MathHistory1a: Pythagoras' theorem
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 .
Lecture I - Beauty and Truth in Mathematics and Science
Robert May, Baron May of Oxford; Professor, Zoology, Oxford University and Imperial College
October 2, 2012
2012 Stanislaw Ulam Memorial Lectures
May explores the extent to which beauty has guided, and still guides, humanity's quest to understand how the world works, with a brief look at the interactions among beliefs, values, beauty, truth, and our expectations for tomorrow's world.
50 Centuries in 50 minutes (A Brief History of Mathematics)
John Dersch (9/19/12)
How did we get the mathematics that is studied today? Who was responsible for major advances in the mathematics that we now take for granted? When and where did this work take place? Such questions will be addressed by tracing the development of mathematics from 3000 B.C. to the dawn of the 21st century. There will be time for questions and suggestions for further study will be made.
MathHistory2a: Greek geometry
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.
The Queen of Mathematics - Professor Raymond Flood
Carl Friedrich Gauss one of the greatest mathematicians, is said to have claimed: Mathematics is the queen of the sciences and number theory is the queen of mathematics. The properties of primes play a crucial part in number theory. An intriguing question is how they are distributed among the other integers. The 19th century saw progress in answering this question with the proof of the Prime Number Theorem although it also saw Bernhard Riemann posing what many think to be the greatest unsolved problem in mathematics - the Rieman Hypothesis.
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 are currently nearly 1,500 lectures free to access or download from the website.
2011 Hagey Lecture: Dr. Ian Hacking - How did mathematics become possible?
In the 2011 Hagey Lecture, Professor Ian Hacking explores how human beings developed the ability to do math.
Monday, October 3, 2011 at 8:00pm, Humanities Theatre, Hagey Hall
Drawing from recent cognitive science, the history of early mathematics, social studies of science, and what has been called the archaeology of mind—how fashioning artifacts has changed the human mind itself—the lecture aims less at building bridges between these different kinds of inquiry, than at highlighting how much we are learning right now, and how little we know.
Ian Hacking is regarded as a leading scholar in the history and philosophy of science, although his work has touched fields as diverse as statistical inference and the emergence of multiple personality disorder. His contributions have earned many awards, including the Killam Prize for Humanities and an appointment to the Order of Canada.
Waterloo's premier invitational public lecture series since 1970, the Hagey Lectures are co-sponsored by the Faculty Association and the University of Waterloo.
History of Mathematics
An animated movie on the development of numbers throughout history.
The Map of Mathematics
The entire field of mathematics summarised in a single map! This shows how pure mathematics and applied mathematics relate to each other and all of the sub-topics they are made from.
If you would like to buy a poster of this map, they are available here:
I have also made a version available for educational use which you can find here:
To err is to human, and I human a lot. I always try my best to be as correct as possible, but unfortunately I make mistakes. This is the errata where I correct my silly mistakes. My goal is to one day do a video with no errors!
1. The number one is not a prime number. The definition of a prime number is a number can be divided evenly only by 1, or itself. And it must be a whole number GREATER than 1. (This last bit is the bit I forgot).
2. In the trigonometry section I drew cos(theta) = opposite / adjacent. This is the kind of thing you learn in high school and guess what. I got it wrong! Dummy. It should be cos(theta) = adjacent / hypotenuse.
3. My drawing of dice is slightly wrong. Most dice have their opposite sides adding up to 7, so when I drew 3 and 4 next to each other that is incorrect.
4. I said that the Gödel Incompleteness Theorems implied that mathematics is made up by humans, but that is wrong, just ignore that statement. I have learned more about it now, here is a good video explaining it:
5. In the animation about imaginary numbers I drew the real axis as vertical and the imaginary axis as horizontal which is opposite to the conventional way it is done.
Thanks so much to my supporters on Patreon. I hope to make money from my videos one day, but I’m not there yet! If you enjoy my videos and would like to help me make more this is the best way and I appreciate it very much.
Here are links to some of the sources I used in this video.
Summary of mathematics:
Earliest human counting:
First use of zero:
First use of negative numbers:
History of complex numbers:
Proof that pi is irrational:
Also, if you enjoyed this video, you will probably like my science books, available in all good books shops around the work and is printed in 16 languages. Links are below or just search for Professor Astro Cat. They are fun children's books aimed at the age range 7-12. But they are also a hit with adults who want good explanations of science. The books have won awards and the app won a Webby.
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Intergalactic Activity Book:
Solar System App:
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LNU-MSU Faculty Lecture Series - "Brief History of Mathematics"
A very condensed brief history of mathematics from ancient times to present.
ARVIND GUPTA - HINDI - MATHS THROUGH ACTIVITIES - Inspire lecture
LECTURE-DEMONSTRATION AT THE PANDIT RAVISHANKAR SHUKLA UNIVERSITY, RAIPUR, INSPIRE CAMP ON 28 FEB 2013 This work was supported by IUCAA and Tata Trust. This film was made by Ashok Rupner TATA Trust: Education is one of the key focus areas for Tata Trusts, aiming towards enabling access of quality education to the underprivileged population in India. To facilitate quality in teaching and learning of Science education through workshops, capacity building and resource creation, Tata Trusts have been supporting Muktangan Vigyan Shodhika (MVS), IUCAA's Children’s Science Centre, since inception. To know more about other initiatives of Tata Trusts, please visit
Math 2B. Calculus. Lecture 01.
"Too much Maths, too little History: The problem of Economics"
This is a recording of the debate hosted by the LSE Economic History Department, in collaboration with the LSESU Economic History Society and the LSESU Economics Society.
Proposition Team - Lord Robert Skidelsky & Dr. Ha-Joon Chang
Opposition Team - Prof. Steve Pisckhe & Prof. Francesco Caselli
Chair - Professor James Foreman-Peck
The LSE is currently the only institution to have a separate EH department. We want to encourage students and academics alike to rethink the methodologies used to explain how our world works.
Do we use the theoretical and econometrical method to create models with assumptions to distil the complexities of human nature and produce measurable results? Or do we use the historical process of considering all factors to provide a more holistic explanation? More importantly, which method should be adopted to better understand increasingly complex economic phenomena in the future?
We are striving to provide our students breadth that exceeds their current theoretical studies. Hence, whilst we recognise the importance of economic history in allowing us to become closer to the truth and produce more intricate portrayal of events, the significance of models and mathematics remains to be emphasised.
Indeed, we wish to have this controversially named debate in order to both highlight the tension between the two disciplines and to produce a more nuanced overview in defence of the future of Economics.
3. The Birth of Algebra
(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.
Stanford Continuing Studies Program:
Stanford University Channel on YouTube:
Lecture 01: Beginning Algebra (Math 70)
Dr. James Curl of Modesto Junior College teaches beginning algebra. This first lesson reviews the course objectives and schedule, homework procedures, tests, grading, and similar protocols. The lecture covers numbers, why we use them, and some common formulas. Included are real numbers, integers, and the properties of real numbers (closure, commutative, associative, identity, inverse, distributive).
Birmingham Popular Mathematics Lectures - the story of Pi
Robin Wilson, Emeritus Professor of Pure Mathematics at the Open University and Emeritus Professor of Geometry at Gresham College, London, relays the history of π, from the ancient Egyptians and Mesopotamians, via Archimedes, China and the Middle Ages, to the Indiana court case and the advances of the modern computer age.
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.
S G Dani delivers History of Mathematics lecture - Diophantine Arithmetic
This is second in the series of Special lecture entitled History of Mathematics. Professor Srikrishna Gopalrao Dani narrates historically the developments of certain aspects of number theory-in particular Diophantine arithmetic and approximations. The History of Mathematics lectures were initiated by Vista Foundation Bangalore at the behest of Prof Ravi S Kulkarni a senior Mathematician working currently at Bhaskaracharya Pratishthan Pune.
Introduction to Higher Mathematics - Lecture 2: Introduction to Proofs
This lecture will introduce you to the language of proofs and show you how the axioms on which you build them are important.
Lec 1 | MIT 18.03 Differential Equations, Spring 2006
The Geometrical View of y'=f(x,y): Direction Fields, Integral Curves.
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