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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    The Map of Mathematics

    11:06

    The entire field of mathematics summarised in a single map! This shows how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.

    If you would like to buy a poster of this map, they are available here:

    I have also made a version available for educational use which you can find here:

    To err is to human, and I human a lot. I always try my best to be as correct as possible, but unfortunately I make mistakes. This is the errata where I correct my silly mistakes. My goal is to one day do a video with no errors!

    1. The number one is not a prime number. The definition of a prime number is a number can be divided evenly only by 1, or itself. And it must be a whole number GREATER than 1. (This last bit is the bit I forgot).

    2. In the trigonometry section I drew cos(theta) = opposite / adjacent. This is the kind of thing you learn in high school and guess what. I got it wrong! Dummy. It should be cos(theta) = adjacent / hypotenuse.

    3. My drawing of dice is slightly wrong. Most dice have their opposite sides adding up to 7, so when I drew 3 and 4 next to each other that is incorrect.

    4. I said that the Gödel Incompleteness Theorems implied that mathematics is made up by humans, but that is wrong, just ignore that statement. I have learned more about it now, here is a good video explaining it:

    5. In the animation about imaginary numbers I drew the real axis as vertical and the imaginary axis as horizontal which is opposite to the conventional way it is done.

    Thanks so much to my supporters on Patreon. I hope to make money from my videos one day, but I’m not there yet! If you enjoy my videos and would like to help me make more this is the best way and I appreciate it very much.

    Here are links to some of the sources I used in this video.

    Links:
    Summary of mathematics:
    Earliest human counting:
    First use of zero:
    First use of negative numbers:
    Renaissance science:
    History of complex numbers:
    Proof that pi is irrational:
    and

    Also, if you enjoyed this video, you will probably like my science books, available in all good books shops around the work and is printed in 16 languages. Links are below or just search for Professor Astro Cat. They are fun children's books aimed at the age range 7-12. But they are also a hit with adults who want good explanations of science. The books have won awards and the app won a Webby.

    Frontiers of Space:
    Atomic Adventure:
    Intergalactic Activity Book:
    Solar System App:

    Find me on twitter, instagram, and my website:



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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.

    Subscribe to this channel -
    Life and Philosophy of Pythagoras -
    Pythagorean Mathematics -
    Pythagoras wiki -
    Pythagorean Theory of Music and Color -
    Pythagoras - Internet Encyclopedia of Philosophy -
    Biography of Pythagoras -
    Greek Mathematics -
    Mathematics of the Pythagoreans -
    Pythagorean History -
    Monochord -
    Pythagorean Monochords -
    Who Was Pythagoras? -
    Greek History -

    As always, use this info to gather more info.

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    History of Mathematics

    7:05

    WEBSITE:

    An animated movie on the development of numbers throughout history.

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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 .

    A screenshot PDF which includes WildTrig1 to 35 can be found at my WildEgg website here:

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    Riemann geometry -- covariant derivative

    10:09

    In this video I attempt to explain what a covariant derivative is and why it is useful in the mathematics of curved surfaces. I try to do this using as many visual arguments as possible; however, some knowledge of differential calculus on the part of the viewer is necessary.

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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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    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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    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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    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.

    Special thanks to the following patrons:

    Music by Vincent Rubinett :

    ------------------

    3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

    If you are new to this channel and want to see more, a good place to start is this playlist:

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    Pythagorean Triplets - An Introduction

    5:32

    A basic introduction into the concepts and patterns of pythagorean triplets

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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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    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 .

    A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here:

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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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    Pythagorean Triplets

    9:55

    Create your personal learning account. Register for FREE at

    DeltaStep is a social initiative by graduates of IIM-Ahmedabad, IIM-Bangalore, IIT-Kharagpur, ISI-Kolkata, Columbia University (USA), NTU (Singapore) and other leading institutes. At DeltaStep, we understand that just like every child has a unique face, a unique fingerprint; he has a unique learning ability as well. Hence we have built an intelligent adaptive learning system that delivers a tailor-made learning solution and helps a student to learn at his own pace because when it comes to learning, one size does not fit all.

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    How to Find Pythagorean Triples

    2:20

    Pythagorean Theorem- A^2+B^2=C^2
    Our equation's for finding Pythagorean triples-
    Pick any two positive integers make the larger one M and the smaller N
    A=M^2 - N^2
    B= 2(MN)
    C= M^2+N^2

    Rob. (1997). Formula for pythagorean triples. Ask Dr. Math, Retrieved from

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    How To Find The Pythagorean Triples/Triads - Awesome Trick

    8:10

    Subscribe Happy Learning :
    Google+ : google.com/+HappyLearning
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    In this video, you will learn how to find the pythagorean triplets if one out of three numbers of triplet/triad is known. you can also determine those value by using the pythagoras theorem.

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    The Birth Of Calculus

    24:44

    A documentary on Leibniz and the calculus.

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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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    Art of Problem Solving: Power of Pythagorean Triples

    9:35

    Art of Problem Solving's Richard Rusczyk (we think) explains why knowing Pythagorean triples can be useful.

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    Interview at CIRM : Peter Scholze

    10:55

    Peter Scholze became known as a mathematician after finishing his Bachelor's degree in three semesters and his Master's degree in two further semesters. Scholze's subsequent PhD-thesis on Perfectoid spaces yields the solution to a special case of the weight-monodromy conjecture.

    He was made full professor shortly after completing his PhD, the youngest full professor in Germany.

    Since July 2011 Scholze is a Fellow of the Clay Mathematics Institute. In 2012 he was awarded the Prix and Cours Peccot. He was awarded the 2013 SASTRA Ramanujan Prize. In 2014 he received the Clay Research Award. In 2015 he will be awarded the Frank Nelson Cole Prize in Algebra, and also the Ostrowski Prize.

    According to the University of Bonn and to his peers, Peter is one of the most brilliant researchers in his field...

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    i^i

    12:39

    complex^complex=real?,
    what is i^i?,
    i^i,
    e^(-pi/2),
    classic math problem,
    complex exponent,
    polar form of complex numbers,





    what is i^i?
    complex analysis,
    math for fun,
    ln(i),
    blackpenredpen,

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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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    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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    Word Study For Bible Journaling Course: Easily Learn Affordably

    1:39

    LEARN MORE:

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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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    Who cares about topology?

    18:16

    An unsolved conjecture, the inscribed square problem, and a clever topological solution to a weaker version of the question, the inscribed rectangle problem (Proof due to H. Vaughan, 1977), that shows how the torus and mobius strip naturally arise in mathematical ponderings.

    Patreon:
    Get 10% your domain name purchace from by using the promo code TOPOLOGY.

    Special shout out to the following patrons: Dave Nicponski, Juan Batiz-Benet, Loo Yu Jun, Tom, Othman Alikhan, Markus Persson, Joseph John Cox, Achille Brighton, Kirk Werklund, Luc Ritchie, Ripta Pasay, PatrickJMT , Felipe Diniz, Chris, Andrew Mcnab, Matt Parlmer, Naoki Orai, Dan Davison, Jose Oscar Mur-Miranda, Aidan Boneham, Brent Kennedy, Henry Reich, Sean Bibby, Paul Constantine, Justin Clark, Mohannad Elhamod, Denis, Ben Granger, Ali Yahya, Jeffrey Herman, and Jacob Young

    ------------------

    3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).

    If you are new to this channel and want to see more, a good place to start is this playlist:

    Various social media stuffs:
    Twitter:
    Facebook:
    Reddit:

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    Riemann hypothesis solved

    12:12

    Euclidean geometry prime numbers math science golden ratio angle trisection Riemann hypothesis zeta function #Riemann millennium prize problem

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    INTERVIEW AT CIRM: PETER SARNAK

    19:28

    Peter Sarnak is a South African-born mathematician with dual South-African and American nationalities. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics. He is known for his work in analytic number theory. Sarnak is also on the permanent faculty at the School of Mathematics of the Institute for Advanced Study. He also sits on the Board of Adjudicators and the selection committee for the Mathematics award, given under the auspices of the Shaw Prize.

    Sarnak graduated University of the Witwatersrand (B.Sc. 1975) and Stanford University (Ph.D. 1980), under the direction of Paul Cohen. Sarnak’s highly cited work (with A. Lubotzky and R. Philips) applied deep results in number theory to Ramanujan graphs, with connections to combinatorics and computer science.

    Peter Sarnak was awarded the Polya Prize of Society of Industrial & Applied Mathematics in 1998, the Ostrowski Prize in 2001, the Levi L. Conant Prize in 2003, the Frank Nelson Cole Prize in Number Theory in 2005 and a Lester R. Ford Award in 2012. He is the recipient of the 2014 Wolf Prize in Mathematics.

    He was also elected as member of the National Academy of Sciences (USA) and Fellow of the Royal Society (UK) in 2002. He was awarded an honorary doctorate by the Hebrew University of Jerusalem in 2010. He was also awarded an honorary doctorate by the University of Chicago in 2015.

    Jean Morlet Chair (Mariusz Lemańczyk - Second Semester 2016)
    'Ergodic Theory and Möbius Disjointness' (5-9 December 2016 at CIRM) : This meeting will focus on the recent progress on Sarnak’s conjecture on Möbius disjointness: methods, results, and the feedback in ergodic theory.

    Sarnak's talk: 'Möbius Randomness and Dynamics six years later'
    There have many developments on the disjointness conjecture of the Möbius (and related) function to topologically deterministic sequences. We review some of these highlighting some related arithmetical questions.

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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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    Genius - Pythagoras

    45:19

    The life of Pythagoras, check more on

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    8th maths ex-6.2 Que no. 2 Pythagorean triplet in Hindi by sativa madam

    17:51

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    WildTrig66: Squares in a pentagon

    9:39

    We use rational trigonometry to solve a problem concerning a pentagon with inscribed touching squares---the question being to find the ratio of the sides of the pentagon to the squares, and also the ratios of two circles, one circumscribing the pentagon, the other inscribed in the squares.

    The Golden spreads of WildTrig 25 play a key role, as they do for many problems with five fold symmetry.

    This video is part of the WildTrig series, which introduces Rational Trigonometry and applies it to many different aspects of geometry.

    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 .

    A screenshot PDF which includes WildTrig36 to 71 can be found at my WildEgg website here:

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    Math Antics - The Pythagorean Theorem

    12:55

    Learn more at mathantics.com
    Visit for more Free math videos and additional subscription based content!

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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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    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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    MathHistory6a: Polynomial equations

    52:41

    We now move to the Golden age of European mathematics: the period 1500-1900, in this course on the History of Mathematics. We discuss hurdles that the Europeans faced before this time and how they emerged, with the help of Arab algebra and translations of Greek works, to harness the Hindu-Arabic number system and a host of novel symbols including Vieta's new use of letters to represent unknowns to tackle new problems.

    Quadratic equations had been solved by almost all earlier mathematical civilizations; cubic equations was a natural step, taken by Tartaglia and Cardano and others. Tartaglia also discovered a formula for the volume of a tetrahedron, and Vieta a trigonometric way of solving cubics.

    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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    Pythagorean Triples Song

    2:25

    Geometry song to help students memorize the Pythagorean triples.

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    Paris_New_Mathematics_part2.avi

    12:02

    A documentary about history of mathematics, focusing on Paris in the 19th century. (Part 2)

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    MathHistory8: Projective geometry

    1:9:42

    Projective geometry began with the work of Pappus, but was developed primarily by Desargues, with an important contribution by Pascal. Projective geometry is the geometry of the straightedge, and it is the simplest and most fundamental geometry. We describe the important insights of the 19th century geometers that connected the subject to 3 dimensional space.

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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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    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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    MathHistory17: Topology

    55:48

    This video gives a brief introduction to Topology. The subject goes back to Euler (as do so many things in modern mathematics) with his discovery of the Euler characteristic of a polyhedron, although arguably Descartes had found something close to this in his analysis of curvature of a polyhedron. We introduce this via rational turn angles, a renormalization of angle where a full turn has the value one (very reasonable, and ought to be used more!!) The topological nature of the Euler characteristic was perhaps first understood by Poincare, and we sketch his argument for its invariance under continuous transformations.

    We discuss the sphere, torus, genus g surfaces and the classification of orientable, and non-orientable closed 2 dimensional surfaces, such as the Mobius band (which has a boundary) and the projective plane (which does not). The interest in these objects resulted from Riemann's work on surfaces associated to multi-valued functions in the setting of complex analysis.

    Finally we briefly mention the important notion of a simply connected space, and the Poincare conjecture, solved recently, according to current accounts, by G. Perelman.

    If you enjoy this subject, you can have a look at my video series Algebraic Topology. This series has now also been continued, so if you go to the Playlist MathHistory, you will find more videos on the History of Mathematics.

    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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    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.

    A screenshot PDF which includes AlgTop0 to 10 can be found at my WildEgg website here:

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    MathHistory30: Ancient astronomy in Babylon and China I

    43:13

    We go back to the beginnings of astronomy, which has had an intimate connection with mathematics for most of recorded history. People have been trying to understand the remarkable occurrences of the night sky for a long time!

    In order to appreciate how ancient civilizations regarded the night sky, we first briefly review the modern heliocentric system, and then go back to reinterpret that along the lines of ancient people's observations and thinking.

    The equator of the rotation of the night sky, the ecliptic and the horizon are then three fundamental great circles on the celestial sphere that play a big role. We discuss equatorial coordinates, the equinoxes, and the role of the Babylonians in setting up the angular measurements that we still use today.

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    MathFoundations57: Polynomials and polynumbers

    9:47

    We begin the important task of defining the fundamental objects of modern algebra. First we review different roles played by polynomials. We are going to base polynomials on something more fundamental called polynumbers, whose arithmetic parallels but is richer than that of the natural numbers and rational numbers.

    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 .

    A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here:

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