Pythagoras' theorem (a) | Math History | NJ Wildberger
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- Опубліковано 2 чер 2024
- 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.
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Screenshot PDFs for my videos are available at the website wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at www.researchgate.net/profile/...
My blog is at njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at / njwildberger Your support would be much appreciated.
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: / playlist
list=PL8403C2F0C89B1333
Year 9 Maths: • Year9Maths
Ancient Mathematics: • Ancient Mathematics
Wild West Banking: • Wild West Banking
Sociology and Pure Mathematics: • Sociology and Pure Mat...
Old Babylonian Mathematics (with Daniel Mansfield): / playlist
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: • MathHistory: A course ...
Wild Trig: Intro to Rational Trigonometry: • WildTrig: Intro to Rat...
MathFoundations: • Math Foundations
Wild Linear Algebra: • Wild Linear Algebra
Famous Math Problems: • Famous Math Problems
Probability and Statistics: An Introduction: • Probability and Statis...
Boole's Logic and Circuit Analysis: • Boole's Logic and Circ...
Universal Hyperbolic Geometry: • Universal Hyperbolic G...
Differential Geometry: • Differential Geometry
Algebraic Topology: • Algebraic Topology
Math Seminars: • MathSeminars
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And here are the Wild Egg Maths Playlists:
Triangle Centres: • ENCYCLOPEDIA OF TRIANG...
Six: An elementary course in pure mathematics: • Six: An elementary cou...
Algebraic Calculus One: • Algebraic Calculus One
Algebraic Calculus Two: • Algebraic Calculus Two
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Of course you can understand this series. It does not mean that everything will be clear the first time around, but the general story you can follow. Just be patient and try to understand everything slowly.
Thanks
Thanks for your work btw
i always appreciate what you do sir because of u i developed my interest in maths thankyou very much..
Sir, before I watch this series, or at least the first part of it to decide whether the presentation form will not conflict with the tight reasoning and insubortination to learn that issues my mind, I must say beforehand that I found the historical chronological order in teaching to be the best way of transcribing the metaphisically observed reality onto the human readable syntax forms with the goal of acquiring knowledge. Kudos
but if you put the subtitles that will be very nice ... thank you
Yes, actually I plan on adding more videos to this series next year!
Dude. Only 20 minutes in and this is already the best math lecture I’ve ever heard. (And I’m old) *”Area is more fundamental than length.”* 🤔😯🤯 Awesome! Thank you for your work!
Thanks Jake! There’s plenty more…
@@njwildberger
Please add Subtitle English, and Google translation automatic all langauge choise
@@njwildberger WHAT IS E=MC2 is taken directly from F=ma, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS the rotation of WHAT IS THE MOON matches the revolution. Consider TIME AND time dilation ON BALANCE. The stars AND PLANETS are POINTS in the night sky ON BALANCE.
The diameter of WHAT IS THE MOON is about one quarter of that of what is THE EARTH. On balance, the density of what is the Sun is believed to be about one quarter of that of what is THE EARTH. Excellent. Consider what is THE EYE ON BALANCE. The TRANSLUCENT AND BLUE sky is CLEARLY (and fully) consistent WITH what is E=MC2. WHAT IS THE EARTH/ground is fully consistent WITH what is E=MC2. CLEAR water comes from what is THE EYE ON BALANCE. Notice what is the fully illuminated (AND setting/WHITE) MOON AND what is the orange (AND setting) Sun. They are the SAME SIZE as what is THE EYE ON BALANCE. Lava IS orange, AND it is even blood red. Yellow is the hottest color of lava. The hottest flame color is blue. What is E=MC2 is dimensionally consistent. WHAT IS E=MC2 is consistent with TIME AND what is gravity. What is gravity is, ON BALANCE, an INTERACTION that cannot be shielded or blocked.
Consider what are the tides. The human body has about the same density as water. Lava is about three times as dense as water. The bulk density of WHAT IS THE MOON IS comparable to that of (volcanic) basaltic lavas on what is THE EARTH/ground. Pure water is half as dense as packed sand/wet packed sand. Now, the gravitational force of WHAT IS THE SUN upon WHAT IS THE MOON is about twice that of THE EARTH. Accordingly, ON BALANCE, the crust of the far side of what is the Moon is about twice as thick as the crust of the near side of what is the Moon. The maria (lunar “seas”) occupy one third of the visible near side of what is the Moon. The surface gravity of the Moon is about one sixth of that of what is THE EARTH/ground. The lunar surface is chiefly composed of pumice. The land surface area of what is the Earth is 29 percent. This is exactly between (ON BALANCE) one third AND one quarter. Finally, notice that the density of what is the Sun is believed to be about one quarter of that of what is THE EARTH. One half times one third is one sixth. One fourth times two thirds is one sixth.
By Frank Martin DiMeglio
@@njwildbergere33333?❤❤❤1
When you focus on education instead of what color everyone is, this is what you get. Sad that most of the bwack students will fail this class since they cant understand math past algebra.
The Greeks had knowledge of zero, but just not an effective symbol for it as a place-holder in their number system. As for their understanding of `irrationals', this was in some ways more advanced and profound than our current understanding.
The Greeks were well aware of the irrationality of things like sqrt(5). Eudoxus gave a thorough treatment of arithmetic with such geometrical irrationalities, and Euclid followed him.
NONSENSE!!.The greeks eg pythogoras studied in Kemet how did he has a theorem when the first pyramid was built on the same theorem A+B=C the step pyramid
WOW!!! I have to say I love your videos thank you for sharing your intellect. As i read things like Euclid's Elements, clifford algebra or about differential forms and calculus on manifolds my imagination runs wild; I feel isolated and yet freed, in powered but humbled. The intellect displayed is astonishing in these books. But the subjects you choose to touch on I find so insightful thank you. What a wonderful collection of thought. The level of thought you display is brilliant!
your work is VERY much appreciated mr njw.
Dr. Wildberger, thank you for your video series. I am an adult that always struggled with advanced mathematics, but have always keep an interest in it. Your explanations are very understandable and I am excited to learn all I can from your lectures.
Thank you professor for sharing your math knowledge. We need.more educators like you at the university level.
AWESOME SET OF VIDEOS! Thank you so much for sharing.
this is the course i am really looking for ,at last i got one ,very well explained and good approach of learning mathematics (my opinion). thank you very much prof. NJ wildberg.
Your videos are so insightful, interesting, and engaging. You make math much more intuitive. Thank you
I really like your presentations. Thanks.
Your approach to teaching helped me to enjoy maths. Thank u sir.
I wanted for a long time a math history class. I'm glad I found one. Thanks.
One of the best methods of learning maths for me
Thank you for making these videos Prof. Wildberger.. Because of these videos,I have now started reading Euclid's Elements....
Does no one come to class on time at UNSW? This class is great; I'd never be late. Thank you Professor Wildberger for posting these wonderful lectures. Your Internet students will always come to class on time.
Maybe crossing campus from least class?
@@AnthonyL0401 students love that excuse.
You have done a wonderful thing for humanity. I am honoured to learn from you
amazing course
I’ve always struggled to really relate to math in any way but for some reason this narrative and understanding of the history is really helping me see the reasoning and application behind a lot of math that I kinda just brute force learned. Thank you very much for the lessons, they help my comprehension immensely!
Dividing education into the four key groups... history, math, science and english... is just a technique used to brainwash low eyequeue ppl into believing whi mehn are evil and the ultimate enemy of humanity. History class and english class have no actual criteria and are merely a medium to indoctrinate. Math and science are split up so that wahmn and bwacks can go into "biology" and get science degrees when they lack the capacity for high end math.
Great lecture. Looking forward to seeing all of them. If my maths teacher was as clear and as clearly enthusiastic as you are about your subject I probably would have shown more interest in maths in my school days.
Like others have been saying: thank you so much for sharing these lectures online. Only now in my adulthood am I realizing the power and elegance of Mathematics, and starting to take a lot of interest again. Unfortunately I had terrible, terrible Math teachers, but excellent material like this is a blessing.
You had wahmn math teachers who couldn't understand advanced math and lack the ability to put themselves in someone elses shoes to explain history accurately. Most of whom were all hopped up on persciptions... just like the rest of us. All in the name of equality.
Don't automatically blame the teachers. I had exellent teachers in the difficult subjects, and many students thought the teachers were bad. Why? Because they weren't doing the work required to understand it!
@@Tommy_007 if you have kids for 36 weeks or whatever it is, and you cant teach them how to solve for x in a basic algebra equation.... either the students have subhuman intelligence or the teachers are bad
@@notallowedtobehonest2539 That is not true. Some student refuse to learn. That is a fact. And the teacher can't spend that amount of time on each topic. And some student have an IQ less thatn 85. And more importantly, not all topics are that easy.
@@Tommy_007 which kids have eyequeues under 85?
Really good and clear presentation using classical technology - chalk! Thank you.
I'm very thrilled having the chance of attend this lecture (even if it's virtual)
you, sir, are really awesome. Thanks for this great course
These are very intresting lectures , thanks sir for your efforts
what can i to say!, i am watching in korea,
ur lecture is so nice....
thanks for uploading these staff, becouse of u i can enjoy math more!
best channel for insight into math.
Thank you for uploading these great lectures, this is very interesting stuff
Thanks for all your great content!
This reminds me of Michael Sugrue’s Philosophy lectures at Yale in the 80s/90s. Incredible work!
Blessed to have Dr. Wildberger.
love what youre doing sir keep up the good work
one of my favourite videos
To everyone nitpicking about year 0 or "still not unresolved" or light years or whatever, cmon, you know what he meant... plus "still not unresolved" is the only real mistake there, year 0 makes perfect sense even if it isn't officially part of the calendars, and he was referring to the distance of the written decimal when he said light years
Thank you SO MUCH for sharing your lectures! These are exactly what I've been looking for to enrich my own math classes in high school!
love this! just what i have been looking for to satiate my love for math and history!!
Literally one of my favourite videos ever
Outstanding series on the subject. Most books on the topic are difficult to read. These video are so easy to learn from and absorb. Congrats and thank you.
Totally agree with MrJosephArthur. Pythagors actually stolen the theorm from the ancient Egyptians while he studied at temples in Egypt.
The lecturer also did not mention that the ancient Egyptians started with math. almost a millennium before 2000 BC.
You are awsome. You 1 lecture solves some of my childhood problems with maths. Thank you for your insights. This is one of the best program in UA-cam.
Especially, starting and ending music is great.
thank you sir for the lecture
I love how this professor teaches.
Frickin awesome. Thanks.
Great lectures!...thank you!!
Thank you sir for your time. I will read the construction again and also check out your lectures and then get back if it doesn't click even after that.
The joy of mathematics - how wonderful!
The only thing missing here, which is actually present. It is the concept of the video itself. The blur of the cultures leading to present day unidentified yet readily identifiable down to the minute right here. Fantastic work here.
beautifully presented
Which piece of classical music is played at the beginning and the end? Sounds like Haydn.
Thank you for the class.
About twenty five years ago, I dropped a history of math course in order to take some crappy poli sci thing. Ended up in history of science (evolution and biology mostly), took a terminal masters, had another life. However, I've always regretted missing that course, so here I am! :) Thanks! (I'm a private tutor -- SAT, ACT, stuff like that; write books, too. So, the stuff on the PT is very helpful. I show the kids the geometric version of c^2 = a^2 + b^2, and it blows their minds. :)
very good lecture. the problem with schools these days is that they don't explain any of this. they just say to the kids 'x+y = z' with absolutely no background, context or description.
This is a really nice playlist. I just finished the first video and I am feeling quite sad because I did not come across this playlist when I was a student at university. Anyhow, it is better to be late than never. I would like to thank you for this playlist😊👏
Outstanding!
Great course series. Got the book :-)
Well, the ancient Egyptian mathematics is really very interesting and provides some excellent ideas to understand arithmetic and geometry to its foundation. A student of history of mathematics would love to read it.😊
The famous Pythagorean Theorem has some sort of oddity in the history of mathematics in reference to the Babylonian clay tablet, 'Plimpton 322'. This tablet provides a strong evidence that the Pythagorean theorem was well known to the Babylonian mathematicians more than a thousand years before Pythagorean was born. Here, then, is an unrewarded anticipation, for doubtless the name of the famous theorem will remain as a true mumpsimus - "the Pythagorean theorem" - for all time. Ref : 'Mathematics in the time of Pharoahs' by Richard J. Gillings.
"The sense of a mathematical, if examined from the reproduced illustration of the original, is completely changed" - Arnold Buffum Chace in 'The Rhind Mathematical Papyrus, Vol. II. '
Well, thank you for the great video. Especially, for summarizing the history of the world at the beginning. It really helped me a lot to develop a good mathematical perspective throughout its history. 👌
You can find the answer to your concerns at 26:00.
thank you very much, A math lover from Morocco
Beautiful...
hi thank you professor for these lectures thank you very much
This is Great!
Thanks for the lectures Proffessor njw
That's Haydn's string quartet in D minor, Op. 76; No. 2.
you read my mind
Nice proof Prof. First I've understood!..
I believe it would be somehow easier to take a shorter step at 28:53 without shifting the triangles.
The larger square has sides x+y where x and y are the lengths of the shorter sides of the triangles while z is the side of the smaller square.
Then the area of the large square is ( x+y)( x+y) = x^2 + y^2 +2xy
The total area of the four triangles is 4xy/2 = 2xy.
So if we subtract the area of the four triangles forms the large area then we have x^2 +y^2 + 2xy--2xy = area of the smaller square.
Bur the area of the smaller square its sides say z multiplied by itself, hence z^2
Hence z^2 = x^2 +y^2 the square of the hypotenuse equals the sum of the squares of the adjacent sides.
Hello, Mr. wildberger. I found your course very interesting, and watched all the lectures. Now I want to read the book of Stillwell, but looking at contents, I found it not so complete, and basically all you already gave in your lectures - even with just my university mathematics for computer engeneer I found no linear algebra, graph theory, multivariable calculus, and smth newer like game theory, which I wanted to read about. I remember in one of lectures you give a note about some larger book, but I cant find it right now. can you post its name/author please. thanks.
i wish we had this course in our uni
Fantastic!
The Math history as dealt with here starts with the Greek period. Besides that it would also be interesting to learn what kind of math the Egyptians maintained. For example, building complex pyramids and temples does not simply require a tape-measure or a ruler.
Good to know length was originally a concept derived from area not the other way around. Thanks professor. Plan to watch this series. One small correction-light year is not a unit of time but distance. :)
I'm not convinced about area preceding distance historically. All sentient beings come pre-equipped to sense close vs. far, approach vs. retreat, and prehistoric man had various ways to measure distance informally employing available inexact units (paces, feet, hands, days of walking, etc.). By comparison, area is a more evolved, more abstract concept. Rectangular area DEPENDS UPON 2 edge-length measurements for its definition and units.
Also, math history is clear that knowledge of the 3-4-5 right triangle goes back much further than the Pythagorean Theorem (to India), and was the specific case that led Greeks to ponder questions about the general case. Do the numbers 3, 4 and 5 NOT represent lengths?...of course they do. The 3-4-5 was likely discovered by laying out 12 same-length bones or sticks on the ground.
I think the Prof. is simply noting that the Pythagoreans, limited to a whole number measuring system, could state the invariance (a^2 + b^2 = c^c) of the right triangle (where the 2 leg-lengths were whole numbers) in terms of areas, but not in terms of distances -- because hypoteneuse-length takes you outside the whole number universe -- and such was blasphemy.
This should be taught as a requirement for any mathematically centric degree.Im a Mechanical Engineering student and my brain is well suited for math,& I feel its very important to understand the principles & reasoning behind the math, rather than just being able to follow predefined rules and systems.If you understand the context, reasoning, and principles involved I feel it exponentially expands your critical thinking ability.Math is the base for so many fields & the context is never pushed?
thank you for this lecture. I need this book
Hi @njwildberger. Thank you for an interesting and well told lecture! I am doing a high school project about greek mathematics and especially the discovery of irrational numbers. The proof that you gave that root 2 is irrational struck me as quite modern (the heavy use of algebra) and I was wondering if you know any sources with original greek proofs? I have been trying to find one in Euclids elements, but so far without luck. Do you know if there is any in this work? I hope that you can help me :) Thank you very much. Asger.
+Asger Andersen The result was known to the Pythagoreans, but the first proof is I believe in Euclid, Book X, Prop 117.
nice. i loved the proof of the theorem
Good question. There are two good reasons to prefer area: first it is a 2 dim concept, more fitting the plane than a 1 dim concept like length, and secondly area can be defined and computed purely rationally, while length is a transcendental concept, meaning infinite approximations are required to set it up in a Euclidean framework.
Length is intuitively simple from everyday life, but it just happens to be hard to encode mathematically. Euclid realized this very well.
It's possible in 3D to arrange things to make area less pleasing: What's the surface area of a cube whose volume is three?
I imagine the same could be said of volume in 4D, etc.
i don't quite get it , areas can be irrational too, right? like area of a rectange with sides 1 and sqrt2 , a circle with radius 1 ,etc?
2D area is measured in quantities inclusive of 1D length, and 'almost all' numbers are transcendental (i.e., unlike the square root of 2 (which is irrational but not transcendental) they cannot be expressed as a root of any non-zero polynomial with rational coefficients), even though the number of non-transcendental numbers is infinite (even though the 'measure' of rational numbers is called zero -- i think that, because it's arguably non-zero, it should be called infinitesimal) , even though we've shown ourselves able to prove only a few transcendental numbers to be by us definable
thanks so much for sharing this!!!!!
Hi. I am from Argentina. Great class
@njwildberger Professor, I got confused around 34min, when you were explaining that the pyth thrm is still a theorem even when your are doing it the cartesian way? Can you elaborate on that point?
Nice lecture. Sir kindly make a playlist of Lectures about Group Theory. I hope you will elaborate it very nicely as you did previously. Kindly oblige the matter!!!
Fiaz (Pakistan)
Do you have a basic math course or book with all this very good historical mathematic information ????
I REALLY hope so!!!
Could please activate in the video the option of automatic subtitles, that depends on the administrator, if they do not activate the other people does not get that option.
Podrían por favor activar en el vídeo la opción de subtitulos automáticos, eso depende del administrador, si no lo activan a las otras personas no le sale esa opción.
Thank you!
Great lecture looking at motivation and intuition in mathematics by Dr Wildberger. You'd think someone in maintenance would fix the rattling of that whiteboard at 16:30 though it's quite distracting. :-)
Thank you so much!
What were the applications for knowing the areas of the squares extending from the sides of a right triangle (He said that originally the side lengths were not considered).
Dear N J Wildberger, is there such a course for the second half of Stillwell's book? If not, do you have plans to make such course available? Thanks
It always reminds that Algebra is congruent to Geometry, algebra for the algorithm, logarithm, while Geometry for the Geometrical sizes and shapes, including the phases, behavior of sciences of sciences according to their class, group, sub group as well as equations ∆=π | ÷ ∆ = { 3 } { 3 } |
Such an excellent lesson. It would be great if you could tell us about the mathematical knowledge of the native pre-columbian cultures, such as the Mayas, Teotihuacans, Aztecs, and Mexicas, for all of these cultures had an accurate astronomical knowledge, building several astronomical observatories. Thank you.
Is there a similar simple way to prove the inverse of this theorem? I.e. If the sum of the two squares are equal to the third one,then the triangle is rectangle. Thank you.
Awesome
24:25 the sqrt root of 5 is especially hard to calculate using Greek or Roman numerals. The invention of the zero and the Hindu numerals are probably the greatest Mathematical discovery in human history.
I think I see what you mean. Dedekind's cuts is a more abstract way of defining the reals which seems to work because the field properties of the reals and the completeness axiom are satisfied. But to make the arguments to 'prove' these properties on needs certain assumptions. and as you say, some may be unjustified.
I'm in heaven! I have always wanted to understand the historical development of Maths
The problem is that when you look in detail at the "method" you describe, you find out that it is immersed in unjustified assumptions and illogical reasoning. In other words, the construction does not work. This important topic is addressed in my UA-cam series MathFoundations, starting around MF87.
Irrationality of √2 is "still not unresolved"
?
?
Didn't you mean, "still not resolved"?
I thought the same. So a double denial means it is resolved, right? 😃
@@Tubemanjac Strictly speaking, yeah.
Fred
I’ve never seen the Diophantus circle problem before or seen it applied in a calculus course. It must have been relegated to a problem or problem set I didn’t solve or think was significant. Does it also figure in Diophantus analysis (number theory?, Diophantine equations?).
thnx 4 sharing
Root 2 => 99/70 (approx). Thank you for this course, Professor.
Too bad I do not speak English, but I loved your lesson to understand the issues perfectly.