A final year project discussing the history and significance of the Pythagorean theorem in the ancient world.
The presentation provides information on the Babylonians, the Egyptians, the Indians and Chinese before moving on to Ancient Greece and Pythagoras himself.
Judaism
- Beliefs in God
- Kerygma (Creed, Proclamation of Faith)
- Diakonia (Concepts and practices)
- Lietorgia (Prayers, devotions, rituals)
- Koinonia (Festivals and celebrations)
- Community (Structure, leadership, and ministry)
For our THEO 1 class | AMDG
All rights reserved (c)
Lesson plan on introduction of trigonometry, students must aware about the history , concepts to be done, what common error they commit and what are the scope of this topic in careers
Judaism
- Beliefs in God
- Kerygma (Creed, Proclamation of Faith)
- Diakonia (Concepts and practices)
- Lietorgia (Prayers, devotions, rituals)
- Koinonia (Festivals and celebrations)
- Community (Structure, leadership, and ministry)
For our THEO 1 class | AMDG
All rights reserved (c)
Lesson plan on introduction of trigonometry, students must aware about the history , concepts to be done, what common error they commit and what are the scope of this topic in careers
This is about the history of the Maurya and Gupta Empire that is commonly not tackled in World History class.
TO DOWNLOAD, PLEASE CLICK THE LINK: https://dlsharefile.com/file/1054610895
THANK YOU!
Pythagoras, since the fourth century AD has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.
This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors
This learner's module talks about the topic Reasoning. It also includes the definition of Reasoning, Types of Reasoning (Inductive and Deductive Reasoning) and Examples of Reasoning for each type of reasoning.
Lecture Presentation on Trigonometry, types of angle, angle measurement, pythagorean theorem, trigonometric function, trigonometric relationship, circle function, co function, reference angle, odd even function,graphing of trigonometric function, special angles and terminology and history of trigonometry
History of mathematics - Pedagogy of MathematicsJEMIMASULTANA32
It includes Prehistory: from primitive counting to Numeral systems, Archaic mathematics in Mesopotamia and egypt, Birth of mathematics as a deductive science in Greece: Thales and Pythagoras and Role of Aryabhatta in Indian Mathematics.
This is about the history of the Maurya and Gupta Empire that is commonly not tackled in World History class.
TO DOWNLOAD, PLEASE CLICK THE LINK: https://dlsharefile.com/file/1054610895
THANK YOU!
Pythagoras, since the fourth century AD has commonly been given credit for discovering the Pythagorean theorem, a theorem in geometry that states that in a right-angled triangle the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides.
This is a school standard presentation for class 10 students .
It will be very helpful to you all.
Hope you all like this .
And pass your exams with flying colors
This learner's module talks about the topic Reasoning. It also includes the definition of Reasoning, Types of Reasoning (Inductive and Deductive Reasoning) and Examples of Reasoning for each type of reasoning.
Lecture Presentation on Trigonometry, types of angle, angle measurement, pythagorean theorem, trigonometric function, trigonometric relationship, circle function, co function, reference angle, odd even function,graphing of trigonometric function, special angles and terminology and history of trigonometry
History of mathematics - Pedagogy of MathematicsJEMIMASULTANA32
It includes Prehistory: from primitive counting to Numeral systems, Archaic mathematics in Mesopotamia and egypt, Birth of mathematics as a deductive science in Greece: Thales and Pythagoras and Role of Aryabhatta in Indian Mathematics.
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A brief description on the history of math, many famous mathematicians and also women mathematicians..
And very huge description ( bio-data, formulas etc.) on famous mathematician S.Ramanujan.
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Instructions for Submissions thorugh G- Classroom.pptxJheel Barad
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The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
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The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Macroeconomics- Movie Location
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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2. Pythagorean Theorem
A simple proof:
Combine 4 triangles to form a square of length (a+b) such that the centre
square has side c. Therefore we obtain:
(a+b)2=2ab + c2
Simplifying we reach the identity:
a2+b2=c2
There are over 400 different proofs
of the Theorem.
4. Babylon
★ There are several clay tablets that show
knowledge of Pythagoras’s theorem.
★ The ones we will look at are: Plimpton 322,
YBC 7289, the Susa tablet and the Tell
Dhibayi tablet.
5. Plimpton 322
★ This tablet dates to around 1800-1650
BCE.
★ The tablet itself is damaged, a large
portion missing.
★ But it is believed to be a list of
Pythagorean triples with a method for
their calculation.
★ Though this is mostly guess work and
some mathematicians believe this
tablet has nothing to do with
Pythagoras at all.
6. YBC 7289
★ This tablet dates to
around 2000-1600 BCE.
★ The contents of the
problem are relatively
simple, but show a clear
use of Pythagoras’s
theorem and a very
accurate estimate of √2.
7. The Susa Tablet
★ Discovered in Susa in 1936 CE, this tablet
dates to the Old Babylonian Period (a similar
time to the previous tablets).
★ The tablet contains a problem about an
isosceles triangle and a circle through the
three vertices.
★ Given the side lengths of the triangle (in this
case 50, 50 and 60), the tablet describes a
method for calculating the radius of the circle
and makes use of Pythagoras’s theorem.
8. The Tell Dhibayi Tablet
★ In 1962 around 500 clay tablets were found, believed
to have originated from around 1750 BCE.
★ One of these tablets contains a problem of finding the
lengths of sides of a rectangle given the area and the
length of the diagonal.
★ For modern day mathematicians, this is a simple
algebraic exercise, but the Babylonians approached
the problem in quite an interesting way, using
Pythagoras’s theorem along the way.
9. Egypt
The ancient Egyptians are well known for their
architecture, but how did they manage to
build so perfectly?
10. Egyptian Rope Constructions
★ It is believed that the ancient
Egyptians used rope to help them
create perfect right angles for their
buildings.
★ The rope was knotted at 12 equally
spaced places and then arranged as
shown on the right to create a
perfect 3-4-5 right-angled triangle.
11. Were 3, 4 and 5 special?
The relationship between the numbers 3, 4 and 5 were
very important to the Egyptians, so much so that it is
believed the ancient Pyramid of Giza is actually a
representation of these numbers.
“The Great Pyramid, ONE Structure, representing the
number THREE by its triangular faces, the number
FOUR by its square base, and the number FIVE by its
apex and four corners.”
12. Were 3, 4 and 5 special?
Another example of the important of these numbers in
ancient Egyptian culture is the Egyptian 3-4-5 triangle, in
which each side of the triangle is given a name:
“The upright, therefore, may be likened to the
male, the base to the female, and the
hypotenuse to the child of both, and so Ausar
(Osiris) may be regarded as the origin, Auset
(Isis) as the recipient, and Heru (Horus) as
perfected result.”
14. China
Sources
The Arithmetical Classic of the Gnomon and
Circular Paths of Heaven (or Zhoubi
Suanjing), circa 1st millennium BC
Nine Chapters on the Mathematical Art (or
Jiuzhang Suanshu), circa 2nd century BC,
and its later commentaries
Sea Island Mathematical Manual (or Haidao
Suanjing), circa 3rd century AD
15. Earliest source: Zhoubi Suanjing
“Therefore fold a trysquare so that the base is three in
breadth, the altitude is four in extension, and the
diameter is five aslant. Having squared its outside, halve
it [to obtain] one trysquare. Placing them round together
in a ring, one can form three, four and five. The two
trysquares have a combined length of twenty-five. This is
called the accumulation of trysquares.”
16. Hsuan-thu
Also contained in the Zhoubi Suanjing is the
famous Hsuan-thu diagram, which displays the
statement in pictorial form.
17. Zhao Shujing’s abstracted proof
A modern translation:
Let the gou be a, the gu be b, and the xian be c.
The area of the triangle is 0.5(ab), therefore the area
of four triangles is 2ab
Then by the second diagram:
c²=2ab+(a-b)²
=2ab+a²-2ab+b²
=a²+b²
Alternatively, by the third diagram:
c²+2ab=(b+a)²
c²+2ab=a²+2ab+b²
c²=a²+b²
a
b c
18. Nine Chapters of the Mathematical Art
Had an influence on Chinese mathematics
similar to Euclid’s “Elements” did in Greece
The final chapter is entitled “gou-gu” and
contains numerous problems requiring use of
the gou-gu theorem to find unknowns using
algebraic methods.
19. Liu Hui
Influential Chinese scholar who
produced a number of works, mainly
extrapolating on knowledge found in
older texts such as the Nine Chapters.
Produced a unique proof of the Gou-gu
based on axiomatic assumptions,
though parts of his proof are now lost.
20. Liu Hui’s applications to surveying
In his “Sea Island
Mathematical Manual”,
written 263AD, Hui provides
applications of the Gou-gu
to problems in land
surveying.
21. Uniqueness of Chinese achievement
China provided both explicit statements and
unique proofs of the theorem.
As Chinese scientific development was
relatively self-contained compared to other
civilisations, it is likely that the Chinese Gou-
gu originated independently of other cultures.
22. India
Sources
The Sulba sutras: Baudhayana sutra (ca. 800BC),
Apastamba sutra (ca. 600AD) and Katyayana sutra
(ca.170BC)
The Brahmanas (ca. 900BC)
23. Sulba Sutras
Roughly translates as “rules of the chords”, a
series of 9 texts dated throughout the first
millennia BC.
Their contents are an amalgam of ritual altar
construction, spiritual knowledge and
mathematics (including approx. of root 2,
circling the square).
The more prominent authors are Baudhayana,
Apastamba, Katyayana and Manava.
24. Baudhayana (ca. 800BC)
Wrote the earliest of the sutras, and in the
first chapter gives the following statement:
1.9: The diagonal of a square produces double the area
[of the square] …
1.12: The areas [of the squares] produced separately by
the lengths of the breadth of a rectangle together equal
the area [of the square] produced by the diagonal
1.13: This is observed in rectangles having sides 3 and 4,
12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.
25. Apastamba and Katyayana
Similar verses are also found in the
later sutras of Apastamba (600-
540BC), in chapter 1 verse 2, and
Katyayana (200-140BC), in chapter 2
verse 7.
26. No Proof?
In contrast to the Chinese mathematicians,
Indian scholars did not actually prove the
theorem until Bhaksara in the 16th century
AD.
However, statements contained in the sutras
were viewed as established rules, i.e.
theorems.
27. Mahavedi
Translating as “great altar” or “entire altar”, the
Mahavedi was used in Vedic ritual practices for centuries.
Procedures for construction of these altars are found in a
number of the sulba sutras.
28. Mahavedi
Analysis on the geometry of the altar reveals
that 18 unique pythagorean triples (plus their
mirrors) are found inherent in its design.
29. The Brahmanas
Dated circa 900BC, the Brahmanas are a commentary on
the Vedas, the underlying texts of the Vedic religion.
The Brahmanas contain methods to construct the
Mahavedi and other altars, although the construction
provided differs from the one found in the later sutras.
Does this mean that the creators of the Mahavedi knew of
our theorem even longer ago? We don’t know.
30. Significance
Despite not producing a proof, the Ancient
Indian scholars most certainly understood the
Pythagorean theorem and were able to
construct numerous pythagorean triples.
They applied this understanding to indigenous
religious practice and more advanced
mathematical constructs.
31. Mutual History: Approximations of pi
The accolade of “best approximation of pi”
passed hands from India to China for
centuries.
Their method of approximation revolved
around repeated uses of the Pythagorean
theorem applied to polygons inscribed within
circles.
33. Pythagoras The Explorer
★ Pythagoras travelled
extensively through Egypt,
Mesopotamia and India.
★ Acquired religious
philosophies as well as
mathematical knowledge,
including geometry and
astronomy.
India
34. Pythagoras The Cultist
★ Pythagoras created a secret society
of mathematicians known as the
Pythagoreans.
★ As the leader, Pythagoras
influenced the followers of his own
beliefs.
★ Any advances made by the
Pythagoreans were attributed to
Pythagoras.
ALL IS NUMBER
35. Pythagoras The Mathematician
★ Pythagoras believed that
numbers were more than just
values and attributed
characteristics to them.
★ He believed that integers could
be used to explain operations in
nature.
★ But he struggled with the notion
of irrational numbers.
36. The First Proof
★ The first known formal proof
appears in Euclid’s Elements.
★ The birth of this theorem resulted
in a new wave of thinking in many
areas of mathematics.
37. Geometry & Arithmetic Link
Arithmetic:
Deals with
the discrete
Geometric:
Deals with the
continuous
Pythagorean
Theorem:
a2+b2=c2
38. Concept of Infinity
★ The Pythagoreans attempted to determine a rational
solution to √2.
★ They produced recurrence relations to generate
solutions to:
x2-2y2=1
★ This implies the notion of limits:
Xn √2 as n ∞
yn
39. Plato & Aristotle
★ Plato founded an academy and
became known as the “maker of
mathematicians”.
★ Made mathematics an essential
part of the curriculum.
★ Aristotle, who joined the
academy, proved the irrationality
of √2.
ALL IS NUMBER
40. Pappus of Alexandria
★ Produced 8 books
known collectively as
the Mathematical
Collection.
★ Discovered a
generalisation of
Pythagoras’ Theorem
known as Pappus’ Area
Theorem.
B
C
A
P
41. Pappus of Alexandria (2)
★ Pappus followed up on
Pythagoras’ belief of mathematics
in nature.
★ He noticed how bees construct
their hives with hexagonal cells,
which have a greater surface area
than that of a square or triangle.
42. “The Pythagorean theorem is mathematically
universal, likely to arise in any sufficiently
advanced civilization. Other such cultural
universals are the concept of π—the ratio of
diameter to circumference in the circle—and the
Euclidean algorithm.”
John Stillwell, Mathematics and Its History, Third Edition (Springer, New York, 2010), p. 70
Editor's Notes
Parallels can be struck up between the rope-geometricians of Egypt and their spiritual assignment of mathematical beauty
As you can see, verse 1.12 gives explicit statement of the theorem, and 1.13 gives examples that result in integer solutions for the hypotenuse (aka Pythagorean triples)
Katyayana was also an influential grammarian, providing groundbreaking work on semantics