History of Mathematics - The Indian Contribution
The Contribution to Mathematics by India can be divided into ten categories:
1. Zero and the place-value notation for numbers
2. Vedic Mathematics and arithmetical operations
3. Geometry of the Sulba Sutras
4. Jaina contribution to Fundamentals of numbers
5. The anonymous Baishali manuscript
6. Astronomy
7. Classical contribution to Indeterminate Equations and Algebra
8. Indian Trigonometry
9. Kerala contribution to Infinite Series and Calculus.
10. Modern Contribution: Srinivasa Ramanujan onwards
Section 1: Zero and the Place-Value Notation
The number zero is the subtle gift of the Hindus of antiquity to
mankind. The concept itself was one of the most significant inventions in
the ascent of Man for the growth of culture and civilization. To it must be
credited the enormous usefulness of its counterpart, the place value system
of expressing all numbers with just ten symbols. And to these two concepts
we owe all the arithmetic and mathematics based upon them, the great ease
which it has lent to all computations for two millennia and the binary
system which now lies at the foundation of communicating with computers.
Already in the first three centuries A.D.. the Hindu ancients were using a
decimal positional system, that is, a system in which numerals in different
positions represent different numbers and in which one of the ten symbols
used was a fully functional zero. They called it 'Sunya'. The word and its
meaning ‘void’ were obviously borrowed from its use in philosophical
literature. Though the Babylonians used a special symbol for zero as early
as the 3rd century B.C., they used it only as a place holder; they did not
have the concept of zero as an actual value. It appears the Maya
civilisation of South America had a zero in the first century A.D. but
they did not use it in a fixed base system. The Greeks were hampered by
their use of letters for the numbers. Before zero was invented, the art of
reckoning remained an exclusive and highly skilled profession. It was
difficult to distinguish, say, 27, 207, 270, 2007, because the latter three
were all written 2 7, with a ‘space’ in between. The positional system is
not possible in the Roman numeral system which had no expression or symbol
for zero. A number, say, 101,000, would have to be written only by 101
consecutive M’s. The Egyptians had no zero and never reached the idea of
expressing all numbers with ten digits. The mathematical climate among the
Hindus, however, was congenial for the invention of zero and for its use as
the null-value in all facets of calculation, due to four factors:
1. A notation for powers of 10 up to the power 17 was already in use
even from Vedic times. Single words have been used to denote the powers of
the number 10. The numbers one, ten, hundred, thousand, ten thousand, … are
given by the sequence of words in the list: eka, dasa, Sata,
sahasra, ayuta, laksha, prayuta, koTi, arbuda, abja, kharva, nikharva,
mahA-padma,{ Paduma nidhi means laKSHAM KOTI] Sankha,{SANKA NIDHI MEANS
KOTI KOTI} jaladhi, antya, mahASankha, parArdha. ( upto 10 power of 12)
Thus the decimal system was in the culture even in the early part of the first
millennium B.C. . The Yajurveda, in its description of rituals and the
mantras employed therein, the Mahabharata and the Ramayana in their
descriptions of statistics and measurements, used all these words, with
total abandon.
2 Section 2. Vedic Mathematics and arithmetical operations
Vedic Mathematics provides an original and refreshing approach to subjects
which are usually dismissed as mechanical and tedious. Bharati Krishna
Tirtha who published his reconstruction of Vedic Mathematics in 1965,
maintains that there are 16 aphorisms and 13 secondary aphorisms which
forms his base of the so-called Vedic Mathematics. Though the origins of
Vedic Mathematics have not yet been historically established, if nothing
else, it provides tremendous insights into the place-value system of
numbers without which it would not work. It is amazing that Vedic
Mathematics does not require of cramming of multiplication tables beyond 5
x 5. One can improvise all the necessary multiplication tables for oneself
and with the aid of the relevant Vedic formulae get the required products
very easily, speedily, and correctly, almost immediately. The formulae can
be used to evaluate determinants, solve simultaneous linear equations,
evaluate logarithms and exponentials. Vedic Mathematics recognises that any
algebraic polynomial may be expressed in terms of a positional notation
without specifying the base. The same algorithmic scheme as applied to
arithmetical operations will easily apply to algebraic problems. And this
brings it to the Modern Algebra of Polynomials. It is difficult, in a
historical introduction like this to get into the details of Vedic
Mathematics. Suffice it to say that with today's over-dependence on
calculators for even simple arithmetical computations, the Vedic methods
have great pedagogical value and, through their revival, the skills of
mental arithmetic may not be lost for posterity.
3 Section 3. Geometry of the Sulba Sutras
Hailing from the times of the Vedas, the ritual literature which gave
directions for constructing sacrificial fires at different times of the
year dealt with their measurement and construction in a systematic and
logical way, thus giving rise to the Sulba Sutras. The construction of
altars (vedi) and the location of sacrificial fires had to conform to
clearly laid down instructions about their shapes and areas in order that
they may be effective instruments of sacrifice. {KR Yajur veda verses
describe so many yagna kunda designs mathematically placed} The Sulba
Sutras provide such instructions for two types of ritual - one for worship
at home and one for communal worship.
The instructions were mainly for the benefit of craftsmen laying out
and building the altars. Baudhayana, Apastamba and Kātyāyana who have
recorded these Sulbasutras were not only priests in the conventional sense
but must have been craftsmen themselves. The earliest of them, The
Baudhayana Sutras , in three chapters, (800 - 600 B.C.) contains a general
statement of the Pythagorean theorem, an approximation procedure for
obtaining the square root of two correct to five decimal places and a
number of geometric constructions. These latter include an approximate
squaring the circle, and construction of rectilinear shapes whose area is
equal to the sum or difference of areas of other shapes. The Baudhayana
version of the Pythagorean theorem sates as follows:
The rope which is stretched across the diagonal of a square
produces an area double the size of the original square.
It is therefore in the fitness of things that the Pythagorean theorem of
Mathematics may be renamed as the Baudhayana theorem.! The other sutras are
two centuries later but all of them are prior to Panini of the fourth
century B.C. The geometry arising from these sutras give several geometric
constructions. Some of these are:
1. To merge two equal or unequal squares to obtain a third square.
2. To transform a rectangle into a square of equal area
3. Squaring a circle and circling a square (approximately)
A remarkable achievement was the discovery of a procedure for evaluating
square roots to a high degree of approximation. The square root of two is
obtained as 1.4142156 …the true value being 1.414213…
The fact that such procedures were used successfully by the Sulbasútra
geometers to operations with other irrational numbers, is clear proof for
negating the western-held opinion that the Sulba sutra geometers borrowed
their methods from the Babylonians. The latter's calculation of the square
root of two is an isolated instance and further they used the sexagesimal
notation for numbers. The achievement of geometrical constructs in Indian
mathematics reached its peak later when they arrived at the construction of
Sriyantra, which is a complicated diagram, consisting of nine interwoven
isosceles triangles, four pointing upwards and four pointing downwards. The
triangles are arranged in such a way that they produce 43 subsidiary
triangles, at the centre of the smallest of which there is a big dot called
the Bindu. The difficult problem is to construct the diagram in such a way
that all the intersections are correct and the vertices of the largest
triangles fall on the circumference of the enclosing circle. In all cases
the base angles of the largest triangles are about 51.5 degrees. This has
connections with the two most famous irrational numbers of Mathematics,
namely p and f. The quantity f, called the golden ratio, has remarkable
mathematical properties and is almost a semi-mystical number.
4 KEY TAKEAWAYS
- The golden ratio is an irrational number that is equal to (1+√5)/2, or
approximately 1.618...
- The ratio is derived from an ancient Indian mathematical formula which
Western society named for Leonardo Fibonacci, who introduced the concept to
Europe.
- Nature uses this ratio to maintain balance, and the financial markets
seem to as well.
- The Fibonacci sequence can be applied to finance by using four main
techniques: retracements, arcs, fans, and time zones.
- Fibonacci numbers have become famous in popular culture, although some
experts say their importance is exaggerated.
The Golden Ratio and Fibonacci numbers were/are being held high in the west
about which few years back I wrote. Here above how Indian was a pioneer was
codified. Later what are these I may write sometime later. Both are for the
time being are GOD’S NUMBERS.
5 Mathematicians, scientists, and naturalists have known about the golden
ratio for centuries. It's derived from the Fibonacci sequence
<https://www.investopedia.com/articles/markets/010515/use-fibonacci-point-out-profitable-trades.asp>,
named after the Pisan mathematician Leonardo Fibonacci, who lived from
around 1175 A.D. until around 1250 A.D.2
Although Fibonacci introduced these numbers to the Western world, they were
actually discovered by Indian mathematicians hundreds of years earlier. The
poet Pingala used them to count the syllables of Sanskrit poetry around 200
B.C., and the method for calculating them was formulated by the Indian
mathematician Virahanka around 800 years later.
{{{Virahanka (Devanagari: विरहाङ्क) was an Indian prosodist who is also
known for his work on mathematics. He may have lived in the 6th century,
but it is also possible that he worked as late as the 8th century. His work
on prosody builds on the Chanda-sutras of Pingala (4th century BCE), and
was the basis for a 12th-century commentary by Pingala. He was the first to
propose the so-called Fibonacci Sequence.}}}
In this sequence, each number is simply the sum of the two preceding
numbers (1, 1, 2, 3, 5, 8, 13, etc.).
*Fibonacci borrowed heavily from Indian and Arabic sources. In his
book Liber Abaci, he described the Hindu-Arabic numeral system represented
by the numbers 0 through 9. He called this the "Modus Indorum," or the
method of the Indians.*
But this sequence is not all that important. The essential part is that as
the numbers get larger, the quotient between each successive pair of
Fibonacci numbers approximates 1.618, or its inverse 0.618. This proportion
is known by many names: the golden ratio, the golden mean, ϕ, and the
divine proportion, among others.
So, why is this number so important? Well, many things in nature have
dimensional properties that adhere to the ratio of 1.618, so it seems to
have a fundamental function for the building blocks of nature.
The exact value of the golden ratio can be calculated by:
ϕ = (1+√5) / 2
*Examples of the Golden Ratio*
Don't believe it? Take honeybees, for example. If you divide the female
bees by the male bees in any given hive, you will get a number around
1.618.4 Sunflowers, which have opposing spirals of seeds, have a 1.618
ratio between the diameters of each rotation.5 This same ratio can be seen
in relationships between different components throughout nature.
The golden ratio also appears in the arts, because it is more aesthetically
pleasing than other proportions. The Parthenon in Athens, the Great Pyramid
in Giza, and Da Vinci's *Mona Lisa* all incorporate rectangles whose
dimensions are based on the golden ratio. It seems to be unavoidable.
But does that mean it works in finance? Actually, financial markets
<https://www.investopedia.com/terms/f/financial-market.asp> have the very
same mathematical base as these natural phenomena. Below we will examine
some ways in which the golden ratio can be applied to finance, and we'll
show some charts as proof.
*Trading and Investing with the Golden Ratio*
The golden ratio is frequently used by traders and technical analysts, who
use it to forecast market-driven price movements. This is because the
Fibonacci numbers and the golden ratio have a strong psychological
importance in herd behaviour. Traders are more likely to take profits or
cover losses at certain price points, which happen to be marked by the
golden ratio.
Curiously, the widespread use of the golden ratio in trading analysis forms
something of a self-fulfilling prophecy
<https://www.investopedia.com/ask/answers/05/selffulfillingprophecy.asp>:
the more traders rely on Fibonacci-based trading strategies, the more
effective those strategies will tend to be.
Thanks to books like Dan Brown's *The Da Vinci Code, *the golden ratio has
been elevated to almost mystical levels in popular culture. However, some
mathematicians have stated that the importance of this ratio is wildly
exaggerated.7
*The Golden Ratio and Technical Analysis*
When used in technical analysis
<https://www.investopedia.com/terms/t/technicalanalysis.asp>, the golden
ratio is typically translated into three percentages: 38.2%, 50%, and
61.8%. However, more multiples can be used when needed, such as 23.6%,
161.8%, 423%, and so on. Meanwhile, there are four ways that the Fibonacci
sequence can be applied to charts: retracements
<https://www.investopedia.com/terms/r/retracement.asp>, arcs, fans, and
time zones. However, not all might be available, depending on the charting
application being used.
6 The Jains were also aware of the theory of indices, though they did
not have the modern notation or any convenient notation for the same.
Calling the successive squares and square roots as the first, the second,
etc. they make the following statement: The first square root multiplied by
the second square root is the cube of the second square root. In modern
notation this is nothing but the identity in the theory of indices:
a1/2 x a1/4 = (a1/4 )3
They have several such rules for working with powers of a number. They also
seem to have had an idea of the logarithm of a number though they don't
seem to have put them to practical use in calculation. Another favourite
topic with them was the study of permutations and combinations. They had
also a great interest in sequences and progressions developed out of their
philosophical theory of cosmological structures. A Jain canonical text
entitled Triloka prajnApati has a very detailed treatment of arithmetic
progressions.
7 Astronomy: The ancients of India have passed on to us 27
mathematical formulae coded in the Sanskrit language, but not very
difficult to remember. In fact, very possibly it has mostly come down to us
by oral transmission from generation to generation. For instance, the
formula krittikA simhe kAyA says that if you see the asterism krittikA
(Pleiades, in modern terminology) on the meridian, that is the time the Leo
(= simha) constellation (of the zodiac) has risen above the horizon by an
amount indicated by the word: kAyA. This latter word interpreted in katapayA
sankhyA {KA …YA], which is the notation used by astronomers, astrologers
and mathematicians to represent numbers, means in this context that the
amount of Leo above the (Eastern) horizon is 27 minutes of time. From this
and the known position of the Sun on the date in question, one mentally
calculates the time of night. On November 7 for example, the Sun is in the
middle of Scorpio. So, if you see krittikA on the meridian it means Leo has
risen 27 minutes before and this means the Sun is behind by 93minutes
(remaining portion of simha) + 2h (full portion of kanyA) + 2h (full
portion of tulA) + 60m (half portion of vrischika) that is 6 hours 33
minutes. In other words, it is 6h 33m before sunrise. So it is 11-27
P.M.(How to compute a jatakam basis}
8 The apex of Mathematical achievement of ancient India occurred
during the so-called classical period of Indian Mathematics. The great
names are: Aryabhata I (b.476 A.D.) ; Brahmagupta (b.598 A.D.); Bhaskara I
(circa 620 A.D.) ; Mahavira (circa 850 A.D.); Sridhara (circa 900 A.D.) ;
Bhaskara II (b.1114 A.D.); Nilakanta Somayaji (1445 - 1545 A.D.).
Aryabhata wrote the famous Aryabhatiyam which is an exhaustive exposition
of Astronomy. In addition, he gave a unique method of representing large
numbers by word forms. He systematized all the knowledge of astronomy and
mathematics prior to him. The first one in Indian mathematics to give the
formula for the area of a triangle was Aryabhata. Several results on
Triangles and circles and on Progressions, algorithm for finding cube roots,
approximation of p, all these give him a unique position in the
development of mathematics. Aryabhata ushered in a Renaissance in Indian
Mathematics and Astronomy, that resulted in a remarkable flourishing of
science and technology in India. Aryabhata, for the first time, secularised
mathematics and astronomy in India and established these as intellectual
disciplines in their own right. Excellent commentators followed Aryabhata
and to them are due several modifications and applications, explanation of
subtle points, and finally, the proofs of results embodied in the sutras of
the Master.
Bhaskara I takes a large share of the credit of explaining the too brief
and aphoristic statements of Aryabhata. On the important topic of
indeterminate equations, the Kuttaka method was introduced by Aryabhata and
elucidated by Bhaskara I.
Brahmagupta is generally known as the Indian mathematician par excellence.
His monumental work Brahma Siddhanta has 24 chapters of which the latter 14
contain original results on arithmetic algebra and on astronomical
instruments. The 12th chapter is on mensuration. The 18th chapter is on
Kuttaka. Among his famous results are those on rational right-angled
triangles, and cyclic quadrilaterals. He is the earliest one, in the
history of world mathematics, to have discussed cyclic quadrilaterals. There
is every reason for us to name cyclic quadrilaterals as Brahmagupta
Quadrilaterals. It was partly through a translation of Brahma-siddhAnta
that the Arabs became aware of Indian astronomy and mathematics.
Bhaskara II's famous work SiddhAnta Siromani has four parts of which the first
two are Mathematics and the latter two are astronomy. The first part,
LilAvati is an extremely popular text dealing with arithmetic, algebra,
geometry and mensuration. The second part, BIja-ganitam is a treatise on
Advanced Algebra. It contains problems on determining unknown
quantities, evaluating
surds and solving simple and quadratic equations. The sheer ingenuity and
versatility of Brahmagupta's approach to indeterminate equations of the
second degree of the form N x2 + 1 = y2 is the climax of Indian work in
this area. Bhaskara II's cakravAla method to solve such equations is
world-famous. By using this powerful method, he solved, as one example, the
above equation with N = 61 and gave the least integral solution as x =
226153980 and y = 1766319049.
The famous French mathematician, Fermat, in 1657 A.D. proposed this
equation with N = 61 for solution as a challenge to his contemporaries. None
of them succeeded in solving the equation in integers. It was not until
1767 A.D. that the western world through Euler, by Lagrange's method of
continued fractions, had a complete solution to such types of equations,
wrongly called Pell's equation by Euler. But the very same equation, though
coincidentally, was completely solved by Bhaskara II *five hundred years
earlier.*
The problem of determining integer solutions of such equations is
called Diophantine
Analysis after the Greek Mathematician Diophantus (3rd cen. A.D.). As soon
as one finds a non-trivial solution (that is, other than the obvious
solution x = 0, y = 1) an infinite number of new solutions can be found by
repeated application of the Principle of Compositions, known as Brahmagupta’s
Bhavana Principle. It is Bhaskara's cakravAla method that makes the
decisive step in determining a non-trivial solution. Under these
circumstances it is appropriate to designate these equations as the
Brahmagupta-Bhaskara
equations.
Bhaskara II introduces also the notion of instantaneous motion of planets. He
clearly distinguishes between sthUla gati (average velocity) and sUkshma
gati (accurate velocity) in terms of differentials. He also gave formulae
for the surface area of a sphere and its volume, and volume of the frustum
of a pyramid. Suffice it to say that his work on fundamental operations,
his rules of three, five, seven, nine and eleven, his work on permutations
and combinations and his handling of zero all speak of a maturity, a
culmination of five hundred years of mathematical progress.
9 Indian Trigonometry
Though Trigonometry goes back to the Greek period, the character of the
subject started to resemble modern form only after the time of Aryabhata.
>From here it went to Europe through the Arabs and went into several
modifications to reach its present form. In ancient times Trigonometry was
considered a part of astronomy. Three functions were introduced: jya,
kojya and ukramajya. The first one is r sin a where r is the radius of the
circle and a is the angle subtended at the centre. The second one is r cos
a and the third one is r (1 – cos a). By taking the radius of the circle to
be 1, we get the modern trigonometric functions. Various relationships
between the sine of an arc and its integral and fractional multiples were
used to construct sine tables for different arcs lying between 0 and 90°.
10 Kerala mathematicians produced rules for second order interpolation
to calculate intermediate sine values. The Kerala mathematician Madhava may
have discovered the sine and cosine series about three hundred years before
Newton. In this sense we may consider Madhava to have been the founder of
mathematical analysis. Madhava (circa 1340 - 1425 A.D. ) was the first to
take the decisive step from the finite procedures of ancient Indian
mathematics to treat their limit-passage to infinity. His contributions
include infinite-series expansions of circular and trigonometric functions
and finite-series approximations. His power series for p and for sine and
cosine functions is referred to reverentially by later writers. Many later
discoveries in European mathematics (for example, the Gregory series for
the inverse tangent) were anticipated by Kerala astronomer-mathematicians.
Neelakanta was mainly an astronomer but his Arya Bhatiya Bhashya and
tantra-sangraha contain work on infinite-series expansions, problems of
algebra and spherical geometry.
K RAJARAM IRS 14 3 24
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