History of Math is a project in which students worked together in learning about historical development of mathematical ideas and theories. They were exploring about mathematical development from Sumer and Babylon till Modern age, and from Ancient Greek mathematicians till mathematicians of Modern age, and they wrote documents about their explorations. Also they had some activities in which they could work "together" (like writing a dictionary, taking part in the Eratosthenes experiment, measuring and calculating the height of each other schools, cooperating in given tasks) and activities that brought out their creativity and Math knowledge (making Christmas cards with mathematical details and motives and celebrating the PI day). Also they were able to visit Museum, exhibition "Volim matematiku" and to prepare (and lead) workshops for the Evening of mathematics (Večer matematike). At the end they have presented their work to other students and teachers.

1. HISTORY OF MATH

2. History of Math
 eTwinning project
 Cooperation between and
 We explored:
 Development of mathematical thought from Sumer till
Modern age
 Distinguished mathematicians from Ancient Greece to
Modern age

3. What did we do?
 Presentations (.ppt) for introducing ourselves:
 Personal
 School
 Hometown

4. What did we do?
 We chose the LOGO of the project
13

5. What did we do?
 We visited exhibition „I Love Math” („Volim matematiku”)/CRO

6. What did we do?
 We organized „The Evening of Mathematics” /CRO

7. What did we do?
 We exchanged Christmas cards
Croatian in Greece
Greek in Croatia

8. What did we do?
 We celebrated the 𝜋 Day

9. What did we do?
 We visited Technical Museum of Ancient Greece in
Thessaloniki /GR

10. What did we do?
 We did measures for calculating the Earth
circumference – Eratosthenes experiment
 We calculated the Earth circumference
 We measured shadow length of schools
 We calculated the height of schools

11. ... and our outcome
for Novska school
is 8.322 m !!!
Greek Team

12. … and our outcome for Edessa school is
𝟏𝟏, 𝟗𝟖 ± 𝟎, 𝟏𝟒 𝒎
Croatian Team

13. What did we do?
 We wrote documents (.doc i .ppt) about given tasks

14. What did we do?
 We presented the project to mathematic teachers of Sisak-
Moslavina County / CRO

15. What did we do?
 We presented the project to teachers and students of our
schools

16. What did we do?
 We set up an exhibition of posters in school hallway / CRO

17.  On TwinSpace Forum we
wrote a dictionary of
mathematical words that
have Greek root in:
 English
 Greek
 Croatian
What did we do?

18. What did we do?
 We wrote seminars about given topics and merge
them in a book /CRO
 http://www.slideshare.net/gordanadivic/povijest-
matematike-history-of-math
 We edited our TwinSpace:
 http://twinspace.etwinning.net/490/home

19. Antonio Jakubek, 4.g

20.  Our first knowledge of math comes from Egypt
and Babylon
 Babylon math dates back to 4000 BC along with
the Sumerians in Mesopotamia

21.  Little is known about Sumer
 It was first inhabited 4500 and 4000 BC
 Today, these people are called Ubaidiansi
 Even less is known about their math

22.  They used cuneiform and wrote on clay tablets
 They used over 2000 signs
Picture 1. Sumerian cuneiform
record

23.  They developed heksagezimal number system
which was taken by the Babylonians
 Babylonians, Assyrians and Hiti inherited
Sumerian law and literature and more
importantly their way of writing
 What we have kept from the Sumerians today
is the division of weeks to 7 days, days to 24
hours, 60 minutes in an hour and 60 seconds in
one minute

24.  With the collapse of the Sumerian civilization
in Mesopotamia Babylon is developed
 They inherited from the Sumerians cuneiform
and heksagezimal number system
Picture 2. The digits of the
Babylonian number system

25.  They used 2 basic forms for numbers:
 They had no symbol for zero or a decimal
point, so it was difficult to interpret the
findings from this era
Picture 3. Babylonian symbol for
the number 10
Picture 4. Babylon symbol for
number 1 or 60

26.  In 1940s German historians Otto Neugebauer
and Abraham Sachs
 Noticed how the verses on the board meet
interesting propertyes
 Decorated triplets of positive integers (a, b, c)
that satisfy a2 + b² = c²

27.  Proof of the existence of Pythagorean triples
thousands of years before the mathematicians
of ancient Greece
Picture 5. Plimpton 322

28.  The site in Nippur - found about 50 000 clay
tablets
 Witnessed considerable knowledge of
mathematics
Picture 7. The site in Nippur

29.  They were building up series of numbers that
include triangular numbers (1, 3, 6, 10, 15 ...),
square numbers (1, 4, 9, 16, 25 ...) and the
pyramidal numbers (1, 5, 14, 30, 50 ...)
Picture 8. i 9. Showing series of
numbers

30.  An example of using a series of numbers is
pyramid stacking of ammunition in Calcutta
and easy calculation of the number of
cannonballs
Picture 10. Pile of ammo in
Calcutta

31. Egypt
Ella Cink, 4.g

32. Moscow papyrus
- discovered in 1893 and the author is unknown
- the greatest achievements of Egyptian geometry
- length is about half a meter and width of less than 8 cm
- kept in the Moscow Museum

33. Moscow papyrus

34. Rhinds papyrus
• In 1858 he was discovered by Scottish Egyptologist
Henry Rhind in Luxor
• It was written by the scribe Ahmes around 1600 BC
• It is 6 meters long, 30 cm wide, preserved in the British
Museum in London

35. • A collection of tables
and exercises with 87
math problems
• It contains the oldest
known written record
number π
Rhinds papyrus

36. Numbers
The Egyptians used a number system with a base 10
 number 1339

37. • Addition
• Subtraction
• Multiplication
• Division
because

38. Fractions
• They only knew fractions
• The exception was 2/3
• Fractions are formed by combining the individual parts of the
symbol Horus eye
the entire symbol of the eye has a value of 1 

39. Geometry
• To build the pyramids and temples they were obliged to
have a well-developed geometry and stereometry
• They knew how to calculate the slope and volume of the
pyramid, and the volume of a truncated pyramid

40. Algebra
• Egyptian algebra was rhetorical
• Problems and solutions are given by words
• They used seven-digit numbers, and their calculations
were a mixture of simplicity and complexity

41. Mathematics of
Ancient Greece
Doroteja Lukić, 3.g

42.  based on Greek texts
 developed from the 7th century BC to
the 4th century AD
 along the eastern shores of the
Mediterranean
 mathematics - Greek Mathematica -
Science
 use general mathematical proofs and
theories

44.  presided crucial and most dramatic
revolution in mathematics ever
 main goal: the understanding of
man's place in the universe
 mathematics has reached the highest
level of development
 began to use papyrus
 Greek contribution to mathematics in
three phases:
 1. Thales and Pythagoras to
Democritus
 2. Euclidean system
 3. phase of Alexandria

45.  Tales - founder of Greek mathematics
 no documentary evidence
 classical philosophy helped to reconstruct
texts a closer period
 editions of Euclid, Archimedes, Apollonius,
etc.
 difficult to follow the course of historical
development
 on Greek mathematics concludes: smaller
components and observations of
philosophers and other authors

46. Greek number system
( About 900 BC - 200 AD)
 The first was based on the initial letters of the
names of numbers

47.  the second used all the letters from
 Greek alphabet and three from the
Phoenician
 Base - 10

48.  the idea of evidence and a deductive
method of using logical steps to confirm or
refute the theory
 gave the mathematics force
 ensures that the proven theories are true
 laid the foundation for a systematic
approach to mathematics
The most important contribution
of the Greeks

49. PYTHAGORA
Petra Kalanja, 2.g

50. General…
 the first "true" mathematician
 born on the Greek island of Samos
 Tales interested him in mathematics
 traveled to Egypt around 535 BC
 founded the Pythagorean school
 Today he is known for the Pythagorean
theorem

51. Through life ...
 philosopher in Egypt
 temple priest in Diospolisu
 captive in Babylon
 married at age 60
 starved to death
 the most perfect number 10
 number - being in philosophy

52. Pythagorean school
 established in Crotona
 emphasis on secrecy and fellowship
 Pythagorean theorem
 the discovery of irrational numbers
 five regular solids

53. Pythagorean theorem
 Surface of the
square on the
hypotenuse of a
right triangle is
equal to the sum
of the squares of
the cathetus

54. Pythagorean triples
 3, 4, 5 9+16=25
 Egyptian triangle
 We can get another infinite number of
Pythagorean triples by making the
numbers 3, 4 and 5 reproduce the same
number
6, 8, 10 36+64=100

55. PLATO (428 - 347 BC)
-lived and worked in Athens
-387 BC founded the
philosophical school
ACADEMY where
mathematics, arithmetic,
trigonometry and
planimetry was taught
Picture 1.Plato
Marija
Kožarić,
4.g

56.  "No entrance for those who
do not know geometry!"
Picture 2.The inscription at the entrance to the Academy

57. Platonic solids
 Described in the work Timaeus
 5 regular polyhedrons:
Picture 3. regular polyhedrons

58. TETRAHEDRON
 4 peaks
 6 edges
 4 sides  equilateral triangles
Picture 4. tetrahedron

59. HEXAHEDRON
 cube
 8 peaks
 12 edges
 6 sides  squares
Picture 5. hexahedron

60. OCTAHEDRON
 6 peaks
 12 edges
 8 sides equilateral triangles
Picture 6. octahedron

61. DODECAHEDRON
 20 peaks
 30 edges
 12 sides  regular pentagon
Picture 7. dodecahedron

62. ICOSAHEDRON
 12 peaks
 30 edges
 20 sides  equilateral triangles
Picture 8. icosahedron

63. Chinese mathematics

64. • in the second millennium BC China had symbols for
numbers
• they counted with sticks until abacus
appeared in the 16th century
Picture 1. chinese
numbers
Picture 2. abacus
• not much is known about the mathematics of ancient China, but it is
fairly certain that the origins of astronomy and mathematics of ancient
China date back to at least the second millennium BC, at that time the
Chinese have already had an elaborate calendar
• oldest surviving mathematical texts originate from the time around 200
BC

65. • Contributions of Chinese
mathematicians:
• The Holy Book of arithmetic (2nd - 12th century) - indirect
talks about the Pythagorean theorem
• Arithmetic in nine books (about 150 BC) - the process of
calculating the area of a triangle, rectangle, circle, circular
section and clip, the volume of prism, pyramid, cylinder,
cone, deprived (truncated) cone and pyramid
• The book of phases (I Ching) - one of the oldest surviving
book - used for fortune telling and divination, contains
elements of the binary notation of numbers

66. • The famous mathematicians:
• Zhang Qiu Jian (5th c.) - Gave the formula for the sum
of the arithmetic series
• Tsu Chung - chih (430 to 500) - the value of the number
π takes a precise six decimal places
• Quin Jiu - Shao (1202 -1261) - Sought the solution of
equations method that is called Horner (William Horner,
1819), although it was known in China 500 years earlier
• Chu Shih - kieh (1270 -1330) - wrote two important
texts that are the pinnacle of Chinese mathematics texts
which contain Pascal's triangle binomial coefficients,
which is known in China for four centuries before it was
discovered by Pascal.

67. INDIAN MATHEMATICS

68. • in ancient Indian mathematics there is no great
works exclusively devoted to mathematics;
mathematics is present only as part of, as a separate
chapter in astronomical or astrological works
• the oldest known mathematical texts are Sulvasutre,
accessories to religious texts
 in them are the rules for measuring and building
temples and altars at the level of elementary
geometry
• characteristics of Indian mathematical texts is that they are generally written
in verse
Picture 1. Indian numbers

69. Ancient India mathematics:
• Aryabhatta (476 to 550) - he knew how to
take out the second and third root of the
division into groups of radikands
• gave the correct formula for the area of a
triangle and a circle, writes about quadratic
equations and potencies
• Brahmagupta (598 – about 670)
• Brahmaguptas formula: a generalization of
Heron's formula in the cyclic quadrilateral;

70. • Mahavira (9th century) - dealt with elementary
mathematics and the first Indian mathematician who
wrote the only math dedicated text
• Bhaskara (1114 – 1185) - the most famous Indian
mathematician to the 12th century, has contributed to
the understanding of numerical systems and solving
equations and proved the Pythagorean theorem
• his main mathematical works of
Lilavati and Bijaganita, dealt with
plane and spherical trigonometry

71. Arab mathematics

72. • Today's western style math is much more similar to what we
find in the Arab mathematics than that of ancient Greeks,
many of the ideas that have been attributed to the Europeans
proved to be actually Arabic
Picture 1. Arabic numbers
Al-Khwarizmi (780 – 850) first great Arab
mathematician

73. Arabic mathematicians:
• Al-Karaji (953 - 1029) - is considered the first person
who completely freed algebra from geometrical
operations and replaced them with arithmetic
• founded the influential algebraic school
that will work successfully for centuries
•Al-Khwarizmi (780 - 850 ) first great Arab mathematician
• he wrote about algebra, geometry, astronomy and he
introduced Arabic numerals in mathematics
• he dealt with linear and square equations. He built tables
for sinus and tangens functions. He gave a general
method for finding two roots of quadratic equations :
𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 𝑥1,2 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎

74. • Al-Haytham (965 - 1040) is probably the first who attempted to
classify the even perfect numbers
• also was the first person that has imposed Wilson's theorem (if p is prime,
then p divides 1 + (p - 1)!), it is unclear if he knew how to prove it
• Omar Khayyam (1048 – 1131) with mathematics also dealt with astronomy,
philosophy and poetry
• gave a complete classification of cubic equations (14 types) and the first to
notice that you do not need to have a unique solution
• Nasir al-Din al-Tusi (1201 - 1274) wrote important works on logic, ethics,
philosophy, mathematics and astronomy
• The most important contribution was his creation of trigonometry as a
mathematical discipline, not a means of astronomical calculations,
and gave the first complete account of plane and spherical
trigonometry.
• This work gave the theorem of the sinuses for planar triangles:
𝑎
sin 𝛼
=
𝑏
sin 𝛽
=
𝑐
sin 𝛾

75. (12th – 13th century)
Picture 1. Fibonacci
Barbara Mašunjac, 4.g

76.  Italian mathematician
 spent his youth in Arabia
 the foundation of his mathematics -
number
 left behind a series of discoveries

77. FIBONACCI SERIES
Picture 2. Fibonacci series

78. FIBONACCI SERIES IN NATURE
Picture 3. Fibonacci series in sunflowe
Picture 4. Fibonacci series in snail Nautilus shell
Picture 5. Fibonacci series
in human body

79. FIBONACCI SERIES IN ART
Picture 6. Fibonacci series
in Mona Lisa portrait Picture 7. Fibonacci series in
Partenon

80.  ”Divine ratio” or the ratio of the golden section
𝜑 =
1 + 5
2
≈ 1.618033989

81. The ratio of the golden section

82. The ratio of the golden section

83. The ratio of the golden section

84. The ratio of the golden section

85. The ratio of the golden section

86. The ratio of the golden section

87. LIBER ABACI
 the most famous work of
arithmetic
 one of the first Western
book that described the
Arabic numerals
 four parts
Picture 8. Liber Abaci

88. JOHN
NAPIER Laura Iličić, 3.g

89. GENERAL:
• Born in Edinburgh 1550, died April 4th 1617
• He enrolled at the University of St. Andrews
• He graduated in Paris, and then stayed in the Netherlands and
Italy
• He is known in mathematical and engineering circles
• He is best known as the inventor of logarithms, Napier's bones,
and the popularization of the decimal point
• He worked in the fields of mathematics, physics, astronomy
and astrology

90. Napier’s bones
Decimal point Logarithms

91. MOST FAMOUS WORKS
• Plaine Discovery of the Whole Revelation of St. John, 1593
• Statistical Account
• Mirifici logarithmorum canonis descriptio, 1614
• Construction of Logarithms, 1619

92. • Mirifici logarithmorum canonis descriptio, Statistical Account i Construction of
Logarithms

93. Henry Briggs
English mathematician
Professor of geometry at Oxford
born in Warleywoodu in
Yorkshire 1561
He studied at St. John's College,
Cambridge
Patricia Kujundžić, 3.g
NO
PICTURE

94. as a professor at Oxford he
learned about Napier
1615 travels to him in Edinburgh
Napier agrees with the proposal
for the Briggs logarithms with
base 10
After Napier's death continues
his work
1624 publishes logarithmic
table Arithmetica
He died in Oxford 1630

95. Blaise Pascal
Antonio Horaček, 4.g

96. Biography
 Blaise Pascal was a French mathematician,
physicist, inventor, writer and Christian
philosopher. He was a 'child prodigy' and was
educated by his father.
 Pascal’s earliest jobs were in applied and
natural sciences, where he contributed to
the study of fluids, and clarified the concepts
of pressure and vacuum by generalizing the
work of EvangelistaTorricelli.

97.  Pascal's first calculating machine

98. Pascal’s contribution to math
 The first significant work, Blaise wrote at age sixteen,
and it was a basic draft of his famous debate on the
sections of the cone.

99.  Blaise Pascal, also created his famous
 mystical hexagram (Pascal's theorem), which has not
survived.
 In his 'Treatise on the arithmetical triangle' '(Traité du
triangle arithmétique), described the convenient,
practical tabulation of binomial coefficients, now called
"Pascal's Triangle'.

100. Pascal’s contribution to physics
 His work in the field of hydrodynamics and hydrostatics has
focused on the principles of hydraulic fluids. His inventions include
the hydraulic press (using hydraulic pressure to multiply force) and
the syringe.
 Hydrostatic pressure increases the depth, acts equally in all
directions and is equal in all places at the same depth.

101.  Pascal’s law
 The fundamental law of hydrostatics:
 The fluid contained in a closed vessel outer pressure p
expands equally in all directions, that is, particles of the
liquid pressure is transmitted equally in all directions.

102. History of infinitesimal
calculus
Stjepan Marijan, 4.g

103. Gottfried Wilhelm Leibniz
• Leipzig 1st July 1646
• Philosopher, mathematician, physicist and diplomat
• The forerunner of George Boole and symbolic logic
• "Differential" and "integral"
• 1559 French Academy of Science
• The first model of the computer machine
Picture 1.1.: Gottfried Wilhelm Leibniz
Picture 1.2.: Leibniz’s mechanical computer

104. Isaac Newton
• Woolsthorpe-by-Colsterworth 4th
January 1643
• Astronomer, mathematician and
physicist
• methods of elimination
• The general law of gravity
• mirror telescope
• The Royal Academy
Picture 2.1.: Isaac Newton
•Picture 2.2.: Mirror telescope

105. Infinitesimal calculus
• Functions, derivation, integral limits and limit
functions
• differential calculus
• integral calculus
Picture 3.1.: Integral
Picture 3.2.: Derivation

106. Newton – Leibniz’s formula
• If 𝐹 is selected primitive functions of function
𝑓 on the interval 𝑎, 𝑏 , following applies:
𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = 𝐹 𝑏 − 𝐹(𝑎)

107. History of infinitesimal calculus
• 5th century BC- Zenon
• 4th century BC- Eudokso
• 225 years BC- Arhimed
• 17th century - Bonaventura Francesco Cavalieri
Picture 4.1.: Method of
ekshaution
Picture 4.2.: Geometria indivisibilibus continuorum nova quadam ratione promota

108. Conflict of Newton and Leibniz
• Isaac Newton- 1671 - De Methodis Serierum et
Fluxionum (published 60 years later), physical
access
• Gottfried Wilhelm Leibniz- 1684 - the first
published results, the geometric approach
Picture 5.2.: De Methodis Serierum et Fluxionum
Picture: 5.3.: Transactions of
the Royal Society of London

109. ABRAHAM DE MOIVRE
Iva Ciprijanović, 4.g

110. DE MOIVRES BEGINIGS:
 was born in Vitry, France, May 26, 1667
 French mathematician famous for the formula
which links complex numbers and trigonometry
 He was a Protestant, and for a while after the
Edict of Nantes (1685) he was in prison, after
which he moved to England, where he lived the
rest of his life.

111.  He was earning money as a private tutor of
mathematics and he taught the students in
their homes, but also in London bars.
 He hoped to one day become a professor of
mathematics, but in every country for some
reason been discriminated against

112. DE MOIVRES ANECTODE:
 His well-known anecdote is that he predicted
the day of his death by determinating that he
sleeps every day 15 minutes longer and
summarizing the corresponding arithmetic
progression, calculated that he would die on
the day that he will sleep for 24 hours and he
was right.

113. THE DOCTRINE OF CHANCE: A METHOD
OF CALCULATING THE PROBABILITIES OF
EVENTS IN PLAY
 Main De Moivre’s work
 In this book we can find the
definition of statistical
independence of events and a
number of tasks related to various
games. Picture1. De Moivre work:
The Doctrine of Chance: A
method of calculating the
probabilities of events in
play

114. DE MOIVRE’S FORMULAS:
Picture 2. Formula for binomial coefficients
Picture 3. The formula which could prove all the integer numbers n
Picture 4. Famous DE MOIVRE’s formula

115. Picture 1. Johann Carl Friedrich Gauss
Marta Ćurić, 3.g

116.  German mathematician (1777-1855)
 except for mathematics, worked in astronomy,
physics, geodesy and topography
 designed "non-Euclidean geometry” at the age of
sixteen
 with twenty-four years he published a masterpiece
Disquisitiones Arithmeticae
 In 1801, according to his calculations discovered
planetoid Ceres
 discovered Kirchhoff's laws
 made primitive telegraph
 created his own newspaper - Magnetischer Verein

117. Picture 2. Disquisitiones Arithmeticae Picture 3. Magnetischer Verein

118.  devised a faster way of solving tasks of adding
numbers from 1 to 100:
 (100 + 1) + (2 + 99) + ... + (50 + 51) = 50 * 101 = 5050
 realized the criteria of constructing proper
heptadecagon
 proved the basic theorem of algebra
 created the Gaussian plane
 created a Gaussian curve that is used in many
sciences, especially in psychology

119. Picture 4. Gaussian plane
Picture 5. Proper heptadecagon
Picture 6. Gaussian curve

120. John Nash
Lana Matičević, 3.g

121. John Nash (1928) is an economist
and mathematician.
He has published several theories
that are used and who have
contributed to the economy.
He won the 1994 Nobel Prize for
economics.
His most famous theory: Nash
Equilibrium (game theory)

122. What is the Nash Equilibrium?
 The concept, which was initially
designed as a tactic for simple games
 It is not the best strategy that can be
used, but it is the best tactic to not use
other players in order to reach the goal

123. Interesting facts
 He was suffering from schizophrenia (up
to 1990)
 Movie Beautiful Mind is based on his
life.

124. From a scientific rationality to the
illusory chaos

125. Equation N.E.