ORIGIN OF MATHEMATICAL CONCEPTS IN THE VEDIC PERIOD

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 .

Meaning:

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.
II. ORIGIN OF FIBONACCI SERIES
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

LGG, GLG, LLG, GGL
LLGL, LGLL, GLLL, LLLLL 8 All Guru 0
One Guru 4
Two Guru 3
All Laghu 1

Six Kalas

GGG
LLGG, LGLG, GLLG, LGGL,
GLGL, GGLL, All Guru 1
LLLLG, LLLGL, LLGLL, LGLLL One Guru 5
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)
III. THE UNIT OF ANGLE MEASURE
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)

IV. BINOMIAL COEFFICIENCTS AND PASCAL TRIANGLE

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

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