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1. HISTORY
OF
MATH
CROATIA-GREECE
2014-15

3.  They developed as the life
became more complicated
(calculations,product exchange)
 They drived in development of
civilization of the Ancient
Egyptians (pyramids,tombs)
 They were written with three
ways: the hieroglyphic,the
hieratic and the demotic.

4.  The system of the Egyptian arithmetic was
based at the simple iterative beginning
according to differently symbols repeated for
the successive forces of ten to form the
requested number.
 With this seven symbols the Egyptians could
wrote any integer number from 1 until
9.999.999

5.  Addition: was a simple process that all you needed
was the replacing ten similar symbols with one
symbol for the next category.
 Deduction: they faced her as otherwise process of
addition.
 Multiplication: was the base of the whole Egyptian
arithmetic and it made with the method of
doubling, halving and with the addition.
 Division: they weren’t consider different process
from the multiplication.(namely 168:12 «multiplied
with 12 until you find 168»)

6. At first Egyptians ignored the metric relations of rectangles
triangle and moreover they weren’t deal with theorems and
evidences. The content of geometry was the computation of
areas and tumors of different shapes based on rules(some
right and some not).The most remarkable results are the
tumor calculation of a bun pyramid and the calculation of
circle areas based on a rule that correspond to formula Ε= [(1
-1/9)d]2 (d the diameter). The formula drive us to the
approximate value of p= 256/81 = 3,16.., (better than the
value p= 3
That used by Babylonians).

7.  Papyrus Rhind, is a collection of 84 problems that copied
approximately 1650 BC
 Papyrus of Moscow , (1850BC ) is a collection of 25
problems.
 The leather roller,(1650 BC) include 26 sum of unit fractons.
 At the end, papyrus the Kahun and the papyrus
Of Berlin, (1850 BC) they include mathematical
operations and problems.
The sources of Egyptian
Mathematics

9.  He was born in AD 90
and died in AD 168
 He was a Greco-Egyptian
writer of Alexandria, known
as:
• a mathematician
• an astronomer
• a geographer
• an astrologer
• and a poet

10.  He wrote in Greek.
 Three of his most important
scientific treatises are:
• the Almagest which is
the only surviving
comprehensive ancient
treatise on astronomy
• the Geography
• the Tetrabiblos or the
Apotelesmatika.

11.  He was an Egyptian Muslim
mathematician during the Islamic
Golden Age.
 He was the first mathematician
to systematically use and accept
irrational numbers as solutions
and coefficients to equations.
 His mathematical techniques were
later adopted by Fibonacci.
 He made important contributions
to algebra and geometry.

12.  He was born in Egypt between 950
and 952 and died in 1009.
 His full name was Abu al-Hasan
'Ali ibn 'Abd al-Rahman ibn Ahmad
ibn Yunus al-Sadafi al-Misri.
 He was an important Egyptian
Muslim astronomer and
mathematician whose works are
noted for being ahead of their
time.
 He was also known as an eccentric
and a poet.

13.  He was born in Alexandria,
Egypt, on November 11, 1911
 In 1944 he began publishing
books and articles in scientific
and other journals. By 1988, he
had written about 120 books
and 500 articles.
 In 1947 he began running
seminars for international
groups which he continued
until his death.
 He died in Paris in 1988.

14.  Founded the International Commission for
the Study and Improvement of Mathematics
Education.
 Founded The Association of Teachers of
Mathematics.
 Founded The Cuisenaire Company in
England.

15.  He was born in Cairo in 1936.
 He is a:
 mathematician,
 philosopher and
 historian of science.
His work:
• focuses on mathematics and physics
of medieval Arab world.
•explores and illuminates the
unrecognized Arab scientific tradition.

16.  Ahmes
 Ismail Mustafa al-Falaki
 Ahmad ibn Yusuf
 Mohammed Reda Madwar
 Menelaus of Alexandria
 Serenus

18.  Al Khwarizmi (780-850) was Persian:
 Mathematician
 Geographer
 Astronomer
He was born in Khwarezm ( Uzbekistan )
Al Khwarizmi was translating Greek and Sanskrit scientific
manuscripts .

19.  He created the foundation in algebra and trigonometry, for
what is considered the "father" of Algebra ( with Diophantus ).
The reason of spread of Indian number system was his book
written around 825, Calculating with Indian numbers.
 His book "The image of the earth" had to corrected values,
the coordinates of Asian Mediterranean and Africa.
He wrote for devices like the sundial and the astrolabe.
 In the 12th century introduced the Latin West the Arabic
numerals, which was based on the decimal system developed
by Indian sources.

20.  "The Compendious Book on Calculation by Completion and
Balancing".(830 CE)
al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wal-muqābala AL-TZAMPR
‫والمق‬ ‫الجبر‬ ‫حساب‬ ‫في‬ ‫المختصر‬ ‫الكتاب‬
The word ALGEBRA from the book’s title
 "The Book of Addition and Subtraction According to the Hindu
Calculation" which refers to Arithmetic.
 He design tables for the trigonometric functions of sines and cosine
and tangent in Zīj al-Sindhind which includes116 tables with
calendrical, astronomical and astrological data and 37 chapters on
calendrical and astronomical calculations.
 "Book of the Description of the Earth“. Refers to the coordinates of
2402 cities and other geographical features. (880 CE)

23. •Fibonacci (September1170-1240)was Italian
Mathematician who was born in Piza and lived in
Bejaia(Algeria).He gained popularity with the Fibonacci
sequence .He was taught accounting and he traveled
together with his father.
Fibonacci Numbers constitute the following sequence
0 1 1 2 3 5 8 13 21 34 55 89 144….or
−21 13 −8 5 −3 2 −1 1 0 1 1 2 3 5 8 13
21
By definition, the first two numbers in the Fibonacci
sequence are either 1 and 1, or 0 and 1, depending on the
chosen starting point of the sequence, and each
subsequent number is the sum of the previous two.

24.  Golden Ratio
 The number F is related to
golden mean. The golden ratio
also is called the golden
mean or golden section .Other
names include ,extreme and
mean ratio ,medial
section, divine proportion, divine
section ,golden
proportion, golden cut,
and golden number.
 The Fibonacci
sequence is named
after Fibonacci. His
book Liber
Abaci introduced
the sequence to
Western European
mathematics .The
book was widely
known after the
introduction of
typography.

25.  The most characteristic experiment that Fibonacci made was the
Rabbit Problem. The question was (if we know that once a pair is two
months old, it bears another pair and from then bears one pair every
month. ) ”Starting with a newborn pair at the beginning of a year,
how many pairs of rabbits will there be at the end of the year?”
In nature

26.  Spirals are patterns that are related to the Fibonacci sequence.
By using quarter-circle arcs inscribed in squares of integer
Fibonacci-number side, shown for square sizes 1, 1, 2, 3, 5,
8, 13, 21, and 34.
 The most famous and beautiful examples of the occurrence of
the Fibonacci sequence in nature are found in a variety of
trees and flowers, generally associated with some kind of
spiral structure. For instance, leaves on the stem of a flower or
a branch of a tree often grow in a helical pattern, spiraling
around the branch as new leaves form further out.
 A completely understandable example is the one about the
daisy. A daisy has a central core consisting of tiny florets
arranged in opposing spirals. There are usually 21 going to
the left and 34 to the right.


28. Birth: on 1540, in Vante which was the kingdom
of France.
Death: in Paris on 1603.
Occupation: Mathematician
Point of interest: law, mathematics
and especially algebra.
Known for: the establishment
of the algebraic symbolism.

29.  Initially he deciphered the
Spanish code of
correspondence.
 Ηe expressed the number π
with the assistance of the
infinity product and he
calculated with accuracy of
nine decimal figures,
improving the Archimedes’
result.
2 2 2 2 2 2 2
2 2 2
  
 

30.  He used the vowels to express the unknown
variable and consonants for the known ones.
 Lastly the formulas known as “Vietas’
formulas” which give many terms between
roots and the quotients of a polynomial.
Especially for the trinomial:
2 0, 0ax bx c a   
b cS Pa a 

31. René Descartes

32. • French philosopher and mathematician.
• He was born in La Haye on 1596 from a wealthy family.
• He was educated at a boarding school and after finishing his
studies in Law, it came his military service.
• In 1628 he moved to the Netherlands, where he devoted
himself to his scientific studies and especially in mathematics.
• He died on 11 February 1650 in Stockholm.

33.  He developed the analytic geometry.
 He was the first to introduce the concept of variable size and
variable function (The dual form of the variable determined the
unity of geometry and algebra).
 He invented the convention of representing unknowns in
equations by x, y, and z, and knowns by a, b, and c and the way of
writing the powers.
 He made the "Cartesian rule" for the determination of the number
of positive and negative roots.
 Showed that the cubic equation is solved by squaring and with
the assistance of diabetes and rule.
 He was involved with the study of the theory of sets and set the
principles of physics and biological determinism.
His works:

34.  “ Discourse on the Method ”, which was published
in 1637,
 “Meditations on First Philosophy”, published in
1641 in Latin,
 "Principles of Philosophy", published in 1644 also
in Latin,
 "Passions of the Soul" (1649)
and his endless work
 "Rules for the Direction of the Mind“ (1626–1628).
All of his works published in Paris from 1897 to 1910
in 12 volumes.
Other works:

35. The Geometry of Descartes with the method of coordinate
system that he created and with the construction of the
vertical tangents and the flat curves, he helped
remarkably the work of Newton, Leibniz, Euler and all of
the other important people after him and he also had an
enormous influence on the development of mathematics.

36. Quotes - Testimonials - Sayings:
«Reading good books is like conversation with the
most perfect people in the past.»
«I think, therefore I live.»
«In the effort of finding the truth it is necessary we
doubt for everything, as much as we can.»
Foundation of his philosophical thoughts is doubt because senses often mislead
people. Since he disputes, he is forced to think, so he justifies his existence as a
human being.

38.  Born: 17 August 1601
 Died: 12 January 1665
 Profession: Lawyer
 Origin: Basque
 Occupied with: Mathematics
as amateur from 1629
 Knowledgeable: French, Latin, Ancient
Greek, Spanish, Italian and perhaps
Basque dialect

39. Project:
He composed various texts about the maximum and
the minimum of functions, which later gave to
Etienne d’Espagnet.
His project on the Analytic Geometry was available
for the public in manuscript in 1636. In one text of
his work, he developed a determination method of
the minimum, maximum and tangent in curves of
different functions, equivalent with the one of the
differential calculus.
He also worked with Pascal. In 1664 they established
the basic foundations of the probability theory.
That is the reason why they are considered to be
co-authors of the probability theory.

40. He invented:
 A technique for the positioning of the center of
gravity of many levels and solids, which has led to
several analyzes on the integrals.
 A type of calculation of the integral attributed to a
sum of geometric progress terms. Terms that were
useful later for many scientists, like Newton.
 The factorization method and the technique of
infinite descent, a special case of the proof of
contradiction, which he used in order to prove the
Last Theorem for the case n=4.

41. He studied:
 A special case of Diophantine equation which was
called ‘the equation of Pell’.
 The perfect numbers, the friendly numbers and the
numbers later known as the Fermat numbers.
where the n is a non negative integer.
2 2
1x n y 
2
2 1
n
nF  

43. Born: 19th of June1623
in Clermont-Ferrand of France
Died: 19th of August 1662,
Considered: one of the most charismatic
mathematics as his contribution was great,
especially in the sections of Probability and Fluid.
Occupied with: mathematics, philosophy,
religion, physics.
Despite the removal of all geometry books from
home by his father , Pascal began at age 12 to
read geometry alone

44. “Pascalina” (invention of his) contained
pinions, which was marked by numbers 1 to
10 and the sum or removing mapped to
rotation angles. When a gear made ​​a full
turn, swept the immediate left of the gear
located and thus conveys the “prisoner”.

45. Formulated those theories:
1. The principle of communicating vessels ,
particularly when in communicating vessels
balances a liquid , all points of the liquid having the
same pressure and the free surface of all containers
located on the same horizontal plane.
2. The Pascal's law is one of the basic laws of
Hydrostatic and determines that any pressure that
may be exerted on the surface of a liquid spread
evenly throughout, in all directions and throughout
the depth of it.
3. The third theory of Pascal , according to which the
points of intersection of the opposite sides of the
hexagon inscribed in a circle ( and generally
recorded on a conic section ) is collinear ( on a line
called a straight Pascal 's hexagon ( the red line in
the figure).

46. The arithmetic triangle, known as
‘Pascal’s Triangle’ is a figure where
every number from third line and
under, except from the units, is the
sum of the numbers from the nearest
line before
Features:
 The first line constitutes from
one number, the second line of
two numbers, the third one of
three numbers, etc.
The n -th line has n numbers.
 The numbers of the n -th line
are coefficients of the blank (a +
b ) n
 The sum of every line’s numbers
equals 2.
 In case the figure is coloured,
the multiples of two form
equilateral triangles.

47. Pascal’s Quotes
 Do not try to add years to your life. You
should better add life to your years.
 The story of mankind would be different if
Cleopatra’s nose had different shape.
 There are two kinds of people:
the fair ones, who consider themselves as
sinful, and the sinful ones, who consider
themselves as fair.
 The greatness of a man is located behind his
ability to think.

49.  A logician
 A philosopher
 A mathematician

50.  He was born on the 4th of August of 1834.
 He died on the 4th of April of 1934 of
unknown causes.
 He was educated by private lesson until 1853
when he went to Gonville and Caius College
of Cambridge.
 In 1857 he graduated and became a member
of the college staff.
 In 1862 he returned to Cambridge as a
lecturer of Ethics while he studied and taught
Logic and the Theory of Probability.

51.  Venn diagrams were invented around 1880.
 They are used in many fields, such as:
Set theory
Probability
Logic
Statistics
Information technology

52. Venn Diagrams are an illustration of sets.
In every Venn Diagram there are:
 A rectangle which symbolizes the biggest
set there could be, depending on what we
want to show, which is usually
symbolized by U.
 Closed lines, usually curves and circles.
The surface they cover symbolizes the set
itself.

53. In a Venn diagram every surface which
is defined by any combination of lines
symbolizes a set.

56.  He was born in 630 in Militos
 He died at the age of 78 around 543 b.C at the
Olympic games due to the heat and thirst.
 He is known as the first philosopher of the seven in
the ancient world
 Except of philosophy, he was also interested in
mathematics, physics, astronomy, engineering,
meteorology
 He set the Milicius school up
 He loved travelling and he made a lot of trips all
over the world

57. He thought that
 The shape of earth was circular disk and
 Water is the starting point of the whole life.
He invented
 electricity and
 the magnetic fields.

58. Some of Thales’ theories are:
 When parallel lines are being
crossed by two other lines then
the parts between the parallel
lines are analogous.
 Ιf A, B and C are points on
a circle where the line AC is
a diameter of the circle, then
the angle∠ABC is a right angle.
Thales' theorem is a special
case of the inscribed angle
theorem, and is mentioned and
proved as part of the 31st
proposition, third book
of Euclid's Elements

59. Τhales’ quotes:
 Σοφώτατον χρόνος· ἀνευρίσκει γὰρ
πάντα.
◦ Time is the wisest of all things that
are; for it brings everything to light.
 Οὔ τι τὰ πολλὰ ἔπη φρονίμην
ἀπεφήνατο δόξαν
◦ A multitude of words is no proof of a
prudent mind.
 Ἐὰν ἃ τοῖς ἄλλοις ἐπιτιμῶμεν, αὐτοὶ μὴ
δρῶμεν
◦ Avoid doing what you would blame
others for doing.
 Μέγιστον τόπος· ἅπαντα γὰρ χωρεῖ
◦ Place is the greatest thing, as it
contains all things.
 Γνῶθι σαυτόν
◦ Know themselves.

61. Pythagoras was a very important Greek :
 Philosopher
 Mathematician
 Geometer
 Theoretical in music
He was born between 592 B.C and 572 B.C . . He died in old age
in Metapontis ,Italy. . He was an intelligent personality and with
his fluency he gained admiration and respect of all his fellow
citizens of that time .He travelled to Egypt and Samos and he
tried to teach other people . He was so intelligent that his fame
reached as far as Miletus and Priene to two of the seven wise
men of antiquity(Thales and Vianta) and in many places people
admired the young Pythagoras.

62.  Pythagorean theorem
 Pythagorean triads
 Asymmetric sizes
 Study regular pentagonal
 Musical scale construction
 Pythagorean school

63. "The square of the hypotenuse (the side
opposite the right angle) of a right-angled
triangle is equal to the sum of the squares of
the two vertical sides."

64. Pythagoras studied and created at least three
(dodecahedron, tetrahedron, cube) of the five
regular polyhedral.

65. Pythagorean school was in Kroton, Italy, whose
founder was Pythagoras.
The Pythagorean school had religious, political
and scientific nature. On the school’s entrance
the following quote was carved
by the Pythagoreans:
«MΗΔΕΙΣ ΑΓΕΩΜΕΤΡΗΤΟΣ ΕΙΣΗΤΟ»
That is no one, who cannot count
every object using human measures,
is allowed to enter or participate
in this fraternity.

66.  “Words are the winds of soul”.
 “Education is a golden wreath ,not only
because of its great value but also of its
benefit that it offers”.
 “ As it seems justice resembles a square all
parts are equal and identical”.

68. • Ancient Greek philosopher (427 b.C.-347b.C)
who died in the age of 80
• Born in Athens
• Socrates is his most known student
• Aristotle’s teacher

69. Plato established Academia in Athens, around 387 b.C., to
organize his educational plans. Academia was named after the
location where it was established; the gardens of Academus.
Ioustinianos, the emperor of Byzantium shut it down in 529 b.C.

70. Tetrahedron Cube Octahedron Dodecahedron Icosahedron
In mathematics he is widely known for the Platonic
Solids which were named like this because they were
studied by Plato's Academia
Platonic solid is a regular regular, convex polyhedron
with congruent faces of regular polygons and the
same number of faces meeting at each vertex

71. According to Plato, geometry relates the world of
ideas with the natural world. The natural world
doesn’t consist perfect cycles, straight lines or points
and the geometric objects don’t exist as eternal and
unstoppable ones. The geometrical knowledge isn’t
conquered with clear thought or with memory of the
soul.

74.  Ancient Greek mathematician
 Geographer
 Astronomer
 Poet
 Historian
 Music theorist
 Born in Cyrene in 276 b.C. (current Libya)
 He became the chief librarian at the Library of
Alexandria in 236 b.C. And he stayed there in
charge for 40 years teaching in its museum.
 In194 b.C. he went blind and a year later he
stopped eating and then he died in Alexandria

75. His most important achievements were the
sieve of Eratosthenes and the measurement of
the Earth's circumference. In mathematics the
sieve of Eratosthenes’ is a simple algorithm for
finding prime numbers.

76. In the 3rd century b.C.
Eratosthenes was informed
that in Swenet during the noon
on the summer solstice, the
sun appears directly overhead
and its reflection heads
towards the bottom of a well.
Simultaneously in Alexandria
the sun rays form an angle 7ο
with the zenith of the location.
Then he measured the distance
between Alexandria and
Swenet and calculated, as it is
shown in the picture below,
with great accuracy the Earth’s
circumference.

78. He lived from : 325 B.C. – 265 B.C
 He born– live-died : In Alexandria, Egupt.
 He was: Mathematician
 He called : “Father’’ of Geometry
 He contributed to: History of maths logic.
 He innovated at: The production of a formal cohesive
total with propositions

79.  Euclid’s contributions at the sector of Geometry and
consequently in Hellenic science is enormous. He wrote
one of the best tasks that named ''elements'‘

80.  ‘’ELEMENTS’’
 Had gather all the geometric knowledge until that time
and he (Euclid) had add his own knowledge in a
marvelous way
 separated at 13 books and conclude 382 theories
 mention Pythagoras an Theoklitos discoveries.
 1-6mention plan meter 7th to 10th arithmetic and the
last three stereometry

81. Based on one of the most important theorems.
Ex. Let’s suppose that there is a straight line ε and an
exterior point Α, there is only one parallel straight
line which can pass from the ε.
His simple perception of the space supports Euclidean
geometry.

82. However, he had written another important tasks:
1.“dedomena”
2. “Peri diairesewn”
3. “Psedaria”
4.“Kwnika”
5. “Optica”
6. “Phenomenika”
7.“Katoptrika”
8. “Katanomi Kanonwn”
9. “Topoi of Epifanion”

83. Nothing is totally sure for his life . Also we know that:
 he related with Alexandria’s library.
 he probably studies at Platoons academy.
What is more he acquired his reputation in Pallada for his
achievements at Math’s due to the fact that Ptolemy
invited him in Alexandria and called him "savior" .
Additionally, his work appreciated by Pharaoh.
For his character we only know he was gentle and
contagious.

85. He lived from 287 B.C – 212 B.C and died at the age of 75
He was at the same time a physician, mathematician and mechanic
He is considered as the greatest Mathematician of his age

86. A weapon that threw rocks and arrows against enemies)
Street meter ( means that could calculate the distance that
traversed a moving object)
Wreckers (a machine that could capture the opponents’ boats and
cause them damage, while they were sieging his town)
Planetarium (a device capable of calculating the exact position of
the sun, the moon and other planets)
A device that could define the concentration and salinity of fluids)
A screw (that was used for pulling water and utilize it in irrigation)

87.  On the measurement of circle
 On the Sphere and the Cylinder
 On spirals
 On floating bodies
 (o) Stomachion
These and many others are only a few of the Writings of Archimedes that were saved,
while the ones that are gone are not a few.
Furthermore he did a lot of research based on the area of circle, ellipse,parabola and
spiral . He also studied the area and volumes of cylinders, cones and especially the
volume of the sphere. Thus he did he calculated satisfactorily the decimals of ‘π’
and that’s why it is called “constant of Archimedes”.

88.  In Euclid Geometry the number π is
a mathematical constant, the ratio of
a circle's circumference to its diameter.

89.  Other scientists argue that ‘π’ is a fully-made
rotation of a circle, diameter 1, on a
horizontal floor.

90.  One day, the king Ieronas called Archimedes to give him
a task. Earlier, Ieronas had ordered a crown to be made
for him. So, he wanted Archimedes to find out whether
the crown was purely made out of gold or not.
 Later,while Archimedes was at his bathroom he realised
that as he was sinking in it, the surface of the water was
emerging! He had finally found the solution of that task!
He immediately jumped out of the bathroom and
started running on the streets claiming “Eureka”. He
thought that if he sank the crown the surface of the
water would emerge too! So he tried that experiment
comparing the crown with pure gold, and discovered
that the crown was fake!

92. His life
 Diophantus is a Greek
mathematician who lived in
Alexandria, Egypt in the 3rd century
AD and he died in old age
 He worked on the solution of
problems that had the form of
equation and this helped the
evolution of Αlgebra .Some people
call him "father" of Algebra.
 The Epigram is a known
mathematical riddle. From the
solution we learn that Diophantus
passed away 84 years.

93.  The Numerically is the best known and oldest Greek textbook.
It includes 130 problems
 Also studied and developed the undefined or Diophantine
equations, namely the equations with multiple solutions. To
these problems required only positive solutions
 The most famous Diophantus problem is the problem 8 of the II
book of numerical: “Analyze a given square number into two
square numbers”

94. Diophantus 1/6 his life he was a child.
1/12 was young and then spent 1/7 his life until he married. Five
years later, his son was born.
The life of his son, was the half life of Diophantus. After the death
of his son, he lived four years in deep sorrow and then he died.
How many years has lived Diophantus?
(The answer is 84 years)

96. She was a mathematician, philosopher ,engineer
and astonomer.
She was born in 370 B.C. and died in 416 B.C.
Her father’s name was Theon
She took part at the coursework in Neoplatoniki
school from Plutarch the younger

97.  She discovered that the movement of the
earth is elliptical.
 Astronomical canon.
 The Apollonian cones.
 She wrote 13 books with comments about the
arithmetic of Diophantus .

98. She had a big interest for engineering and practical
technologies
So she made several organs such as :
 One astrolabe used for measuring the positions of
stars, planets and the sun.
 Developed a device for refining water
 And the hydrometer to measure a liquid for
specific gravity

99.  Hypatia was a woman who separated the
community in two parts.
• Those who considered it a miracle of light.
• Those who was seeing it like a apostle of darkness.
 Her action considered dangerous for the spread of
Christianity, gradually has cultured a climate which
was versus her and has led to her violent murder
from a mob or from fanatic monks teams.

100. C R O A T I A - G R E E C E