This document provides an overview of the history of mathematics, beginning with ancient civilizations like Babylonia, Egypt, and Greece. It discusses important mathematicians and their contributions, including Pythagoras, Euclid, Archimedes, Brahmagupta, Fibonacci, Descartes, Newton, Euler, Gauss, and Ramanujan. Key advances and discoveries are highlighted, such as the development of algebra, calculus, complex numbers, and non-Euclidean geometry. The document traces the evolution of mathematics from ancient times through the modern era.

1. Great Mathematicians and
their contributions

2. Mathematics is nearly as old as humanity
itself:
evidence of a sense of geometry and
interest in geometric pattern has been
found in the designs of prehistoric pottery
and textiles and in cave paintings.

4. Primitive counting systems were almost
certainly based on using the fingers of one
or both hands, as evidenced by the
predominance of the numbers 5 and 10 as
the bases for most number systems today.

5.  ANCIENT MATHEMATICS
 MEDIEVAL AND RENAISSANCE MATHEMATICS
 WESTERN RENAISSANCE MATHEMATICS
 MATHEMATICS SINCE THE 16TH CENTURY

6. ANCIENT MATHEMATICS
 Greek Mathematics
 Mesopotamian(Babylonian )
 Egyptian Mathematics

7. Ancient Mathematical Characters
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8. Babylonian Mathematics
There mathematics was dominated
by arithmetic, with an emphasis on
measurement and calculation in
geometry and with no trace of
later mathematical concepts such
as axioms or proofs.

9. Babylonian Mathematics
In the Babylonian system, using clay tablets consisting of
various wedge-shaped marks, a single wedge indicated 1
and an arrow-like wedge stood for 10 Numbers up through
59 .
The number 60, however, was represented by the same
symbol as 1, and from this point on a positional symbol was
used.
For example, a numeral consisting of a symbol for 2
followed by one for 27 and ending in one for 10 stood for 2 ×
602 + 27 × 60 + 10.

10. Babylonian Mathematics
The Babylonians in time developed a sophisticated
mathematics by which they could find the positive roots
of any quadratic equation.
The Babylonians had a variety of tables, including tables
for multiplication and division, tables of squares, and
tables of compound interest.
They could solve complicated problems using
Pythagoras' theorem; one of their tables contains integer
solutions to the Pythagorean equation, a2 + b2 = c2,
The Babylonians were also able to sum not only
arithmetic and some geometric series.
. In geometry, they calculated the area of rectangles,
triangles, and trapezoids, the volumes of simple shapes
such as bricks and cylinders.
No copy right –Mrs.P.Nayak,K.V.Fort william

11. Egyptian Mathematics
The earliest Egyptian texts,
composed about 1800 BC, reveal a
decimal numeration system with
separate symbols for the
successive powers of 10 (1, 10, 100,
and so forth) , just as in the system
used by the Romans. Numbers were
represented by writing down the
symbol for 1, 10, 100, and so on, as
many times as the unit was in a

12. Egyptian Mathematics
The Egyptians were able to solve all problems of
arithmetic that involved fractions, as well as some
elementary problems in algebra.
In geometry, the Egyptians arrived at correct rules
for finding areas of triangles, rectangles, and
trapezoids, and for finding volumes of figures such
as bricks, cylinders, and, of course, pyramids.
To find the area of a circle, the Egyptians used the
square on of the diameter of the circle, a value
close to the value of the ratio known as pi, but
actually about 3.16 rather than pi's value of about
3.14

13. Greek Mathematics
The Greeks adopted elements of mathematics from both
the Babylonians and the Egyptians.
The new element in Greek mathematics was the
invention of
an abstract mathematics founded on a logical structure
of
definitions, axioms, and proofs.
According to later Greek accounts, this development
began in
the 6th century BC with Thales of Miletus and

14. Culver Pictures
Pythagoras is Considered the
first true mathematician 6th-
century BC.
The followers of this
movement, Pythagoreans, were
the first to teach that the Earth
is a sphere revolving around
the Sun.

15. •Studies of odd and even numbers and of prime and
square numbers, essential in number theory..
• In geometry the great discovery of the school was
the hypotenuse theorem, or Pythagoras' theorem,
which states that the square of the hypotenuse of a
right-angled triangle is equal to the sum of the
squares of the other two sides
He gave the idea about Perfect Numbers

16. The Greek mathematician Euclid , who
lived around 300 bc, wrote Elements, a
13-volume work on the principles of
geometry and properties of numbers. His
work was rediscovered in the 15th
century, when it was translated from
Arabic, and until recent years has been
the principal source for the study of
geometry.

17. Aryabhata, also spelt Aryabhatta (476-c. 550),
Hindu astronomer and mathematician, born in
Pataliputra (modern Patna), India.
He was known to the Arabs as Arjehir, and his
writings had considerable influence on Arabic
science. Aryabhata held that the Earth rotates on
its axis, and he gave the correct explanation of
eclipses of the Sun and the Moon. In mathematics
he could solve quadratic equations, although
many of his geometric formulas were incorrect.
His only extant work is the Aryabhatiya, a series
of astronomical and mathematical rules and

18. Culver Pictures
Archimedes made extensive
contributions to theoretical
mathematics, in particular geometry.
Through his study of conic sections he
derived formulas for the areas of circles
and parabolas, and his work became the
basis for the development of calculus in
the 17th century.

19. MEDIEVAL AND RENAISSANCE MATHEMATICS
 Al-Karaji completed Muhammad al-
Khwarizmi's algebra of polynomials to
include even polynomials with an
infinite number of terms.
 Ibrahim ibn Sinan continued
Archimedes' investigations of areas and
volumes,
 Kamal al-Din and others applied the
theory of conic sections to solve optical
problems.

20. MEDIEVAL AND RENAISSANCE MATHEMATICS
 Thus mathematicians extended the
Hindu decimal positional system of
arithmetic from whole numbers to
include decimal fractions.
 In 12th-century Persian mathematician
Omar Khayyam generalized Hindu
methods for extracting square and cube
roots to include fourth, fifth, and higher
roots.

21. MEDIEVAL AND RENAISSANCE MATHEMATICS
 Finally, a number of Muslim mathematicians made
important discoveries in the theory of numbers,
while others explained a variety of numerical
methods for solving equations.
 Together with translations of the Greek classics,
these Muslim works were responsible for the
growth of mathematics in the West during the late
Middle Ages.
 Italian mathematicians such as Leonardo
Fibonacci and Luca Pacioli depended heavily on
Arabic sources

22. The Italian mathematician Leonardo
Fibonacci was largely responsible for
introducing the advances made by Arabic
and Indian mathematicians to Europe. His
Liber Abaci, published in 1202, helped
spread this knowledge and promoted the
Arabic numerals that we use today.

23. Bhaskara (1114-c. 1160), one of the most
outstanding of Indian mathematicians. His
major works were :
Lilavati, Bijaganita, Siddanta Siromani.
The Bijaganita analyses algebraic
expressions and explores solutions to
quadratic equations..

24. WESTERN RENAISSANCE MATHEMATICS
The discovery, an algebraic formula for the
solution of both the cubic and quartic
equations, was published in 1545 by the
Italian mathematician Gerolamo Cardano in
his Ars Magna. The discovery drew the
attention of mathematicians to complex
numbers and stimulated a search for
solutions to equations of degree higher than

25. WESTERN RENAISSANCE MATHEMATICS
The 16th century also saw the
beginnings of modern algebraic and
mathematical symbols, as well as the
remarkable work on the solution of
equations by the French
mathematician François Viète.

26. During the 17th century, the greatest
advances were made in mathematics
since the time of Archimedes and
Apollonius. The century opened with the
discovery by the Scottish mathematician
John Napier of logarithms,

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The science of number theory, which had lain
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Fermat to advance the theory of numbers greatly.
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28. Isaac Newton, one of the greatest
scientists of all time, revolutionized
mathematics in the 17th century. He
was responsible for the invention of
calculus and for advances in
algebra, analytic geometry, and the
theory of equations.

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 Two important developments in pure
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 The first was the publication, in Discourse on
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of analytic geometry, which showed how to
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Renaissance to investigate the geometry of
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curves.
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The second development in geometry was the
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publication by the French engineer Gérard
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Desargues in 1639 of his discovery of
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projective geometry.
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30. René Descartes founded analytic
geometry, which uses algebra to
represent geometric lines and
curves in terms of axes and
coordinates. He also contributed
to the theory of equations .

31. The 17th-century thinker
Gottfried Leibniz made many
contributions to mathematics.
He formulated the theory of
calculus

32.  French mathematician Gaspard Monge invented
differential geometry.
 Also in France, Joseph Louis Lagrange gave a
purely analytic treatment of mechanics in his great
Analytical Mechanics (1788), in which he stated the
famous Lagrange equations for a dynamical system
 His contemporary, Laplace, wrote The Analytic
Theory of Probabilities (1812) and the classic
Celestial Mechanics (1799-1825), which earned him
the title of the “French Newton”.

33. This illustration shows the Swiss mathematician
brothers Jakob ( left ) and Johann ( right )
Bernoulli discussing a geometrical problem. The
brothers both made important contributions to
the early development of calculus.

34. French astronomer and mathematician Pierre Simon
Laplace was best known for applying the theory of
gravitation.
The mathematical procedures Laplace developed to
make his calculations laid the foundation for later
scientific investigation of heat, magnetism, and
electricity.

35.  The greatest mathematician of the 18th
century was Leonhard Euler, a Swiss, who
made basic contributions to calculus and to
all other branches of mathematics.
 He wrote textbooks on calculus, mechanics,
and algebra that became models of style for
writing in the areas of Newton's ideas based
on kinematics and velocities, Leibniz's
explanation , based on infinitesimals, and
Lagrange's the idea of infinite series.
No copy right –Mrs.P.Nayak,K.V.Fort william

36. Although hindered by loss of sight,
Leonhard Euler was an important
contributor to both pure and applied
mathematics. Euler is best known for his
analytical treatment of mathematics and
his discussion of concepts in calculus,
but he is also noted for his work in
acoustics, mechanics, astronomy, and
optics.

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In a
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logically satisfactory approach to calculus.
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He based his approach only on finite
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38. Augustin Louis Cauchy was
one of the most brilliant
mathematicians of the 19th
century, making important
contributions to the fields of
functions, calculus, and
analysis.

39. In the 18th century the Swedish
astronomer Anders Celsius invented the
centigrade or Celsius scale of 100 degrees
between the freezing and boiling points of
water for the measurement of
temperature. The Celsius scale is one of
the most commonly used measurement
scales in the world.

40. . Early in the century, Carl Friedrich Gauss gave a
satisfactory explanation of complex numbers, and these
numbers then formed a whole new field for analysis,
Another important advance in analysis was Fourier's
study of infinite sums whose terms are trigonometric
functions. Known today as Fourier series, they are still
powerful tools in pure and applied mathematics.
In addition, the investigation of which functions could be
equal to Fourier series led Cantor to the study of infinite
sets and to an arithmetic of infinite numbers.

41. Gauss was one of the greatest mathematicians who ever
lived. Diaries from his youth show that this infant
prodigy had already made important discoveries in
number theory, an area in which his book Disquisitiones
Arithmeticae (1801) marks the beginning of the modern
era. While only 18, Gauss discovered that a regular
polygon with m sides can be constructed by straight-edge
and compass when m is a power of two times distinct
primes of the form 2n + 1.

42. German mathematician Carl
Friedrich Gauss contributed
to many areas of
mathematics, including
prime numbers, probability
theory, algebra, and
geometry.
Gauss also applied his
mathematical work to
theories of electricity and
magnetism. The magnetic
unit of intensity is named in
his honour.

43. Klein applied it to the classification of geometries
in terms of their groups of transformations
Lie applied it to a geometric theory of differential
equations by means of continuous groups of
transformations known as Lie groups. In the 20th
century, algebra was also applied to a general form
of geometry known as topology.

44. Ramanujan, Srinivasa (1887-1920),
Indian mathematician known for his
work on number theory, whose genius
brought him from obscurity to a brief
but remarkable collaboration with G. H.
Hardy at Cambridge.

45. Ramanujan was born at Erode, in Tamil
Nadu state, south India December 22,
1887, into a poor Brahmin family. His
father was an accountant with a cloth
merchant; his mother earned a few
rupees singing bhajans at the temple.
The young Ramanujan quickly showed
a single-minded love for mathematics.

46. However, his neglect of other subjects in
college led him to fail and lose his
scholarship. With no money, he gave up his
studies and eventually found a small job at
the Madras Port Trust in 1911. Before leaving
school, Ramanujan had bought himself a
copy of G. S. Carr's A Synopsis of
Elementary Results in Pure and Applied
Mathematics.

47. He worked his way through this
systematically and began his own research,
publishing articles in Indian mathematical
journals and soon becoming recognized as a
remarkable mathematician.

48. It was his letter to the English mathematician G. H.
Hardy, at the University of Cambridge, discussing
and questioning some of Hardy's published work,
that brought him, after considerable efforts by Hardy
and his colleagues, to Cambridge in 1914, on a
research scholarship. Here he was able to study and
research freely. His health was frail, however, and in
1917 he became very ill, probably from tuberculosis.
He was elected Fellow of Trinity College, and a
Fellow of the Royal Society, in 1918, at the age of 31 .

49. Ramanujan's main interest had been the
study of numbers, and his most remarkable
results were in the partitioning of numbers.
He also worked on identities, modular
equations, and mock-theta functions. The
notebooks he left, full of the fevered work of
his last days, are still being studied. His
extraordinary intuition, and unorthodox
methods, led to some of the strangest and
most beautiful formulae in mathematics.

50. He had proved that zero divided
by zero was neither zero nor one,
but infinity.

51. Another subject that was transformed
in the 19th century, notably by
English mathematician George
Boole's Laws of Thought (1854) and
Cantor's set theory, was the
foundations of mathematics. Towards
the end of the century, however, a
series of paradoxes was discovered
in Cantor's theory. One such paradox,
found by English mathematician
Bertrand Russell, aimed at the

52. The German mathematician Georg
Cantor was renowned for his
developments in the field of set
theory. His line of inquiry led in
the 20th century to the
fundamental investigation of the
nature of mathematical logic.

53. In the 19th century the British
mathematician George Boole
developed a form of algebra,
known as Boolean algebra, which
today is very important to
computer operations, such as in
the use of Internet search
engines.

54. Current Mathematics
harles Babbage in 19th-century England who designed
machine that could automatically perform
portant calculations.
ilbert could not have foreseen seems destined to play
even greater role in the future development
mputations based on a programme of instructions
ored on cards or tape.
abbage's imagination outran the technology of his day,
d it was not until the invention of the relay, then of the
cuum tube, and then of the transistor, that large-scale,
ogrammed computation became feasible with

55. The inventor of the Difference Engine, a
sophisticated calculator, the
mathematician Charles Babbage is also
credited with conceiving the first true
computer. With the help of Augusta Ada
Byron, Babbage created a design for the
Analytical Engine, a machine remarkably
like the modern computer, even
including a memory.

56. THANK YOU