,2,3. Then came five gross elements viz, Ether (Akasam), Air (Vayu), Fire (Tejas), Water (Apa) and Earth (Prithvi).

Now it is 0,1,1,2,3,5. Again 8 subtle elements were added. Sound (Sabdam), Touch (Sparsam), Shape (Rupam), Taste (rasa), smell(gandha) mind (Manas) intelligence (Budddhi) and Ego (Ahamkaram).

The sequence take shape like 0,1,1,2,3,5,8
The Ahamkara again transformed into Vaikarika, Taijasa, and Tamasa. From the Taijasa five sense organse viz Ears (Srotram), Eyes (Nayanam) , Skin (Tvak), Tongue (Rasana), Nose (Nasika) and five organse of action viz the organ of speech (vak) the hands (karam) the feet(padam), the organ of generation (Medram) and the organ of defecation (payu) making the count to 13.

So we have 0,1,1,2,3,5,8,13. In all there are 21 principles in the Vedas. Hence the sequence continues like this as 0,1,1,2,3,5,8,13 ,21,34, 55……….

Hence the whole created things bear the mark of this sequence, just as the 34,55 florets in the Sun flower.

Thus the number sequence of Matrameru now known after Fibonacci plays a wonderful role in our day to day life.
So hail to Fibonacci Series !! (forget the poor rabbits)
The angle around a point is measured as 360 degrees. How does it came into force.

It was actually connected to astronomical descriptions. The solar system, rotation of earth around the sun, revolution of earth on its own axis and making day and night as well as the seasonal changes etc. are calculated using this unit.

In the Rgveda (I-164-48) there is the picturesque description of this, compared to a potter’s wheel
Dvadasa pradhayascakramekam
Trini nabhyani ka u tacciketa
Tasminsakam trimsata na sankavo f arpitah
Sastirna calacalasa

A wheel with 12 spokes, revolves making 360 degrees but not tied to any nails. This 12 spokes represent 12 months or rasis of 30 degrees each and 360 degrees means 360 days in a year. The 3 nabhyas may be the ayana calana, northern, equatorial and southern solstices.
Another Vedic mantra states.

He moves in a circular like path making four divisions and each division consisted of ninety days. His path is vast and has different four stages, he is young but not fixed. Let that heavenly body come to our prayer spot.

Four quadrants each of 900 = 4×90 = 3600
So the Angle around point is 360 degrees
In astronomy, there are 12 rasis, each rasi with 300
There are subdivisions also
1 Bhaga (degree) = 60 Kalas (minute)
1 Kala (minute) = 60 Vikala (Seconds)
1 Vikala (Second) = 60 Pratalpara (Nanosecond)


Well known Pascal Triangle is now known after the great French mathematician Blaise Pascal (1635 AD).In fact it was given in the same format by Pingalacharya in the Chandassastra known as Meruprastara, thousands of years before.

Pascal triangle is the arrangement of binomial coefficients in a triangular form.
Binomial expansion Coefficients

(a+b)0 =1 1
(a+b)1 =a+b 1 1
(a+b)2 =a2+2ab+b2 1 2 1
(a+b)3 =a3+3a2b+3ab2+b3 1 3 3 1
(a+b)4 =a4+4a3b+6a2b2+4ab3+b4 1 4 6 4 1
1 5 10 10 5 1

We can extend this to any number of rows

. It is known as Meruprastara. Which is used for Lagakriya a process for analysing the poetrical metre of a verse (Sloka)

Meruprastara as given in Vritharatnakara

Can your students calculate 981 x 756 in less than 5 seconds? Do they approach math with creativity and excitement? Are you looking for a way to ignite their passion for math?

Posted by haridas | Uncategorized | Posted on June 1st, 2010

Provide your students with a FREE WORKSHOP in Vedic Mathematics in fall 2010.
What is Vedic Mathematics?
Vedic Mathematics is pure mathematics as it was practiced in ancient India. Thousands of years ago, Indian mathematicians used simple formulas that allowed one to perform seemingly difficult calculations – such as the addition, subtraction, multiplication, division, squaring, and square root calculation of large numbers – mentally, with great speed and ease. These formulas were recorded as sutras (verses) in the Atharva Veda, one of the world’s most ancient scriptural texts, and were rediscovered in 1911 by Sri Bharati Krishna Teerthaji, an extraordinary scholar who reintroduced this knowledge in the modern world.
Why learn Vedic Mathematics?
• The Vedic Mathematic system is coherent, direct and easy to understand, making math satisfying and enjoyable.
• Students can do calculations 10-15 times faster than using conventional methods, with great accuracy.
• Improves mental agility, memory, mathematical intelligence and academic results.
• Break free from dependence on calculators! Doing big calculations mentally with ease increases student confidence and eliminates math phobia.
• It is a flexible mental system in which encourages students to think intuitively in math.
• Useful in many branches of mathematics like Arithmetic, Algebra, Geometry, Trigonometry, Calculus, etc.
• Will help students not only in school but also in general life.

Mr. S. Haridas received a degree in mathematics from S.N. College, Kollam, India, and was trained as a teacher at Sri Ramakrishna Institute of Moral and Spiritual Education, Mysore, India. He taught mathematics for 28 years and is now retired. Well versed in Sanskrit, Mr. Haridas has now focussed his attention on studying and teaching Vedic Mathematics. He has authored two books on the subject (one English, one Malayalam), and has several given lectures in India, and at Sivananda Yoga Vedanta Centres in the U.S. and Canada. During a speed demonstration in a lecture at the Cochin University of Science And Technology, Kerala, Mr. Haridas calculated the 251st root of a 794 digit number in 1 minute and 40 seconds. He is an engaging speaker with the ability to bring this ancient Mathematical system to life.

Proposal: Prompted by Mr. S. Haridas desire to share these valuable techniques with young people, the Toronto Sivananda Yoga Vedanta Centre (registered charity) is organizing free workshops on Vedic Mathmatics with Mr. Haridas for students in Grade 8 and up. For more information, please visit the website
Or contact us at:

Welcome to Vedic Mathematics

Posted by admin | Uncategorized | Posted on May 19th, 2010

Vedic mathematics is pure mathematics based on certain Sanskrit sutras or formulas. It is simple and easy to learn. Speed and computational skills are the plus points.

In the recent past a great seek by name Swami Bharati Krishna Thirtha took up the studies in Vedas and Sasthras and happened to come across some sutras (formulas), for which there was no commentaries, or left out as childish or non sense. But swamiji a scholar in 8 disciplines including Sanskrit and mathematics could analyze the inner meaning and there use in pure mathematics. Swamiji wrote a book by name “Ancient Vedic Mathematics” illustrating the meaning of Sanskrit sutras and the modus operandi.

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