Math 114-powerpoint
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1) A discovery is recognizing something that already exists for the first time, such as Christopher Columbus discovering the Americas or Fibonacci sequences in nature. 2) An invention is creating something totally new through one's own ideas and development, such as Thomas Edison inventing the light bulb or the development of mathematical concepts. 3) Generalization is reaching a general conclusion through repeated reasoning or making something more widespread, such as the rule that all triangles have three angles or that all numbers except 0 raised to 0 is equal to 1.
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What impact did Pythagoras have on EuclidSolutionPythagorasP.pdf
What impact did Pythagoras have on Euclid?
Solution
Pythagoras
Probably the most famous name during the development of Greek geometry is Pythagoras, even
if only for the famous law concerning right angled triangles. This mathematician lived in a secret
society which took on a semi-religious mission. From this, the Pythagoreans developed a number
of ideas and began to develop trigonometry. The Pythagoreans added a few new axioms to the
store of geometrical knowledge.
1)The sum of the internal angles of a triangle equals two right angles 180*.
2)The sum of the external angles of a triangle equals four right angles 360*
3)The sum of the interior angles of any polygon equals 2n-4 right angles, where n is the number
of sides.
4)The sum of the exterior angles of a polygon equals four right angles, however many sides.
5)The three polygons, the triangle, hexagon, and square completely fill the space around a point
on a plane - six triangles, four squares and three hexagons. In other words, you can tile an area
with these three shapes, without leaving gaps or having overlaps.
6)For a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of
the other two sides.
Most of these rules are instantly familiar to most students, as basic principles of geometry and
trigonometry. One of his pupils, Hippocrates, took the development of geometry further. He was
the first to start using geometrical techniques in other areas of maths, such as solving quadratic
equations, and he even began to study the process of integration. He solved the problem of
Squaring a Lune and showed that the ratio of the areas of two circles equalled the ratio between
the squares of the radii of the circles.
Euclid
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gathered the work of all of the earlier mathematicians and created his landmark work, \'The
Elements,\' surely one of the most published books of all time. In this work, Euclid set out the
approach for geometry and pure mathematics generally, proposing that all mathematical
statements should be proved through reasoning and that no empirical measurements were
needed. This idea of proof still dominates pure mathematics in the modern world.
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theorem shaped the ideas of western philosophers right up until the present day. Great
philosopher mathematicians such as Descartes and Newton presented their philosophical works
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concepts. Abraham Lincoln was a fan, and the US Declaration of Independence used Euclid\'s
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Its all about history of math
The document provides a summary of the history of Greek mathematics from Thales in the 6th century BC to the collapse of the University of Alexandria in the 5th century AD. It discusses how Thales founded the Ionian school of mathematics and influenced Pythagoras. It then outlines the development of Greek mathematics through figures like Euclid and Archimedes, and the establishment and destruction of the University of Alexandria, which marked the end of the peak of Greek mathematical achievements.
Math and chamistry
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Earlier a place value notation number system had evolved over a leng.pdf
Earlier a place value notation number system had evolved over a lengthy period with a number
base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to
be the foundation of more high powered mathematical development.
Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied
from at least 1700 BC. Systems of linear equations were studied in the context of solving number
problems. Quadratic equations were also studied and these examples led to a type of numerical
algebra.
Geometric problems relating to similar figures, area and volume were also studied and values
obtained for π.
The Babylonian basis of mathematics was inherited by the Greeks and independent development
by the Greeks began from around 450 BC. Zeno of Elea\'s paradoxes led to the atomic theory of
Democritus. A more precise formulation of concepts led to the realisation that the rational
numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers
arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius.
Further mathematical discoveries were driven by the astronomy, for example the study of
trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress
continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This
work did not match the progress made by the Greeks but in addition to the Islamic progress, it
did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later
Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into
Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with
Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic
equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of
the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical
research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in
France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a
calculatory science with his discovery of logarithms. Cavalieri made progress towards the
calculus with his infinitesimal methods and Descartes added the power of algebraic methods to
geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the
mathematical study of probability. However the calculus was to be the topic of most significance
to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow,
developed the calculus into a tool to push forward the study of nature. His work contained a
wealth of new discoveries showing the interaction between mathemat.
History of Calculus
Calculus is a branch of mathematics dealing with continuous change and rates of change. In the 17th century, both Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus. While Newton developed many of its theorems and applications earlier, Leibniz is credited with inventing calculus. Calculus has since become essential in fields like physics, chemistry and biology for solving problems involving instantaneous rates of change and accumulation over infinite components.
Mathopolis15 finalsfinal
X denotes the end of a proof in mathematics. It was popularized but not invented by mathematician Paul Halmos, who first used it in mathematical context in place of QED. Halmos wrote that the symbol appeared in popular magazines before he used it mathematically.
17th Century Mathematics
The document summarizes key developments and figures from the Scientific Revolution period including:
- The development of modern scientific notation and concepts like calculus, logarithms, and probability by figures like Descartes, Newton, Leibniz, Napier, and Pascal.
- Advancements in fields like astronomy, physics, and mathematics through inventions like the telescope and discoveries in mechanics, projective geometry, and analytic geometry.
- The establishment of organizations and institutions like the Royal Society of London that helped facilitate scientific progress during this era.
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Similar to Math 114-powerpoint
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2)The sum of the external angles of a triangle equals four right angles 360*
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statements should be proved through reasoning and that no empirical measurements were
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concepts. Abraham Lincoln was a fan, and the US Declaration of Independence used Euclid\'s
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base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to
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from at least 1700 BC. Systems of linear equations were studied in the context of solving number
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algebra.
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The Babylonian basis of mathematics was inherited by the Greeks and independent development
by the Greeks began from around 450 BC. Zeno of Elea\'s paradoxes led to the atomic theory of
Democritus. A more precise formulation of concepts led to the realisation that the rational
numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers
arose. Studies of area led to a form of integration.
The theory of conic sections shows a high point in pure mathematical study by Apollonius.
Further mathematical discoveries were driven by the astronomy, for example the study of
trigonometry.
The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress
continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This
work did not match the progress made by the Greeks but in addition to the Islamic progress, it
did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later
Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into
Europe.
Major progress in mathematics in Europe began again at the beginning of the 16th Century with
Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic
equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of
the universe.
The progress in algebra had a major psychological effect and enthusiasm for mathematical
research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in
France.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a
calculatory science with his discovery of logarithms. Cavalieri made progress towards the
calculus with his infinitesimal methods and Descartes added the power of algebraic methods to
geometry.
Progress towards the calculus continued with Fermat, who, together with Pascal, began the
mathematical study of probability. However the calculus was to be the topic of most significance
to evolve in the 17th Century.
Newton, building on the work of many earlier mathematicians such as his teacher Barrow,
developed the calculus into a tool to push forward the study of nature. His work contained a
wealth of new discoveries showing the interaction between mathemat.
History of Calculus
Calculus is a branch of mathematics dealing with continuous change and rates of change. In the 17th century, both Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus. While Newton developed many of its theorems and applications earlier, Leibniz is credited with inventing calculus. Calculus has since become essential in fields like physics, chemistry and biology for solving problems involving instantaneous rates of change and accumulation over infinite components.
Mathopolis15 finalsfinal
X denotes the end of a proof in mathematics. It was popularized but not invented by mathematician Paul Halmos, who first used it in mathematical context in place of QED. Halmos wrote that the symbol appeared in popular magazines before he used it mathematically.
17th Century Mathematics
The document summarizes key developments and figures from the Scientific Revolution period including:
- The development of modern scientific notation and concepts like calculus, logarithms, and probability by figures like Descartes, Newton, Leibniz, Napier, and Pascal.
- Advancements in fields like astronomy, physics, and mathematics through inventions like the telescope and discoveries in mechanics, projective geometry, and analytic geometry.
- The establishment of organizations and institutions like the Royal Society of London that helped facilitate scientific progress during this era.
Philosophy of science
Mathematics has developed over thousands of years through contributions from many early civilizations. Key ideas emerged from cultures like the Babylonians with their number systems, Egyptians with practical applications of math, Indians who developed the base-10 system, and Greeks who made advances in geometry, astronomy, and formalizing math. Islamic mathematicians also made important contributions, especially in algebra. The study of math history can help students by putting a human face on the subject and showing how concepts developed to further understanding. It also provides motivation and opportunities to investigate. Philosophy of math examines questions around what math is, what is means for objects to exist in math, and how knowledge is acquired in the field.
Mathematical problems130i q
The document discusses Hilbert's view that unsolved mathematical problems are essential for advancing the field, as they test new methods and ideas and can lead to new discoveries and applications. He cites examples throughout history where difficult problems inspired major developments, such as the problem of the shortest line influencing many areas of math and Klein's work on the icosahedron connecting geometry, group theory, and more. Hilbert argues that unsolved problems keep mathematics alive and ensure its continued independent development.
Ptolemy, Copernicus, and the Church
The document discusses the transition from the geocentric model of the universe proposed by Ptolemy and Aristotle, in which Earth is the center, to the heliocentric model proposed by Copernicus, in which the Sun is the center. It also provides context about the Scientific Revolution that began in Europe in the late 1500s and early 1600s, when scientists began systematically testing theories through experiments and observation rather than relying on traditional explanations. Some of the key figures who helped establish the modern scientific method are mentioned, including Galileo, Vesalius, Harvey, Boyle, Bacon, and Descartes.
Ptolemy, Copernicus, and the Church
The document discusses the transition from the geocentric model of the universe proposed by Ptolemy and Aristotle, in which Earth is the center, to the heliocentric model proposed by Copernicus, in which the Sun is the center. It also provides context about the Scientific Revolution that began in Europe in the late 1500s and early 1600s, when scientists began systematically testing theories through experiments and observation rather than relying on traditional explanations. Some of the key figures who helped establish the modern scientific method are mentioned, including Galileo, Vesalius, Harvey, Boyle, Bacon, and Descartes.
Ansatz 2021 finals
The document provides information about the Ansatz Finals math competition including 24 questions in a pounce and bounce format. Teams can pounce to answer a question for points, and if the originally chosen team gets it wrong it bounces to others. Example trial questions are provided about Jack Dorsey's first tweet and Ulam spirals. The document encourages participants not to search for answers online and states that the question master's (QM) decisions are final.
ppt Listening
1) Albert Einstein was a famous German-born physicist. He published several groundbreaking papers on photoelectricity, Brownian motion, special relativity, and mass-energy equivalence.
2) His mass-energy equivalence equation (E = mc2) introduced the concept that mass and energy are equivalent and interchangeable. This equation influenced the modern world.
3) Einstein was awarded the Nobel Prize in Physics in 1921 for his services to theoretical physics, and his explanation of the photoelectric effect. He is regarded as one of the most influential scientists of the 20th century.
Listening
แผนการจัดการเรียนรู้เพื่อพัฒนาทักษะการฟัง
กลุ่มสาระการเรียนรู้ภาษาต่างประเทศวิชาภาษาอังกฤษ ชั้นมัธยมศึกษาปีที่ 4
Unit: Community
Topic: Famous People
Presentation (plg)
1) Albert Einstein was a famous German-born physicist. He published several groundbreaking papers on photoelectricity, Brownian motion, special relativity, and mass-energy equivalence.
2) His mass-energy equivalence equation (E = mc2) has had a huge influence on the modern world. For this work, Einstein was awarded the Nobel Prize in Physics in 1921.
3) After returning to Germany in 1914, Einstein assumed the role of director of the Kaiser Wilhelm Institute for Physics in Berlin. He later became a figurehead for Zionism and was offered the presidency of Israel in 1952.
Math riddles
This document contains math riddles, trivia questions, jokes and information about mathematicians René Descartes, Adrien-Marie Legendre, Blaise Pascal, and Charles Hermite. It includes 3 riddles with answers, 3 trivia questions with answers, 2 jokes with answers, and biographies of the 4 mathematicians and their contributions to mathematics such as analytical geometry, least squares method, probability theory, and elliptic functions.
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Math 114-powerpoint
- 1. Presented by: ROLLY B. REYES BSED III B Math Major Student UNIT 1 C: Discovery, Invention, and Generalization Math 114: Seminar on Problem Solving in Mathematics
- 3. A discovery is recognizing something that already exists for the first time, that
- 5. Other examples of discovery are:
- 6. 1. Fibonacci Sequence in Nature
- 7. 1. Pi in Nature
- 8. 3. There are no negative square roots.
- 11. An invention is creating something totally new with one’s own ideas and development.
- 12. Creating all the material into something with the concepts of + − × and ÷ is what invention is all
- 13. e.g. how Thomas Edison invented the light bulb.
- 14. All mathematical concepts, equations, and formulas are invented.
- 32. By grasping and implementing these concepts, anyone could give life to a new invention.
- 36. e.g. All triangles have three angles.
- 37. Making something more widespread or common, or to use repeated reasoning.
- 38. e.g. All numbers except 0 raised to 0 is equal to 1. (6462372348436+9/9+8*2+2%+7862 34674327674316278674687392949 ²+654278437823697435734538657 43349849385437435933926743659 38743534899832938983765376538
- 39. Think about this: Differentiate discovery and invention. How is generalization related to discovery and invention?
- 40. Discovery and Invention If there’s a number of trees on the forest but no one’s there to count them, does that