From the Indus Civilization to Srinivasa Ramanujan - A History of Indian Mathematics

From the Indus Civilization to
   Srinivasa Ramanujan
     A History of Indian
        Mathematics

Indus Civilization 3000 BC

A highly sophisticated urban civilization
that died mysteriously around 3000 BC.

Indus Civilization 3000 BC
It stretched over a region of more
than a million square kilometers

                    (c)MSF

Indus Civilization 3000 BC
Lothal

  An artist’s imagination based on archaeological inputs

Indus Civilization 3000 BC

Kalibangan

Indus Civilization 3000 BC

It comprised baked clay brick buildings,
  highly developed sea and river ports

Indus Civilization 3000 BC
The degree of
advancements in science
and technology can be
gauged from the highly
evolved system of
plumbing and public baths
as well as from the
constructions of the cities
and the sophisticated seals
and sculptures that have
been found.

Practical Mathematics in the Indus Civilization
The people of the IVC demonstrated the use
of mathematics in their daily life such as:

   They manufactured bricks
    with dimensions in the
    proportion 4:2:1 to lend
    stability  of   a   brick
    structure

 They used a standardized system of weights based on
  the ratios: 1/16, 1/8, 1/4, 1/2,    1, 2, 4, 10, 20, 40, 100,
  200, 400, 500 and 800, with         the unit weight being
  approximately 13.63 grams.           The heaviest known
  weight was about 10.9 Kg            and the lightest 85.1
  centigrams.

  The weights were made in regular geometrical shapes like
  hexahedra, barrels, cones, and cylinders, thereby demonstrating
  knowledge of basic geometry

How did they measure length?
One evidence is a scale with nine parallel
equidistant markings that is also indicative of
the decimal system.

Out of the graduated scales that were prepared,
only three survive. These scales do not conform to
each other indicating that probably, different
systems of linear measurement were in vogue.

Units of length
    The Indus Foot

    The Indus Yard

Both these lengths were a little more than their present
day versions.

The Great Bath at Mohenjo-Daro

  Aerial View      Ground View

CITIES
The ruins of Mohenjo-Daro in the
   Sindh province of Pakistan

SEALS

The problems
1. The script has not been deciphered

The problems
2. A great deal of plundering took place
in some of the most important sites

What has been found?
The sophistication of the civilization-ports,
magnificent brick buildings and cities.
                                  Lothal

Dancing Girl

Mohenjo-Daro. 2500 B.C.

Ornaments

This collection of gold and agate ornaments includes objects
          found at both Mohenjo-daro and Harappa.

Just around the time that the Indus
Civilization    disappeared   the Vedic
Civilization began to evolve.

It is not clear as to what the links are between
the two but there is a great deal of
speculation.

The Vedic Civilization
and Mathematical
Discoveries
     3000 BC onwards

The Vedic Civilization evolved around the
  development and the practice of the Vedas.

Our interest is in the growth of
 mathematics during that period.

The Sathpatha Brahmana: 900 BC

This text records the theorem
   of Pythagoras 400 years
      before Pythagoras.

The Sulba Sutras
Sulba – means pieces of chord or string

Sutra – formula or aphorism
The Sulba Sutras are the mathematical discoveries
made by famous mathematicians at about 1000 BC
to 200 BC, using a piece of chord for constructions
of various fire sacrifice altars.

                       (c)MSF

The Sulba Sutras

Baudhayana : 800 BC
  Baudhayana was the author of one of the
           earliest Sulbasutras.

  Baudhayana discovered a proof of the
        Theorem of Pythagoras.
It is a recorded fact that Pythagoras who lived
around 500 BC visited India and interacted with
Indian Mathematicians and scholars. In fact Indian
tradition records that he turned vegetarian under
the influence of Jainism after visiting India.

François-Marie Arouet de Voltaire (1694-1774)
               I am convinced that everything has
               come down to us from the banks of the
               Ganga      –    Astronomy,    Astrology,
               Spiritualism, etc.
               It is very important to note that 2,500
               years ago at least Pythagoras went
               from Samos to the Ganga to learn
               Geometry …..
               But he would certainly not have
               undertaken such a strange journey had
               the reputation of the Brahmins’ science
               not been long established in Europe
Pythagoras     …..
 of Samos

Katyayana : 200 BC
Katyayana was the author of one of
        the Sulba Sutras

The Sulba Sutras

 From the Katyayana Sulba Sutra
Click here for a proof of The Theorem of Pythagoras

Irrational Numbers in the Sulba Sutras
In all the extant Sulba Sutras, we find clear mention
of the following numbers

 2 , 3 , 1 These num bers were given specific nam es
            3
They had also calculated a value of
          1    1          1
 2  1                        1.4142156
          3 3  4 3  4  34
which is correct to 5 decim al places

Panini 520 BC

                            A postal ticket
                         released after Panini

Panini was a Sanskrit grammarian who gave a
comprehensive and scientific theory of phonetics,
phonology, and morphology.
 Panini invented a perfect Sanskrit grammar.
 He used the concept of zero.

Pingal : 200 BC
 Pingal was a prosodist.

 He was trying to invent new meters from the
  known Vedic meters such as the Gayatri,
  Anustabh, Brhati by varying the syllables through
  permutations and combinations of long and short
                      sounds.

 This led him to discover the Meru Prastara in 200
  BC which is now known as Pascal’s Triangle,
  discovered by Pascal 1800 years after Pingal. He
  wrote this in his book the Chandah Sutra

Meru Prastara: Pingala’s discovery in 200 BC

Halayudh : 1200 AD
Halayudh wrote a commentary on Pingal’s
work and in the process discovered the
Binomial Theorem 400 years before Newton.

                n  n 1 1      n  n 1 n
(a  b)  a   a b  ...  
      n    n
                                       ab  b
                1              n  1

Jain and Buddhist Mathematics Before Christ
Returning to the time of the advent of Christ, it is well
known that Jaina and Buddhist mathematics had
evolved with sophistication.
   They could represent large numbers-unlike
    the Greeks-such as
                          10 53 , (8400000 ) 28
   They (Jain’s) knew the laws of indices
                           m n
  a   m
          a    n
                    a            ; (a )  a
                                         m    n           mn

There is evidence that the Jain mathematicians understood
different orders of infinity thereby anticipating in many ways the
great discoveries of Georg Cantor.

The Bakshali Manuscript
• An early mathematical manuscript, written on birch
bark and found in the summer of 1881 near the village
of Bakhshali then in India and now in Pakistan.

• A large part of the manuscript had been destroyed
and only about 70 leaves of birch bark, of which a few
were mere scraps, survived to the time of its discovery.

• The manuscript is dated to 300 AD and is a
commentary on an earlier mathematical work .

• Gives clear evidence of the use of the decimal system

The Discovery of the Bakhshali Manuscript
•A tenant on a piece of land discovered the manuscript while
digging a stone enclosure in a ruined place.

• This was during the time of the British rule in India so eventually
the manuscript landed in England and now resides in Oxford.

• A photo image of the Bakhshali manuscript is housed in the
National Museum in New Delhi

Evolution of Numerals:
     India’s Contribution

Brahmi Numerals: Around the Time of Christ

Progression of Brahmi number forms through the centuries
    (column far left showing forms in use by 500 AD)

Numeral forms found in Bakhshali Manuscript
    showing place value and use of zero

Fractions in Bakshali Manuscripts

             3 1+   means   3 - 1
             4 2            4   2

An unusual feature is the sign + placed after a
number to indicate a negative.

Some Contents of the Bakhshali Manuscript
• Solution of linear equations with as many as five unknowns.

• Quadratic equations with solutions.

• Progressions: Both arithmetic and geometric.

• Simultaneous equations.

• Fractions and other advances in notation including use of zero
and negative sign.

• A remarkable method to compute square roots (useful in
approximating irrational numbers)

Square root in Bakshali Manuscript

√Q = √(A2 + b) = A + b/2A - (b/2A)2/(2(A + b/2A))

This is stated in the manuscript as follows:-

In the case of a non-square number, subtract the nearest
square number, divide the remainder by twice this nearest
square; half the square of this is divided by the sum of the
approximate root and the fraction. This is subtracted and will
give the corrected root.

                              (c)MSF

Aryabhata I : 476 AD
Aryabhata I wrote the
“Aryabhatiya” which
summarizes Hindu
mathematics up to that 6th
Century.
He was 21 years of age when
he wrote the Aryabhatiya. In
this monumental work he
records many important
discoveries in mathematics
and astronomy.

Aryabhat’s Discoveries:
1. The earth is round and revolves around its axis.
   In fact Aryabhat said that the earth is as spherical
   in shape as the kernel of the flower of the
   Kadamb Tree.

2. He also said that the sun appears go around the
   earth when in fact it is the earth that revolves
   and he said this was similar to a situation where a
   passenger in a boat on a river will feel that the
   trees on the banks of the river will appear to be
   moving in opposite direction whereas it is the
   boat that is moving.

3. He discovered the value of pi correct to four
   decimal places.

3. Aryabhat gave methods of finding square roots
   and cube roots.

4. He was the first mathematician in the world to
   give a general solution of the indeterminate
   equation of first degree:
   by = ax c, where x and y are unknowns.

5. He put Indian astronomy on firm mathematical
   foundation.

Brahmagupta : 598 AD – 670 AD
Brahmagupta was the
foremost Indian
mathematician of his time.

He made advances in
astronomy and most
importantly in number
systems including algorithms
for square roots and the
solution of quadratic
equations.

Some of the achievements of Brahamagupta are:

1. He solved the equation

                     y  Nx  1
                      2        2

   where x and y are unknown. This equation was later
   named by Euler as the Pell’s equation.

2. Discovered a formula for the area of a cyclic
   quadrilateral.

3. Discovered formulae for determining the diagonals of a
   cyclic quadrilateral in terms of its sides.

4. He was the first to obtain formulas for the sum of
   squares and cubes of first n natural numbers in the
   form:

                nn  1 2n  1
  1  2  n 
   2       2             2

                   2       3
                 nn  1 
                                              2

  1  2  n  
   3       3             3
                           
                    2     

Trigonometry in India (300 A.D. onwards)

Surya Siddhanta – unknown author; Founder of
  modern trigonometry

It makes distinctive uses of the modern
trigonometric functions:
 Sine (Jya)
 Cosine (kojya)
 Inverse sine (otkram jya)
 Tangent
 Secant

 A sidereal year was computed as
  365.2563627 days, which is only 1.4
  seconds longer than the modern
  value of 365.2563627 days.

 A tropical year was computed as
  365.2421756 days, which is only 2
  seconds shorter than the modern
  value of 365.2421988 days.

Bhaskara II
              1114 AD - 1185
Bhaskara II or
Bhaskaracharya was
one of India’s greatest
mathematicians who
made numerous
important discoveries
including the discovery
of the Calculus

Bhāskara II was a distinguished mathematician and
an astronomer who worked in major areas of
mathematics. His contributions to various areas of
mathematics include :

 Proof for division by zero being infinity
 Was the first to observe that a positive
  number has two square roots
 Properties of Surds
 Operations with products of several
  unknowns

The solutions of Quadratic, Cubic
 and Quartic equations
 Equations with more than one
 unknown
 Quadratic equations with more than
 one unknown

 Discovery of the derivative
 Called the derivative tatkalika gati
  (instantaneous velocity)
 He showed that the derivative of the sine
  function is cosine
 Discovered Rolle's theorem
        -a special case of the mean value
theorem

 Gave the chakravala (cyclic) method to
  solve the general form of Pell's equation.
 Was the first to use symbols for unknowns
  in algebra.
 Computation of π, correct to 5 decimal
  places
 Discovered the trigonometric formulae
   sina  b   sina  cos b   cos a  sinb 
   sina  b   sina  cos b   sinb  cos a 

Vasco da Gama
    An adventurer
    not a navigator

Sailed to India from Europe : 1497 - 1499

     His navigator was an Indian :
           Named “ Kanha”

Vasco da Gama & Kanha
 Used an instrument called “Kamal” or “ Rapalagai”
          to determine the latitude at sea

This instrument involves a harmonic scale with knots on a string to
   measure angular elevation of pole star above the horizon for
                     measuring local latitude
                                (c)MSF

“Bhaskara I” knew the method of determining
       longitude from time difference

                      Mechanical Clock
                      Revolutionized the
                       sea navigation in
                         17th century
                     This lead to the invention of
                      CHRONOMETER

Madhava of Sangamagramma
          1350 AD - 1425 AD
Madhavacharya was a
mathematician          from
South India. He made
major discoveries in
calculus          including
important advances in
infinite series expansions
for          trigonometric
functions.

There are a number of credits to
Madhava’s name like:
 The discovery of the          power series
  expansion of tan 1  x      which today is
  called the Gregory’s series
 The expression
               1 1 1
             1   
          4     3 5 7
 Computed      an     extremely     close
  approximation of π as 3.14159265359

Srinivas Ramanujan
  1887 – 1920 AD

          House of Srinivas Ramanujan

Ramanujan was a self taught genius.

He made numerous deep and extra
ordinary discoveries.

We will present only two of them over
here.

We all know the prime numbers.

2, 3, 5, 7, 11, 13, …

Prime numbers are extremely important in Number
Theory.

Long ago mathematicians realized that one can not
have a polynomial formula that gives prime numbers.

From the time of Gauss they began to see how prime
numbers occur amongst all natural numbers.

Gauss looked at the following function:
                      (n)
Which counts the number of primes less than or equal
to n.
                                     n
He conjectured that        ( n) ~
                                   log n
It took some of the greatest European mathematicians
to finally prove in 1898 that the conjecture was true.

 What was Ramanujan’s connection with this theorem
 known as the Prime Number Theorem?

Partitions and Ramanujan
A partition of a natural number n is a representation of n
as a sum of positive integers.

The order of the summands is not relevant.

Example:

5=5
 =4+1
 =3+2
 =3+1+1
 =2+2+1
 =2+1+1+1
 =1+1+1+1+1

So,

P(5) = 7

P(n) increases very rapidly with n.

P(200) = 3,972,999,029,388

Ramanujan showed

P (5n + 4) is always a multiple of 5.

In fact Hardy and Ramanujan actually gave a precise
formula for p(n).

G. H. Hardy : 1877
G. H. (Godfrey Harold)
Hardy was a prominent
English Mathematician,
known      for     his
achievements        in
number theory and
mathematical analysis.

1729

           Taxi Cab Number
The smallest number representable in two
ways as a sum of two cubes. It is given by

     1729 = 13 + 123 = 93 + 103

Thanks