MathHistory1a: Pythagoras theorem
48:55
Pythagoras' theorem is both the oldest and the most important non-trivial theorem in mathematics.
This is the first part of the first lecture of a course on the History of Mathematics, by N J Wildberger, the discoverer of Rational Trigonometry. 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 segment, or perhaps more precisely as the proportion or ratio between two segments, 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. This is a valuable and under-appreciated insight which high school students ought to explicitly see.
In fact young people learning mathematics should really see more of the history of the subject! 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.
This series has now been extended a few times--with more than 35 videos on the History of Mathematics.
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 .
The Pythagorean Theorem : A Presidential Proof
9:37
In this video I show two proofs of the famous Pythagorean Theorem.
The second proof is due to the 20th President of the United States, President James A. Garfield, who was a passionate lover of mathematics.
5. How Did Human Beings Acquire the Ability to do Math?
1:54:24
(October 29, 2012) Keith Devlin concludes the course by discussing the development of mathematical cognition in humans as well as the millennium problems.
Originally presented in the Stanford Continuing Studies Program.
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How Einstein Discovered E=MC2 - A Brief History of Mathematics Documentary
1:49:56
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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 .
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.
1. General Overview and the Development of Numbers
1:44:17
(October 1, 2012) Keith Devlin gives an overview of the history of mathematics. He discusses how it has evolved over time and explores many of its practical applications in the world.
Originally presented in the Stanford Continuing Studies Program.
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MathHistory27: Sets, logic and computability
53:01
In this video we give a very quick overview of a highly controversial period in the development of modern mathematics: the rise of set theory, logic and computability in the late 19th and early 20th centuries.
Starting with the pioneering but contentious work of Georg Cantor in creating Set Theory arising from questions in harmonic analysis, we discuss Dedekind's construction of real numbers, ordinals and cardinals, and some of the paradoxes that this new way of thinking led to. We also explain how the Schools of Logicism, Intuitionism and Formalism all tried to steer a path around these paradoxes.
I should qualify this lecture by stating clearly that in fact I don't really ascribe to any of the theories presented here. My objections will be laid out at length in my MathFoundations series. In this video I am mostly overviewing--rather briefly to be sure!-- the standard thinking, even though I have very little sympathy with it.
But it is important to understand this historical period, since it impacts so heavily on the mathematics that we currently believe in, teach and apply to the world. We are part of a trajectory of human thought, and not necessarily on the pinnacle or high point of that trajectory--much as we would like to think so! In particular, there is much to be learnt by a study of the issues here that so captured the imagination of the late 19th century and early 20th century mathematical and philosophical thinkers.
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 .
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:
An Evening with Leonhard Euler
1:25:40
A talk given by William Dunham, Professor of Mathematics at Muhlenberg College.
MathHistory21b: Galois theory II
29:56
We continue our historical introduction to the ideas of Galois and others on the fundamental problem of how to solve polynomial equations. In this video we focus on Galois' insights into how extending our field of coefficients, typically by introducing some radicals, the symmetries of the roots diminishes. We get a correspondence between a descending chain of groups of symmetries, and an increasing chain of fields of coefficients. This was the key that allowed Galois to see why some equations were solvable by radicals and others not, and in particular to explain Ruffini and Abel's result on the insolvability of the general quintic equation.
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 .
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.
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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 .
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.
MathHistory13: The number theory revival
57:12
After the work of Diophantus, there was something of a lapse in interest in pure number theory for quite some while. Around 1300 Gersonides developed the connection between the Binomial theorem and combinatorics, and then in the 17th century the topic was again taken up, notably by Fermat, and then by Euler, Lagrange, Legendre and Gauss. We discuss several notable results of Fermat, including of course his famous last theorem, also his work on sums of squares, Pell's equation, primes, and rational points on curves. The rational parametrization of the Folium of Descartes is shown, using the technique of Fermat.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
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 .
Genius of Pythagoras Science Documentary
3:22:26
Genius of Pythagoras This Documentary describes Pythagoras. It was produced by Cromwell Productions in 1996. Pythagoras (fl. 530 BCE) must have been one of t.
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MathHistory28: Computability and problems with Set theory
47:05
We look at the difficulties and controversy surrounding Cantor's Set theory at the turn of the 20th century, and the Formalist approach to resolving these difficulties. This program of Hilbert was seriously disrupted by Godel's conclusions about Inconsistency of formal systems. Nevertheless, it went on to support the Zermelo-Fraenkel axiomatic approach to sets which we have a quick look at.
Then we introduce Alan Turing's ideas of computability via Turing machines and some of the consequences.
The lecture closes with a review of historical positions on the contentious idea of completed infinite sets, quoting illustrious mathematicians from Aristotle to A. Robinson, along with G. Cantor himself.
In summary, it appears that this is not a closed chapter in the History of Mathematics. For those interested in a more in depth discussion of these and other interesting issues, see my MathFoundations series of YouTube videos--also at this channel.
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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.
MathFoundations197: Modular arithmetic with Fermat and Euler
37:21
There are two important theorems that make the job of understanding powers in modular arithmetic much simpler. These go back to Fermat and Euler. We apply these to the nice problem of deciding z mod 13. Fermat's result helps us understand powers to a prime modulus. Euler's result relies on understanding the interesting Euler phi function, and is a generalization of Fermat's. As usual we like to illustrate theorems with explicit examples.
History of Mathematics in 50 Minutes
54:22
GRCC Mathematics Professor John Dersch reviews many historical innovations in math.
Polynomials and their Roots - Professor Raymond Flood
54:14
We are familiar with the formula for solving a quadratic equation where the highest power of the unknown is a square. The quest for a similar formula for equations where the highest power is three, four five or more led to dramatic changes in how this question was regarded. Powerful techniques in algebra were developed following work by Abel and Galois in the 19th century to show that there is no such formula when there are powers higher than four.
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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MathHistory9: Calculus
1:00
Calculus has its origins in the work of the ancient Greeks, particularly of Eudoxus and Archimedes, who were interested in volume problems, and to a lesser extent in tangents. In the 17th century the subject was widely expanded and developed in an algebraic way using also the coordinate geometry of Descartes. This is one of the most important developments in the history of mathematics.
Calculus has two branches: the differential and integral calculus. The former arose from the study by Fermat of maxima and minima of functions via horizontal tangents.
The integral calculus computes areas and volumes beyond the techniques of Archimedes. It was developed independently by Newton and Leibnitz, but others contributed too. Newton's focus was on power series, for which differentiation and integration can be done term by term using a formula of Cavalieri, and which gave remarkable new formulas for pi and the circular functions. He had a dynamic view of the subject, motivated in large part by physics.
Leibnitz was more interested in closed forms, and introduced the notation which we use today. Both used infinitesimals, in the form of differentials.
FamousMathProbs19d: Dedekind cuts and computational difficulties with real numbers
1:10:07
In this final video on the most fundamental and important problem in mathematics [which happens to be: How to model the continuum?] we tackle the seriously unfortunate developments leading to the current misunderstandings about the so-called 'real numbers'. Of course this name is a complete misnomer: they are not 'real' at all; rather they constitute a desperate attempt to enforce the existence in mathematics of objects which are actually unattainable without resorting to an infinite number of computational steps (whatever that might actually mean!)
In this video we give a bird's eye view of the various misguided attempts at establishing 'real numbers' and sketch some of the logical and technical difficulties that students are usually shielded from. The basic construction arises from Stevin's decimal numbers extended, using a dollop of wishful thinking, to arbitrary infinite decimals, not just the repeating decimals encoded by rational numbers. Understanding the difficulties with this approach is not that hard, and in essence the same kinds of problems resurface in the various variants which we also discuss: infinite sequences of nested intervals of rational numbers, monotonic and bounded sequences of rationals, Cauchy sequences of rationals, equivalence classes of Cauchy sequences, and finally the icing on the cake of irrationality: Dedekind cuts.
Students of mathematics! Listen carefully: none of these approaches work. This is the reason why not one of these 'theories' are properly laid out in front of you when you begin work in calculus or even analysis. To those who would try to convince you otherwise, via appeals to authority or numbers, name-calling, or by special pleading on behalf of all those lovely 'results' that supposedly follow from the required beliefs: ask rather for explicit examples and concrete computations.
These are the true coin in the realm of mathematics, and will not lead you astray.
This is perhaps a place to thank my many contributors, subscribers and online friends. We are on our way to a more beautiful and logically coherent mathematics, but there is a long ways to go from here to there! Your support is a big help.
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 .
MathHistory19: Complex numbers and curves
57:56
In the 19th century, the study of algebraic curves entered a new era with the introduction of homogeneous coordinates and ideas from projective geometry, the use of complex numbers both on the curve and at infinity, and the discovery by the great German mathematician B. Riemann that topological aspects of complex curves were intimately connected with the arithmetic of the curves.
In this lecture we look at the use of homogeneous coordinates, stereographic projection and the Riemann sphere, circular points at infinity, Laguerre's projective description of angle, curves over the complex numbers and the genus of Riemann surfaces.
This meeting of projective geometry, algebra and topology led the way to modern algebraic 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 .
Another Pythagorean theorem proof | Right triangles and trigonometry | Geometry | Khan Academy
10:23
Visually proving the Pythagorean Theorem
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Geometry on Khan Academy: We are surrounded by space. And that space contains lots of things. And these things have shapes. In geometry we are concerned with the nature of these shapes, how we define them, and what they teach us about the world at large--from math to architecture to biology to astronomy (and everything in between). Learning geometry is about more than just taking your medicine (It's good for you!), it's at the core of everything that exists--including you. Having said all that, some of the specific topics we'll cover include angles, intersecting lines, right triangles, perimeter, area, volume, circles, triangles, quadrilaterals, analytic geometry, and geometric constructions. Wow. That's a lot. To summarize: it's difficult to imagine any area of math that is more widely used than geometry.
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Calculus: What Is It?
46:17
This video shows how calculus is both interesting and useful. Its history, practical uses, place in mathematics and wide use are all covered. If you are wondering why you might want to learn calculus, start here!
MathHistory20: Group theory
58:54
Here we give an introduction to the historical development of group theory, hopefully accessible even to those who have not studied group theory before, showing how in the 19th century the subject evolved from its origins in number theory and algebra to embracing a good part of geometry.
Actually the historical approach is a very fine way of learning about the subject for the first time.
We discuss how group theory enters perhaps first with Euler's work on Fermat's little theorem and his generalization of it, involving arithmetic mod n. We mention Gauss' composition of quadratic forms, and then look at permutations, which played an important role in Lagrange's approach to the problem of solving polynomial equations, and was then taken up by Abel and Galois.
The example of the symmetric group is at the heart of the subject, and so we examine S_3. In the 19th century groups of transformations became to be intimately tied to symmetries of geometries, with the work of Klein and Lie. A nice example that ties together the algebraic and geometric sides of the subject is the symmetry groups of the Platonic solids.
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:
MathHistory23: Simple groups, Lie groups, and the search for symmetry I
51:10
During the 19th century, group theory shifted from its origins in number theory and the theory of equations to describing symmetry in geometry. In this video we talk about the history of the search for simple groups, the role of symmetry in tesselations, both Euclidean, spherical and hyperbolic, and the introduction of continuous groups, or Lie groups, by Sophus Lie.
Along the way we meet briefly many remarkable mathematical objects, such as the Golay code whose symmetries explain partially the Mathieu groups, the exceptional Lie groups discovered by Killing, and some of the other sporadic simple groups, culminating with the Monster group of Fisher and Greiss.
The classification of finite simple groups is a high point of 20th century mathematics and the cumulative efforts of many mathematicians.
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Tadashi Tokieda || Toys in Applied Mathematics || Radcliffe Institute
45:41
Tadashi Tokieda RI '14 invents, collects, and studies toys—simple objects from daily life that can be found or made in minutes, yet which, if played with imaginatively, exhibit behaviors so surprising that they intrigue scientists for weeks. In this video, he explores toys and their relevance to applied mathematics.
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.
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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.
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.
MathHistory22: Algebraic number theory and rings I
48:27
In the 19th century, algebraists started to look at extension fields of the rational numbers as new domains for doing arithmetic. In this way the notion of an abstract ring was born, through the more concrete examples of rings of algebraic integers in number fields.
Key examples include the Gaussian integers, which are complex numbers with integer coefficients, and which are closed under addition, subtraction and multiplication. The properties under division mimic those of the integers, with primes, units and most notably unique factorization.
However for other algebraic number rings, unique factorization proved more illusive, and had to be rescued by Kummer and Dedekind with the introduction of ideal elements, or just ideals.
This interesting area of number theory does have some foundational difficulties, as in most current formulations it rests ultimately on transcendental results re complex numbers, notably the Fundamental theory of algebra. Sadly, this is not as solid as it is usually made out, and so very likely new purely algebraic techniques are needed to recast some of the ideas into a more solid framework.
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 .
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.
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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.
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3. The Birth of Algebra
1:44:24
(October 15, 2012) Professor Keith Devlin looks at how algebra, one of the most foundational concepts in math, was discovered.
Originally presented in the Stanford Continuing Studies Program.
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Mathematics Documentary | Decoding the Language of the Universe | History Films
53:26
Mathematics Documentary | Decoding the Language of the Universe | History Films.
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers),[2] structure,[3] space,[2] and change.[4][5][6] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[7][8]
Mathematicians seek out patterns[9][10] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.
Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.[11]
Galileo Galilei (1564–1642) said, The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth.[12] Carl Friedrich Gauss (1777–1855) referred to mathematics as the Queen of the Sciences.[13] Benjamin Peirce (1809–1880) called mathematics the science that draws necessary conclusions.[14] David Hilbert said of mathematics: We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.[15] Albert Einstein (1879–1955) stated that as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.[16] French mathematician Claire Voisin states There is creative drive in mathematics, it's all about movement trying to express itself. [17]
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.[18]
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The Geometry of Relativity and why your GPS works
59:13
A brief introductory lecture on the Global Positioning System (GPS), and how both Einstein's special and general theories of relativity need to be considered to ensure that it works properly.
We describe the satellite set-up of GPS, the use of cesium atomic clocks to transmit time and position information, and the geometry of why four satellite positions determine where we are. Precursors to this system include LORAN (Long Range Navigation system) of beacons used at sea, and earlier audio systems used in WW1 to try to determine enemy gun positions by measuring differences of distances from observers.
We discuss the accuracy of the clocks, and why Einstein's Special theory (SR) and General theory (GR) of relativity both are affecting this situation. We present simplified views of SR and GR, starting with the Michaelson-Morley experiment which showed that the speed of light was constant in different uniform motion coordinate systems, then Einstein's remarkable conclusion that there is no fixed reference coordinate system for the world: the laws of physics are the same for different observers moving at uniform motion with respect to each other. A bizarre consequence is that simultaneity of events is a relative notion.
GR is explained in terms of the Equivalence Principle relating observers in accelerated frames and in gravitational fields: this allows us to apply the (relativistic) Doppler effect to conclude that clocks higher in a gravitational field appear to run faster.
Both of these effects have effects on the running of the atomic clocks in the GPS satellites--in fact they work in opposite directions, and their cumulative effect must be taken into account by the engineers who manage the system.
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:
DiffGeom25: Manifolds, classification of surfaces and Euler characteristic
46:17
Here we give an informal introduction to the modern idea of `manifold', putting aside all the many logical difficulties that are bound up in this definition: difficulties associated with specification, with the use of `infinite sets', with the notions of `functions' etc.
Even those students who aspire to understanding mathematics correctly ought to be at least aware of the standard formulations, and if one is teaching a course at a major university these days one is limited by the curriculum and the orientation of students and other lecturers in the level of directness that one may address these foundational problems.
I will eventually be discussing the difficulties with these concepts in the MathFoundations series.
In this lecture we talk about charts, manifolds, orientation, and then look more carefully at the two dimensional case of compact surfaces, where things are more concrete and explicit, largely through the classification of Dehn and Heegard which utilizes in a major way the Euler characteristic.
MathFoundations204: Euclid and the failure of prime factorization for z
27:48
We summarize the argument why the number z has no possible prime factorization. The more prime factors we know of z, the more complex is the quotient of z by those factors. We can be rather specific about how big this complexity grows, using our understanding of reciprocals of natural numbers.
So how do we reconcile this with Euclid's theorem of Unique Factorization? Was Euclid seriously wrong? Or perhaps are we incorrect in our interpretation of Euclid's thinking? The reality is that the ancient Greeks spent, relatively speaking, a lot more time thinking about foundational issues than we do, so the answers here should not really be a surprise!
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.
FamousMathProbs 6: Archimedes squaring of a parabola
43:29
Archimedes was the greatest mathematician of all time. In this video we give his solution to one of the very first problems in calculus: to calculate the area of a section of a parabola. This requires some understanding of elementary but beautiful properties of a parabola.
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 .
College Algebra - Lecture 2 - Language of mathematics
27:10
College Algebra with Professor Richard Delaware - UMKC VSI - Lecture 2 - Language of mathematics. This Lecture gives an overview of the Language of Mathematics.
MathHistory10: Infinite series
1:11:01
We discuss various uses of infinite series in the 17th and 18th centuries. In particular we look at the geometric series, power series of log, the Gregory-Newton interpolation formula, Taylor's formula, the Bernoulli's, Eulers summation of the reciprocals of the squares as pi squared over 6, the harmonic series, product expansion of sinx, the zeta function and Euler's product expansion for it, the exponential function, complex values and finally the circular functions too!
MathHistory23b: Simple groups, Lie groups, and the search for symmetry II
26:26
This is the second video in this lecture on simple groups, Lie groups and manifestations of symmetry.
During the 19th century, the role of groups shifted from its origin in number theory and the theory of equations to its role in describing symmetry in geometry. In this video we talk about the history of the search for simple groups, the role of symmetry in tesselations, both Euclidean, spherical and hyperbolic, and the introduction of continuous groups, or Lie groups, by Sophus Lie. Along the way we meet briefly many remarkable mathematical objects, such as the Golay code whose symmetries explain partially the Mathieu groups, the exceptional Lie groups discovered by Killing, and some of the other sporadic simple groups, culminating with the Monster group of Fisher and Greiss.
The classification of finite simple groups is a high point of 20th century mathematics and the cumulative efforts of many mathematicians.
FamousMathProbs 1: Factoring large numbers into primes
42:03
This is the first video of a new series, which will discuss a wide variety of famous (and perhaps not so famous) mathematical problems, ranging from antiquity to modern times. I am N J Wildberger, a pure mathematician at UNSW, the discoverer of Rational Trigonometry and Universal Geometry, and a big fan of YouTube, the future of free on-line education!
After introducing the series, we begin with FMP #1: how to factor a particular number z (10 to the 10 to the 10...altogether ten times, plus 23) into primes. We remind viewers about primes and the Fundamental theorem of Arithmetic, and have a look at explicit factorizations related to our basic problem.
Then we introduce briefly modular arithmetic and the theorems of Fermat and Euler. This is used to show that 7 does not divide z. Some comments on difficulties with arithmetic with big numbers follows, and then some exercises give some challenges for the viewer.
Thanks to Professor Saad El-Zanati from Illinois State University for suggesting this series. Thanks also to Steve Sexton from Exchange for giving permission to use their music track Highlights.
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 .
MathHistory7a: Analytic geometry and the continuum
56:35
The development of Cartesian geometry by Descartes and Fermat was one of the main accomplishments of the 17th century, giving a computational approach to Euclidean geometry. Involved are conics, cubics, Bezout's theorem, and the beginnings of a projective view to curves. This merging of numbers and geometry is discussed in terms of the ancient Greeks, and some problems with our understanding of the continuum are observed; namely with irrational numbers and decimal expansions. We also discuss pi and its continued fraction approximations.
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Ghosts of Departed Quantities: Calculus and its Limits - Professor Raymond Flood
56:31
In 1734 Bishop Berkeley published a witty and effective attack on the foundations of the calculus as developed by Newton and Leibniz. But it took nearly 90 years for the calculus to be given a rigorous foundation through the work of the prolific mathematician, Augustin-Louis Cauchy, who formalised the concept of a limit and created the specialism now called analysis.
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. There is currently over 1,300 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 .
MathHistory15: Complex numbers and algebra
1:7:16
Complex numbers of the form a+bi are mostly introduced these days in the context of quadratic equations, but according to Stillwell cubic equations are closer to their historical roots. We show how the cubic equation formula of del Ferro, Tartaglia and Cardano requires some understanding of complex numbers even when only real zeroes appear to be involved.
The use of imaginary numbers in calculus manipulations is illustrated with some computations of Johann Bernoulli relating the inverse tan function to complex logarithms, and the connections bewteen tan (na) to tan(a).
The geometrical planar representation of complex numbers goes back to Cotes, Euler and DeMoivre in some form, and then more explicity at the end of the 18th century to Wessel and Argand, and then Gauss.
The Fundamental theorem of algebra is a key undergraduate result that often proves elusive---it was so also for the pioneers of the subject. Euler, Gauss and d'Alembert all struggled with the result, but made progress. Here we outline the ideas behind the proofs of d'Alembert and Gauss.
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: