MathHistory1a


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    MathHistory1b: Pythagoras theorem

    23:26

    Pythagoras' theorem is both the oldest and the most important non-trivial theorem in mathematics.

    This is the second part of the first lecture of a short course on the History of Mathematics, by N J Wildberger at UNSW (MATH3560 and GENS2005). We will follow John Stillwell's text Mathematics and its History (Springer, 3rd ed). Generally the emphasis will be on mathematical ideas and results, but largely without proofs, with a main eye on the historical flow of ideas. A few historical tidbits will be thrown in too...

    In this first lecture (with two parts) we first give a very rough outline of world history from a mathematical point of view, position the work of the ancient Greeks as following from Egyptian and Babylonian influences, and introduce the most important theorem in all of mathematics: Pythagoras' theorem.

    Two interesting related issues are the irrationality of the 'square root of two' (the Greeks saw this as a length, but not as a number), and Pythagorean triples, which go back to the Babylonians. These are closely related to the important rational parametrization of a circle, essentially discovered by Euclid and Diophantus.

    The Greeks thought of mathematics differently than we do today, and all students can benefit from a closer appreciation of the difficulties which they saw, but which we today largely ignore.

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    Pythagoras Theorem Explained - Mathemagic with Bawa

    6:39

    - Mathemagic - (a+b)²=a²+2ab+b² - But Why?

    One of the elementary formula in high school mathematics, most of us know this formula. But do we actually know why? In this video Khurshed Batliwala, fondly called as Bawa, explains the reason behind this. With a gold medal in Mathematics from IIT, Bombay, India, Khurshed is a faculty of The Art of Living founded by His Holiness Sri Sri Ravi Shankar. According to Bawa he thought it is better to teach people how to breathe and make them happy than teach them Mathematics and make them miserable. :)

    You can follow Khurshed on

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    BBC - History of Indian Mathematics Part-1 of 2

    10:03

    Part-2:

    Marcus du Sautoy looks at the contribution of Ancient Indians like Aryabhata, Brahmagupta, Bhaskara, Madhava, etc in the field of Mathematics.

    Programme: BBC - The Story of Maths
    Presenter: Marcus du Sautoy

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    WildTrig1: Why Trig is Hard

    9:48

    The usual trigonometry is overly complicated, inaccurate and logically dubious. This is the first of a series that shows you a better way---rational trigonometry! Rational trigonometry replaces distance and angle with more algebraic notions called quadrance and spread. All of those complicated formulas become much simpler, you don't need a calculator any more, and the theory extends Euclidean geometry to arbitrary fields!

    What mathematics teacher or student could ask for more?
    Assoc. Prof. N J Wildberger from UNSW is also the creator of the MathFoundations series, the WildLinAlg series, and a more advanced video series on Algebraic Topology. Enjoy!

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    Pythagoras - Secret Teachings of All Ages

    13:30

    Pythagoras - Secret Teachings of All Ages

    Although best known for his Pythagorean Theorem, many strange legends have been preserved concerning the birth of Pythagoras. Some maintained that he was no mortal man: that he was one of the gods who had taken a human body to enable him to come into the world and instruct the human race. Pythagoras was one of the many sages and saviors of antiquity for whom an immaculate conception is asserted. In his Anacalypsis, Godfrey Higgins writes: The first striking circumstance in which the history of Pythagoras agrees with the history of Jesus is, that they were natives of nearly the same country; the former being born at Sidon, the latter at Bethlehem, both in Syria. The father of Pythagoras, as well as the father of Jesus, was prophetically informed that his wife should bring forth a son, who should be a benefactor to mankind. They were both born when their mothers were from home on journeys, Joseph and his wife having gone up to Bethlehem to be taxed, and the father of Pythagoras having traveled from Samos, his residence, to Sidon, about his mercantile concerns. Pythais [Pythasis], the mother of Pythagoras, had a connection with an Apolloniacal spectre, or ghost, of the God Apollo, or God Sol, (of course this must have been a holy ghost, and here we have the Holy Ghost) which afterward appeared to her husband, and told him that he must have no connection with his wife during her pregnancy--a story evidently the same as that relating to Joseph and Mary. From these peculiar circumstances, Pythagoras was known by the same title as Jesus, namely, the son of God; and was supposed by the multitude to be under the influence of Divine inspiration.

    This most famous philosopher was born sometime between 600 and 590 B.C., and the length of his life has been estimated at nearly one hundred years.

    The teachings of Pythagoras indicate that he was thoroughly conversant with the precepts of Oriental and Occidental esotericism. He traveled among the Jews and was instructed by the Rabbis concerning the secret traditions of Moses, the lawgiver of Israel. Later the School of the Essenes was conducted chiefly for the purpose of interpreting the Pythagorean symbols. Pythagoras was initiated into the Egyptian, Babylonian, and Chaldean Mysteries. Although it is believed by some that he was a disciple of Zoroaster, it is doubtful whether his instructor of that name was the God-man now revered by the Parsees. While accounts of his travels differ, historians agree that he visited many countries and studied at the feet of many masters.

    Pythagoras was said to have been the first man to call himself a philosopher; in fact, the world is indebted to him for the word philosopher. Before that time the wise men had called themselves sages, which was interpreted to mean those who know. Pythagoras was more modest. He coined the word philosopher, which he defined as one who is attempting to find out.

    After returning from his wanderings, Pythagoras established a school, or as it has been sometimes called, a university, at Crotona, a Dorian colony in Southern Italy. Upon his arrival at Crotona he was regarded askance, but after a short time those holding important positions in the surrounding colonies sought his counsel in matters of great moment. He gathered around him a small group of sincere disciples whom he instructed in the secret wisdom which had been revealed to him, and also in the fundamentals of occult mathematics, music, and astronomy, which he considered to be the triangular foundation of all the arts and sciences.

    Secret Teachings: A series of videos inspired by Manly P. Hall's Secret Teachings of All Ages.

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    As always, use this info to gather more info.

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    Pythagorean Triplets

    9:55

    Create your personal learning account. Register for FREE at

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    The Birth Of Calculus

    24:44

    A documentary on Leibniz and the calculus.

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    Pythagorean Triplets- Amazing trick

    4:32

    An awesome method to find the corresponding two integer numbers which satisfy pythagoras theorem on them, if we are given the smallest integer.

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    MathHistory2a: Greek geometry

    50:41

    The ancient Greeks loved geometry and made great advances in this subject. Euclid's Elements was for 2000 years the main text in mathematics, giving a careful systematic treatment of both planar and three dimensional geometry, culminating in the five Platonic solids.
    Apollonius made a thorough study of conics. Constructions played a key role, using straightedge and compass.

    This is one of a series of lectures on the History of Mathematics by Assoc. Prof. N J Wildberger at UNSW.

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    MathFoundations1: What is a number?

    9:55

    The first of a series that will discuss foundations of mathematics. Contains a general introduction to the series, and then the beginnings of arithmetic with natural numbers. This series will methodically develop a lot of basic mathematics, starting with arithmetic, then geometry, then algebra, then analysis (calculus) and will also treat so called set theory.

    It will have a lot of critical things to say once we get around to facing squarely up to the many logical weaknesses of modern pure mathematics.

    The series is meant to be viewed sequentially. We spend a lot more time and effort than usual on fundamental issues with number systems. If you are a more advanced student, or a fellow mathematician, then the first few dozen videos might be a bit slow. But they are none-the-less important!

    Screenshot PDFs from my videos can be found at These give you a concise overview of the contents of each lecture.

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    History of Mathematics

    7:05

    WEBSITE:

    An animated movie on the development of numbers throughout history.

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    MathHistory2b: Greek geometry

    24:40

    The ancient Greeks loved geometry and made great advances in this subject. Euclid's Elements was for 2000 years the main text in mathematics, giving a careful systematic treatment of both planar and three dimensional geometry, culminating in the five Platonic solids.
    Apollonius made a thorough study of conics: ellipse, parabola and hyperbola. Constructions played a key role, using straightedge and compass.

    This is one of a series of lectures on the History of Mathematics by Assoc. Prof. N J Wildberger at UNSW.

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    Man, Myth, Mathematician - Pythagoras of Samos - Genius

    45:19

    - The story of Pythagoras is one of innovation, change, determination and sheer genius. As an accurate picture of his life emerges, it is clear that there was more to this great man than one single, simple truth -- here was a great mathematician, philosopher and political leader.

    This Documentary describes Pythagoras. It was produced by Cromwell Productions in 1996.

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    UPSC: Mathematics : Simple Method To Find Pythagorean Triplets.

    11:33

    This lecture is useful for UPSC civil services examinations that is IAS, IPS, IFS etc. CDS, NDA, MPSC, Olympiad, CBSC ,ICSE , etc.
    This lecture is a part of lecture set available on the website upsc-guru.com. In order to get all videos (Paid version) Please visit upsc-guru.com or call 9821562575

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    The Pythagorean Theorem and Pythagorean Triples: Examples

    4:09

    Discover more at ck12.org:

    Here you'll learn the Pythagorean Theorem and how to apply it in order to find missing sides of right triangles and determine whether or not triangles are right triangles.

    This video shows how to work step-by-step through one or more of the examples in Pythagorean Theorem and Pythagorean Triples.

    This is part of CK-12’s Geometry: Right Triangle Trigonometry. See more at:

    1. Pythagorean Theorem and Pythagorean Triples:


    2. Applications of the Pythagorean Theorem:


    3. Inscribed Similar Triangles:


    4. 45-45-90 Right Triangles:


    5. 30-60-90 Right Triangles:


    6. Sine, Cosine, Tangent:


    7. Trigonometric Ratios with a Calculator:


    8. Trigonometry Word Problems:


    9. Inverse Trigonometric Ratios:


    10. Laws of Sines and Cosines:


    Watch the whole series of CK-12's Geometry videos:

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    A Tribute to Euler - William Dunham

    55:08

    A Tribute to Euler

    William Dunham
    Truman Koehler Professor of Mathematics, Muhlenberg College

    Tuesday, October 14, 2008, at 6:00 PM

    Harvard University Science Center, Hall D

    The fall 2008 Clay Public Lecture will be held at Harvard on October 14, in association with the Harvard Mathematics Department. Known for his writings on the history of mathematics, Professor William Dunham will examine the genius of one of the world's most prolific mathematicians in his talk A Tribute to Euler in Hall D of the Harvard Science Center at 6 pm.

    Among history's greatest mathematicians is Leonhard Euler (1707-1783), the Swiss genius who produced an astonishing 25,000 pages of pure and applied mathematics of the very highest quality.

    In this talk, we sketch Euler's life and describe a few of his contributions to number theory, algebra, and other branches of mathematics. Then we examine a particular Eulerian theorem: his simple but beautiful proof that there are as many ways to decompose a whole number as the sum of distinct summands as there are ways to decompose it as the sum of (not necessarily distinct) odd summands.

    Condorcet, in his Eulogy to Euler, wrote that All mathematicians now alive are his disciples. It should be clear to those who attend the Clay Public Lecture that these words are as true today as when they were first set down, over two centuries ago.

    William Dunham, who received his B.S. (1969) from the University of Pittsburgh and his M.S. (1970) and Ph.D. (1974) from Ohio State, is the Truman Koehler Professor of Mathematics at Muhlenberg College. In the fall term of 2008 he is visiting at Harvard University and teaching a course on the work of Leonhard Euler.

    Over the years, he has directed NEH seminars on the history of mathematics and has spoken on historical topics at dozens of U.S. colleges and universities, as well as at the Smithsonian Institution, the Swiss Embassy in Washington, and on NPR's Talk of the Nation: Science Friday.

    In the 1990s, Dunham wrote three books on mathematics and its history: Journey Through Genuis: The Great Theorems of Mathematics (1990), The Mathematical Universe (1994), and Euler: The Master of Us All (1999). In the present millennium, he has written The Calculus Gallery: Masterpieces from Newton to Lebesgue (2005) and edited The Genius of Euler: Reflections on His Life and Work (2007). His expository writing has been recognized by the Mathematical Association of America with the George Pólya Award in 1992, the Trevor Evans Award in 1997, the Lester R. Ford Award in 2006, and the Beckenbach Prize in 2008. The Association of American Publishers designated The Mathematical Universe as the Best Mathematics Book of 1994.

    Our thanks to the Harvard Mathematics Department for hosting this event.



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    MathHistory3a: Greek number theory

    42:04

    The ancient Greeks studied squares, triangular numbers, primes and perfect numbers. Euclid stated the Fundamental theorem of Arithmetic: that a natural number could be factored into primes in essentially a unique way. We also discuss the Euclidean algorithm for finding a greatest common divisor, and the related theory of continued fractions. Finally we discuss Pell's equation, arising in the famous Cattle-problem of Archimedes.

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    What is Math About?: Masao Morita at TEDxKyoto 2012

    9:28

    Kyoto Mathematician Morita Masao unlocks the hidden mysteries and real-world applications of numbers.

    Masao Morita
    Independent scholar
    The word mathematics may delight some people and terrify others. Masao Morita, while still enrolled with the Tokyo University Mathematics Department in 2010, began his explorations as an independent researcher by founding the Kai An Mathematics Dojo in Fukuoka. In the years since, Masao has been continuing his independent research outside the university system, conducting practical workshops that explore the concept of thinking under constraints that go beyond our thoughts. His main research interests include Category Theory and the Theory of Computation. Masao seeks to fuse the theoretic and practical aspects of mathematics, and to share the insights they offer with the world. To this end, Masao regularly conducts unique mathematics concerts around the country, both conducting and performing in a demonstration of Mathematics that will leave you breathless.



    森田 真生
    独立研究者
    「数学の演奏会」をご存知ですか?そのステージで拍手に包まれる演奏家=Playerこそが森田真生さんです。2010年、東京大学理学部数学科在学中に、福岡県糸島市に数学道場「懐庵」を立ち上げたのが、独立研究者としての第一歩。以来、「思考を超えた制約の中に思考を投げ出す」ことをテーマに、さまざまな実験的ワークショップを開催しながら、大学制度の外側で独自の研究活動を展開してきました。主な関心は「圏論」「計算論」。現在は京都に拠点を構え、自然と「ともに-考える(com-putare)」という、言葉本来の意味での計算(computation)ということを理論的、実践的に追求しています。同時に、数学の世界の数々「名作」を「演奏する」ことをコンセプトに、「数学の演奏会」を全国各地で開催しています。



    About TEDx, x = independently organized event
    In the spirit of ideas worth spreading, TEDx is a program of local, self-organized events that bring people together to share a TED-like experience. At a TEDx event, TEDTalks video and live speakers combine to spark deep discussion and connection in a small group. These local, self-organized events are branded TEDx, where x = independently organized TED event. The TED Conference provides general guidance for the TEDx program, but individual TEDx events are self-organized.* (*Subject to certain rules and regulations)

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    Copyrighted Shortcut to find Pythagorean Triplets - Mr. Ajay Kumar

    8:16

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    All possible pythagorean triples, visualized

    15:56

    Can we describe all right triangles with whole number side lengths using a nice pattern?

    Check out Remix careers:

    Regarding the brief reference to Fermat's Last Theorem, what should be emphasized is that it refers to *positive* integers. You can of course have things like 0^3 + 2^3 = 2^3, or (-3)^3 + 3^3 = 0^3.

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    Pythagorean triplets - Squares and square roots - Math Lessons

    3:26

    In this video, we have studied Pythagorean triplets. A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.

    Subscribe to our videos and get fresh quick Math lessons daily.

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    WILD EGG MATHEMATICS COURSES

    2:05

    This is a heads-up short video on my new channel Wild Egg Mathematics Courses on which I hope to present full courses in mathematics, including the Algebraic Calculus series. To kick things off, I am starting a Playlist at that channel called Triangle Geometry--a fascinating and so rich modern subject, and this video gives you a brief intro.

    So please head on over and check it out! Here is the url of the Wild Egg Mathematics Courses channel:



    and here is the Playlist on Triangle Geometry



    Please subscribe!

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    Art of Problem Solving: Pythagorean Triple Warning!

    4:13

    Art of Problem Solving's Richard Rusczyk(?) gives a warning about using Pythagorean triples without thinking about what you're doing.

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    MathHistory4: Infinity in Greek mathematics

    54:08

    We discuss primarily the work of Eudoxus and Archimedes, the founders of calculus. Archimedes in particular discovered formulas that are only found in advanced calculus courses, concerning the relations between the volumes and surface areas of a sphere and a circumscribing cylinder. We also discuss his work on the area of a parabolic arc, Heron's formula (improved using ideas of Rational Trigonometry), hydrostatics, and the Principle of the Lever. He was a true genius.

    If you are interested in supporting my YouTube Channel: here is the link to my Patreon page:
    You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    Non-Euclidean Geometry Topics in the History of Mathematics

    24:26

    Another Open University oldie. This one's a bit more hxc (and considerably older - the 1970s public were apparently considered far smarter than we are today!), but it's mostly easy enough to grasp if you put your mind to it.

    Non-Euclidean Geometry is relevant for the Riemann curvature of space-time in General Relativity and all that. It's also interesting to watch logic (or a bearded professor) decimate what was once considered to be a fundamental truth of mathematics and reality, if you're into that sort of thing. (maybe that's a bit over-dramatic).

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    MathHistory5a: Number theory and algebra in Asia

    49:46

    After the later Alexandrian mathematicians Ptolemy and Diophantus, Greek mathematics went into decline and the focus shifted eastward. This lecture discusses some aspects of Chinese, Indian and Arab mathematics, in particular the interest in number theory: Pell's equation, the Chinese remainder theorem, and algebra. Most crucial was the introduction of the Hindu-Arabic number system that we use today.

    We also discuss the influence of probably the most important problem of the mathematical sciences from a historical point of view: understanding the motion of the night sky, in particular the planets. This motivated work in trigonometry, particularly spherical trigonometry, of both Indian and Arab mathematicians.

    Prominent mathematicians whose work we discuss include Sun Zi, Aryabhata, Brahmagupta, Bhaskara I and II, al-Khwarizmi, al-Biruni and Omar Khayyam.

    If you are interested in supporting my YouTube Channel: here is the link to my Patreon page:
    You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.

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    Pythagorean Triples - MathHelp.com- Geometry Help

    4:04

    For a complete lesson on Pythagorean Triples, go to - 1000+ online math lessons featuring a personal math teacher inside every lesson!

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    MathFoundations93: The magic and mystery of pi

    41:33

    The number pi has been a fascinating object for thousands of years. Intimately connected with a circle, it is not an easy object to get hold of completely rigourously. In fact the two main theorems associated to it--the formulas for the area and circumference of a circle of radius pi--are usually simply assumed to be true, on the basis of some rather loose geometrical arguments in high school which are rarely carefully spelt out.

    Here we give an introduction to some historically important formulas for pi, going back to Archimedes, Tsu Chung-Chi, Madhava, Viete, Wallis, Newton, Euler, Gauss and Legendre, Ramanujan, the Chudnovsky brothers and S. Plouffe, and culminating in the modern record of ten trillion digits of Yee and Kondo. And I also throw in a formula of my own, obtained from applying Rational Trigonometry to Archimedes' inscribed regular polygons.

    It should be emphasized that the formulas here presented are not ones that can easily be rigorously justified, relying as they do on a prior theory of real numbers and often Euclidean geometry. The lecture ends with some speculations about the future role that pi might play in our understanding of the continuum--a huge problem which is not properly appreciated today.

    This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    History of Mathematics

    29:38

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    Pythagoras in 60 Seconds

    1:04

    Short film by Alan Kitching, made in 1986. Apart from explaining the famous theorem, it is also conceived as a visual proof... ie, whereas a still diagram can show a single instance of something, animation can show all possible instances... and that can make the difference between a simple illustration and an actual proof.

    The music is Ponchielli's Dance of the Hours, from the opera La Gioconda.

    For today's Antics 2-D animation software, see Antics Workshop at antics1.demon.co.uk.

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    Universal Hyperbolic Geometry 0: Introduction

    23:13

    Hyperbolic geometry, in this new series, is made simpler, more logical, more general and... more beautiful! The new approach will be called `Universal Hyperbolic Geometry', since it extends the subject in a number of directions. It works over general fields, it extends beyond the usual disk in the Beltrami Klein model, and it unifies hyperbolic and elliptic (and other) geometries.

    This introduction outlines the differences between this course and that found in all the standard textbooks and courses at universities. We avoid ``real numbers'', avoid ``axioms'', avoid transcendental functions such as cos, sin, log, tanh, avoid absolute values. And treat square roots more algebraically. Our approach uses high school algebra, and is essentially projective geometry combined with a distinguished conic---the unit circle in the plane. We will also later see that this approach ties in very closely with the geometry of Einstein's special theory of relativity.

    The series is developed and presented by N J Wildberger, also the originator of Rational Trigonometry.
    Please consider supporting this Channel bringing you high quality mathematics lectures by becoming a Patron at

    Screenshot pdf's for the lectures are available at

    My research papers can be found at my Research Gate page, at

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    Catastrophic Electrical Damage took place - let’s check the Math to see why! By Roger Berry

    54:22

    All protection from damage or injury due to electrical faults begins with determining the amount of potential fault current. Mechanical and thermal energy released in less than 4 milliseconds can produce catastrophic results. Electrical equipment and protective devices must be tested and rated to withstand the potential forces involved.

    Electrical designs are evaluated for worst-case conditions to insure that equipment and people are adequately protected. Fault currents on both sides of the decimal point can result in destructive forces. Short circuits can result in destructive currents over 200,000 amps producing a blast that results in equipment damage and potential injury to personnel due to flash burns and sound, while ground faults as low as five milliamps can produce fibrillation resulting in death. In addition, lower level arcing faults can destroy electrical equipment or start a fire.

    How do we do the math?

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    Mathematics Gives You Wings

    52:28

    October 23, 2010 - Professor Margot Gerritsen illustrates how mathematics and computer modeling influence the design of modern airplanes, yachts, trucks and cars. This lecture is offered as part of the Classes Without Quizzes series at Stanford's 2010 Reunion Homecoming.

    Margot Gerritsen, PhD, is an Associate Professor of Energy Resources Engineering, with expertise in mathematical and computational modeling of energy and fluid flow processes. She teaches courses in energy and the environment, computational mathematics and computing at Stanford University.

    Stanford University:


    Stanford Alumni Association:


    Department of Mathematics at Stanford:


    Margot Gerritsen:


    Stanford University Channel on YouTube:

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    New Theories Reveal the Nature of Numbers

    1:11:15

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    Emory math professor Ken Ono explains major breakthroughs in our understanding of partition numbers, the basis for adding and counting. Ono and his colleagues discovered that partition numbers behave like fractals, and they devised the first finite formula to calculate the partitions of any number. These new theories were hundreds of years in the making, and answer some famous old questions in math. To learn more, visit: emory.edu/esciencecommons.

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    MathHistory12: Non-Euclidean geometry

    50:52

    The development of non-Euclidean geometry is often presented as a high point of 19th century mathematics. The real story is more complicated, tinged with sadness, confusion and orthodoxy, that is reflected even the geometry studied today. The important insights of Gauss, Lobachevsky and Bolyai, along with later work of Beltrami, were the end result of a long and circuitous study of Euclid's parallel postulate. But an honest assessment must reveal that in fact non-Euclidean geometry had been well studied from two thousand years ago, since the geometry of the sphere had been a main concern for all astronomers.

    This lecture gives a somewhat radical and new interpretation of the history, suggesting that there is in fact a much better way of thinking about this subject, as perceived already by Beltrami and Klein, but largely abandoned in the 20th century. This involves a three dimensional linear algebra with an unusual inner product, looked at in a projective fashion. This predates and anticipates the great work of Einstein on relativity and its space-time interpretation by Minkowski.

    For those interested, a fuller account of this improved approach is found in my Universal Hyperbolic Geometry (UnivHypGeom) series of YouTube videos.

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    James Clerk Maxwell: The Greatest Victorian Mathematical Physicists - Professor Raymond Flood

    52:32

    James Clerk Maxwell (1831-1879) was one of the most important mathematical physicists of all time, after only Newton and Einstein. Within a relatively short lifetime he made enormous contributions to science which this lecture will survey. Foremost among these was the formulation of the theory of electromagnetism with light, electricity and magnetism all shown to be manifestations of the electromagnetic field. He also made major contributions to the theory of colour vision and optics, the kinetic theory of gases and thermodynamics, and the understanding of the dynamics and stability of Saturn's rings.

    This talk was a part of the conference on '19th Century Mathematical Physics', held jointly by Gresham College and the British Society of the History of Mathematics. The transcript and downloadable versions of all of the lectures are available from the Gresham College website:


    Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website. There is currently nearly 1,500 lectures free to access or download from the website.
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    MathHistory16: Differential Geometry

    51:32

    Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.

    Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.

    If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    WildTrig73: Spreads, determinants and chromogeometry I

    19:26

    We give a review of some formulas for the spread between two lines, and rewrite some of them using vector language and the theory of determinants. We get the important understanding that a spread between two vectors is just a renormalization of the squared area of the parallelogram spanned by them.

    We are moving towards a more general understanding that will extend to chromogeometry, which we have discussed quite a while ago in this series.

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    Genius - Pythagoras

    45:19

    The life of Pythagoras, check more on

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    WildTrig74: Spreads, determinants and chromogeometry II

    20:26

    We continue exploring spreads from a linear algebra point of view, introducing a dot product, or inner product between vectors, along with our cross product of planar vectors (a number in this case!) We review how the quadrance of a vector is defined in terms of the dot product, as well as the spread.

    We explain the relation between the dot product and perpendicularity. And then we introduce the beautiful trilogy of dot products: blue, red and green!

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    AlgTop0: Introduction to Algebraic Topology

    30:01

    This is the full introductory lecture of a beginner's course in Algebraic Topology, given by N J Wildberger at UNSW. The subject is one of the most dynamic and exciting areas of 20th century mathematics, with its roots in the work of Riemann, Klein and Poincare in the latter half of the 19th century. This first lecture will outline the main topics, and will present three well-known but perhaps challenging problems for you to try.

    The course is for 3rd or 4th year undergraduate math students, but anyone with some mathematical maturity and a little background or willingness to learn group theory can benefit. The subject is particularly important for modern physics. Our treatment will have many standard features, but also some novelties.

    The lecturer is Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW, Sydney, Australia, well known for his discovery of Rational Trigonometry, explained in the series WildTrig, the development of Universal Hyperbolic Geometry, explained in the series UnivHypGeom, and for his other YouTube series WildLinAlg and MathFoundations. He also has done a fair amount of research in harmonic analysis and representation theory of Lie groups.

    Exciting news: screenshot pdfs of these lectures will shortly be available at They will allow you to summarize, review and study these lectures in greater detail.

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    MathHistory18: Hypercomplex numbers

    59:31

    In the 19th century, the geometrical aspect of the complex numbers became generally appreciated, and mathematicians started to look for higher dimensional examples of how arithmetic interacts with geometry.

    A particularly interesting development is the discovery of quaternions by W. R. Hamilton, and the subsequent discovery of octonians by his friend Graves and later by A. Cayley. Surprisingly perhaps the arithmetic of these 4 and 8 dimensional extensions of complex numbers are intimately connected with number theoretical formulas going back to Diophantus, Fibonacci and Euler.

    If you are interested in supporting my production of high quality math videos, why not consider becoming a Patron of this channel? Here is the link to my Patreon page:

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    Infinity: does it exist?? A debate with James Franklin and N J Wildberger

    42:58

    Infinity has long been a contentious issue in mathematics, and in philosophy. Does it exist? How can we know? What about our computers, that only work with finite objects and procedures? Doesn't mathematics require infinite sets to establish analysis? What about different approaches to the philosophy of mathematics--can they guide us?

    In this friendly debate, Prof James Franklin and A/Prof Norman Wildberger of the School of Mathematics and Statistics, Faculty of Science, UNSW, debate the question of `infinity' in mathematics.

    Along the way you'll hear about Jim's new book: `An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure', published this year by Palgrave MacMillan.

    Unfortunately, the microphone could not pick up audience questions and responses very well. The correct answer to Norman's question at the end of the game he described was given by Roberto Riedig: `any number you want'! As for this interesting game itself, Norman seems to remember getting the idea from Wolfgang Mueckenheim, who also ventures into heretical waters: see for example his paper Physical Constraints of Numbers, Proceedings of the First International Symposium of Mathematics and its Connections to the Arts and Sciences, A. Beckmann, C. Michelsen, B. Sriraman (eds.), Franzbecker, Berlin 2005, p. 134 - 141.

    For those interested in this kind of non-standard position, they can also look for Norman's paper: `Set Theory: Should you Believe?'

    You can also check out my blog entry at



    Thanks to Nguyen Le for videoing.

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    THE BINOMIAL DISTRIBUTION: a GeoGebra applet

    3:18

    A quick introduction to the general binomial theorem through a GeoGebra applet. What is the probability of getting a total of k heads out of n tosses of a weighted coin, when the probability of a single head is p? That is B(n,p,k), the binomial probability.

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    MathHistory21: Galois theory I

    43:54

    Galois theory gives a beautiful insight into the classical problem of when a given polynomial equation in one variable, such as x^5-3x^2+4=0 has solutions which can be expressed using radicals. Historically the problem of solving algebraic equations is one of the great drivers of algebra, with the quadratic equation going back to antiquity, and the discovery of the cubic solution by Italian mathematicians in the 1500's. Here we look at the quartic equation and give a method for factoring it, which relies on solving a cubic equation. We review the connections between roots and coefficients, which leads to the theory of symmetric functions and the identities of Newton.

    Lagrange was the key figure that introduced the modern approach to the subject. He realized that symmetries between the roots/zeros of an equation were an important tool for obtaining them, and he developed an approach using resolvants, that suggested that the 5th degree equation was perhaps not likely to yield to a solution. This was confirmed by work of Ruffini and Abel, which set the stage for the insights of E. Galois.


    If you are interested in supporting my production of high quality math videos, why not consider becoming a Patron of this channel? Here is the link to my Patreon page:

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    MathHistory29: Combinatorics

    41:01

    We give a brief historical introduction to the vibrant modern theory of combinatorics, concentrating on examples coming from counting problems, graph theory and generating functions. In particular we look at partitions and Euler's pentagonal theorem, Fibonacci numbers, the Catalan sequence, the Erdos Szekeres theorem, Ramsey theory and the Kirkman Schoolgirls problem.

    My research papers can be found at my Research Gate page, at I also have a blog at where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at Of course if you want to support all these bold initiatives, become a Patron of this Channel at .

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    The Memoirs and Legacy of Évariste Galois - Dr Peter Neumann

    53:24

    Évariste Galois was born 200 years ago and died aged 20, shot in a mysterious early-morning duel in 1832. He left contributions to the theory of equations that changed the direction of mathematics and led directly to what is now broadly described as 'modern' or 'abstract' algebra. In this lecture, designed for a general audience, Dr Peter Neumann will explain Galois' discoveries and place them in their historical context. Little knowledge of mathematics is assumed - the only prerequisite is sympathy for mathematics and its history.

    The transcript and downloadable versions of the lecture are available from the Gresham College website:


    Gresham College has been giving free public lectures since 1597. This tradition continues today with all of our five or so public lectures a week being made available for free download from our website.

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    WildTrig81: Rheticus and 17th century trig tables

    28:07

    The heart of classical trigonometry does not lie in trigonometric identities, or in the many formulas such as the Sine law or Cosine law. Rather, the heart of the matter lies in tables which give the values of the circular functions for different angles. Such has been the story for two thousand years: spanning Greek, Hindu, Arabic, Chinese and then European work. In this video we look at the remarkable work of Rheticus which established the dominant 17th century trig tables. Unfortunately this critical underpinning to the entire subject is almost completely invisible to high school students and indeed undergraduates.

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    What is a Manifold? - Mikhail Gromov

    53:56

    Manifolds are a bit like pornography: hard to define, but you know one when you see one.
    S. Weinberger

    --------------------------------------------------------------------------------
    2010 Clay Research Conference

    What is a Manifold?
    Mikhail Gromov

    Clay Mathematics Institute

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