Introduction to Mathematical Thinking & History #Stanford #MIT #TEDx #Lectures
Published on Mar 18, 2014
Richard Brown is the Director of Undergraduate Studies in the Department of Mathematics at Johns Hopkins University. However, it wasn't until after he completed his undergraduate studies in architecture that he chose to enter the field of mathematics. Brown is the editor of the book 30-Second Mathematics and has been awarded and nominated for multiple teaching awards.
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)
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)
Published on Dec 11, 2012
(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.
Stanford University:
http://www.stanford.edu/
Stanford Continuing Studies Program:
https://continuingstudies.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Originally presented in the Stanford Continuing Studies Program.
Stanford University:
http://www.stanford.edu/
Stanford Continuing Studies Program:
https://continuingstudies.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Published on Dec 11, 2012
(October 8, 2012) Professor Keith Devlin dives into the topics of the golden ratio and fibonacci numbers.
Originally presented in the Stanford Continuing Studies Program.
Originally presented in the Stanford Continuing Studies Program.
Published on Dec 11, 2012
(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.
Originally presented in the Stanford Continuing Studies Program.
Published on Dec 11, 2012
(October 22, 2012) Professor Keith Devlin discusses how calculus is truly one of the most useful discoveries of all time.
Originally presented in the Stanford Continuing Studies Program.
Stanford University:
http://www.stanford.edu/
Stanford Continuing Studies Program:
https://continuingstudies.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Originally presented in the Stanford Continuing Studies Program.
Stanford University:
http://www.stanford.edu/
Stanford Continuing Studies Program:
https://continuingstudies.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Published on Dec 11, 2012
(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.
Stanford University:
http://www.stanford.edu/
Stanford Continuing Studies Program:
https://continuingstudies.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Originally presented in the Stanford Continuing Studies Program.
Stanford University:
http://www.stanford.edu/
Stanford Continuing Studies Program:
https://continuingstudies.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Uploaded on Oct 27, 2010
Growth Rates & Log Graphs
Instructor: Gilbert Strang
http://ocw.mit.edu/highlights-of-calc...
License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu
Subtitles are provided through the generous assistance of Jimmy Ren.
Instructor: Gilbert Strang
http://ocw.mit.edu/highlights-of-calc...
License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu
Subtitles are provided through the generous assistance of Jimmy Ren.
Uploaded on Nov 8, 2010
October 23, 2010 - Professor Margot Gerritsen illustrates how mathematics and computer modeling influence the design of modern airplanes, yachts, trucks and cars. This lecture is offered as part of the Classes Without Quizzes series at Stanford's 2010 Reunion Homecoming.
Margot Gerritsen, PhD, is an Associate Professor of Energy Resources Engineering, with expertise in mathematical and computational modeling of energy and fluid flow processes. She teaches courses in energy and the environment, computational mathematics and computing at Stanford University.
Stanford University:
http://www.stanford.edu/
Stanford Alumni Association:
http://www.stanfordalumni.org/
Department of Mathematics at Stanford:
http://math.stanford.edu/
Margot Gerritsen:
http://margot.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Margot Gerritsen, PhD, is an Associate Professor of Energy Resources Engineering, with expertise in mathematical and computational modeling of energy and fluid flow processes. She teaches courses in energy and the environment, computational mathematics and computing at Stanford University.
Stanford University:
http://www.stanford.edu/
Stanford Alumni Association:
http://www.stanfordalumni.org/
Department of Mathematics at Stanford:
http://math.stanford.edu/
Margot Gerritsen:
http://margot.stanford.edu/
Stanford University Channel on YouTube:
http://www.youtube.com/stanford
Published on May 30, 2014
Margot Gerritsen is a professor of energy resources engineering and the director of the Institute for Computational and Mathematical Engineering. After receiving her master's degree in applied mathematics in the Netherlands, Gerritsen moved to the United States in search of "hillier and sunnier places." She received her doctorate in scientific computing and computational mathematics at Stanford, and later became a faculty member at the University of Auckland in New Zealand. Gerritsen specializes in renewable and fossil energy production and in computational mathematics. She is also active in coastal ocean dynamics and yacht design.
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)
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)
Published on Sep 10, 2013
Margot Gerritsen (Stanford Computational Math) on "Linear Algebra: the incredible beauty of a branch of math with a bad reputation" at a USF LASER - with special thanks to Tim Davis and his beautiful matrix collection
Published on Jun 21, 2012
Ein Porträt der Mathematikerin Olga Holtz, Professorin für angewandte Mathematik in Berlin. Dieses Video wurde produziert für WeltWissen - 300 Jahre Wissenschaften in Berlin, Martin-Gropius-Bau, Berlin, von uncertainty-film, d2010.
History of Mathematics in 50min
MATH HISTORY SERIES OF LECTURES
Published on Sep 21, 2012
GRCC Mathematics Professor John Dersch reviews many historical innovations in math.
Uploaded on Mar 13, 2011
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.
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.
Uploaded on Mar 10, 2011
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.
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.
Uploaded on Mar 17, 2011
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.
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.
Uploaded on Mar 18, 2011
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.
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.
Uploaded on Mar 24, 2011
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.
Uploaded on Mar 28, 2011
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.
Uploaded on Apr 3, 2011
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:
https://www.patreon.com/njwildberger?... You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.
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:
https://www.patreon.com/njwildberger?... You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.
Uploaded on Apr 3, 2011
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.
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.
Uploaded on Apr 10, 2011
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.
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.
Uploaded on Apr 10, 2011
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.
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.
Uploaded on May 8, 2011
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.
Uploaded on May 9, 2011
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.
Uploaded on May 9, 2011
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.
Uploaded on May 30, 2011
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.
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.
Uploaded on Jun 6, 2011
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!
Uploaded on Jun 5, 2011
The main historical problem in the history of science is: to explain what is going on with the night sky, in particular what the planets are doing. The resolution of this was the greatest achievement of the 17th century.
The key figures were Copernicus, Galileo, Brahe, Kepler and most famously Isaac Newton. This interesting story, culminating with Kepler's Laws and their explanation by Newton's Laws of Motion and Law of Gravitation, ought to be studied in depth by all undergraduate students of mathematics!
It is notable that the story involves classical geometry in a major way, and gives a great impetus to our study of conic sections and their many remarkable properties.
The key figures were Copernicus, Galileo, Brahe, Kepler and most famously Isaac Newton. This interesting story, culminating with Kepler's Laws and their explanation by Newton's Laws of Motion and Law of Gravitation, ought to be studied in depth by all undergraduate students of mathematics!
It is notable that the story involves classical geometry in a major way, and gives a great impetus to our study of conic sections and their many remarkable properties.
Uploaded on Jun 5, 2011
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.
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.
Published on Apr 29, 2012
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.
We also state Fermat's little theorem using the modular arithmetic language introduced by Gauss.
Published on Apr 29, 2012
The laws of motion as set out by Newton built upon work of Oresme, Galileo and others on dynamics, and the relations between distance, velocity and acceleration in trajectories. With Newton's laws and the calculus, a whole new arena of practical and theoretical investigations opened up to 17th and 18th century mathematicians, such as the Bernoulli family (Johann, Jacob, Daniel, Nicholas, etc), Euler, Huygens and others. Non-algebraic curves played a prominent role, such as the catenary, the shape of a hanging chain, the cycloid, which become famous as both the curve of quickest descent and the curve of equal time descent, and the lemniscate which would play a major role in the theory or elliptic integrals, and gives us our sign for infinity.
We also discuss some other curves that played a role in mechanics, in particular the vibrating string studied by d'Alembert, and the elastica of Euler.
Moving ahead a few centuries, we show that important progress in the theory of curves still happens in modern times, with the discovery of de Casteljau and Bezier, around 1960, of a new way of thinking about curves in terms of control points.
We also discuss some other curves that played a role in mechanics, in particular the vibrating string studied by d'Alembert, and the elastica of Euler.
Moving ahead a few centuries, we show that important progress in the theory of curves still happens in modern times, with the discovery of de Casteljau and Bezier, around 1960, of a new way of thinking about curves in terms of control points.
Published on Apr 29, 2012
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.
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.
Published on May 6, 2012
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.
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.
Published on May 13, 2012
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.
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.
Published on Mar 5, 2014
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.
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.
Published on Apr 16, 2014
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.
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.
Published on May 8, 2014
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.
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.
Published on May 11, 2014
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.
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.
Published on May 15, 2014
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.
Published on May 18, 2014
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.
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.
Published on May 18, 2014
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 serious 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.
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 serious 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.
Published on May 15, 2014
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.
Published on Jun 3, 2014
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.
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.
Published on Mar 18, 2015
We review some of the development of number systems from the ancient Greeks, followed by the Indian and then Arabic development of our Hindu-Arabic numeral system. Then we focus on the new directions forged by the European mathematicians of the 15th and 16th centuries, culminating in the work of Simon Stevin, who shaped our current view towards decimal number arithmetic. We examine the idea that Stevin was the father of `real numbers' (this is not really credible) and also look at some of his other achievements, for example in music and physics.
If you are interested in supporting my YouTube Channel: here is the link to my Patreon page:
https://www.patreon.com/njwildberger?... You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.
If you are interested in supporting my YouTube Channel: here is the link to my Patreon page:
https://www.patreon.com/njwildberger?... You can sign up to be a Patron, and give a donation per view, up to a specified monthly maximum.
