amazing, every time i watch a lecture on relatively simple mathematical concepts from a slightly different perspective i always learn a lot of new things. probably shows just how bad my foundations truly are. thank you

Mind blown. I just started studying calculus at a local community college after graduating from college almost 20 years ago. Watching this video I think I have experienced what people mean when they say the beauty of math

This parametrization of a unit circle is a very important concept. I talk about it also in some of my other videos, in particular MF29, also WildTrig 14 and 15, and it will be featuring also in the next FMP lecture (13).

@Maheonehooestse-HolyFireMan

4 жыл бұрын

Can you figure the math out on this one? kzread.info/dash/bejne/Z6GlyMaIqqTcfZM.html

@robharwood3538

3 жыл бұрын

@@Maheonehooestse-HolyFireMan There's no math there, only superstition.

Do you have any idea how much help this work you are sharing is to many of us around the world? Thank you so much.

You've plucked a gem from the Greeks, for modern-day calculus! Bravo!

i decided to use the spare time this pandemic season to my advantage. i landed on Dr. Wildberger's videos. Thank you almighty and Dr Wildberger, thks 4 this grand effort

Professor Wildberger, thank you for all these videos!

That’s fascinating! The connections between the Pythagorus’ Therom and the parametrisation of the unit circle, which is essentially a mapping from a circle to a straight line, which then reminds me of the same trick of Riemann in his mapping from a sphere to a plane! It’s so beautiful. Thank you professor.

this is the most interesting corner of the internet -- thank you for sharing your insights Dr Wildberger

My name is Barry Allen, and I'm the fastest man alive 10:27

18:29 in case someone wonders how he got x(zero)+1=2/(1+t^2) x(zero)=(1-t^2)/(1+t^2) Add 1 to both sides x(zero)+1=(1-t^2)/1+t^2)+1 On the right side, multiply the second term (1) with (1+t^2)/(1+t^2) x(zero)+1=(1-t^2)/1+t^2)+(1+t^2)/(1+t^2) t^2-t^2 cancels out and he's left with 1+1 in the numerator

@Bluedune86

7 жыл бұрын

THANK YOU!

Never did well in school with math - I just don't have the natural mindset to comprehend it easily, but I find these lectures absolutely fascinating. Thanks so much for uploading them.

When the first time I read this I was stunned how Diophantus thought about and how he used slope of the line to paramterize the circle. Amazing Work Professor. Thanks

Great Course ! Thanks Dr. Wildberger !

A parametrization of a curve is a specific way to get hold of it concretely. It might even be argued that without a parametrization we do not have a curve.

@glimpseofanpilgrimage5323

4 жыл бұрын

can you give a more of simple explaination on what is parametrization as i am not much familiar with notations in maths

@electric_sand

2 жыл бұрын

@@mathewbriguglio9922 Great. Thanks

I like a lot the course of the course, I mean that it starts for some very simple and well known mathematical concepts and theorems and after a few steps- in less than a half hour - it comes to advanced mathematics. It is great!

Lecture was great. I've always been weak on Parametrization & really liked this treatment. Stilwell text arrived today. It's also very nice. Every section seems to be one or two pages of text followed by a few problems.

Muchas gracias por subir éstos videos!

Mr. Wildberger It was the best of its kind - Parametrization of unit circle. Thanks

Sincere thanks for this. As a child, math fascinated me. However, that interest waned quickly after high school. Setting this with historical/philosophical context has drawn me in. We shall see how long my math knowledge allows me to follow you. Cheers.

Thank you for the uploads! The last math class I took was pre-algebra and that was over a decade and a half ago. Searching for material to continue with algebra this came up. I have to poke around on the net sometimes to get the terminology used in places but this...is making too much sense. Any other Americans out there who did not get much math should watch vids on finding square roots the old fashion way. It's easy, also might make a fun project for neophyte programmers.

As it is evident, the name "Pythagorean triple" derives from the Pythagoras' theorem which states that every right triangle has side lengths satisfying the basic formula for this kind of triples. That is to say, a Pythagorean triple describes the three integer side lengths of a right triangle. But, as it is usual in mathematics, there are also extensions or generalizations of this result, as for example, the Pythagorean quadruple, and much more general, the Pythagorean n-tuple. Another type of generalization of this triple is when the corresponding exponents are positive integers strictly greater than two like it appears in the Fermat's Last Theorem. The rational parametrization of the unit circle was made in order to find rational solutions for the basic equation of the circle by obtaining rational points on the curve. It is interesting to know that this result was obtained from previous algebraic works done by Diophantus about quadratic equations.

these videos are awesome. I've just decided I'm going to go through them in order and be math history literate! :D Thank you for posting these, truly a helping step toward an autodidactic society :)

Very interesting lesson, thanks a lot!

Just amazing !!!👏👏👏👏

Thanks Mr. Wildberger

Please keep the good work Amazing lecture prof Berger

@morpheus6749

6 жыл бұрын

The name is Wildberger.

he looks so proud up there when he does a math problem all on his own

Great lecture!

Cool, I'm gonna check those out right now. Thanks NJ.

I love your expression at 20:55, you can see your enjoyment of the math. Thank you :)

This is magical.

Very, very nice. Thank you!

Awesome video!!

Sir you great. Thanks for these interesting video

Sometimes negative values of distance are better interpreted as displacements. For example, the x and y coordinates of a point are not really distances, but displacements, so can be either positive or negative. Does this help?

Fascinating videos. They are very helpful for Project Euler as well..

Beautiful!

wonderful lecture :)

Good greats Dr Wildberger. I have a question: The Pythagoras' theorem serves to find a dihedrical angle between two oblique planes line intersection not given? Is it possible? Congratulations for you site. Good memories from ICGG 2014, Innsbruck. With my best regards from Colombia. Germán Valencia García - Architect.

I'd really need to study calculus (all aspects of it) as well as trigonometry/geometry (all aspects of this too). Those are the main areas of mathematics which I have NOT studied well and planning to study 'em more. But I'm so interested in other areas too, bahh, I don't know what to study. How to best spend my time?! So hard, so hard. Thanks for this lecture Doctor Norman!

@MichaelGoldenberg

8 жыл бұрын

+Keni Angervo Check out Norm Gross' free lectures from 1970 (MIT Open Courses) on KZread: Calculus Revisited.

Excellent professor

i'm studying math just for fun !!! but i feel like a genius XD

Mr Wildberger, that is really an elegant way of thinking about the parametrization of unit circle. Also, it is an excellent presentation about where tangent half-angle substitution comes from, i.e. t=tan(theta/2). If I am not mistaken, it is also called Weierstrass substitution, right? Elegant in mathematics means that the derivation is simple, but it leads to profound results, right? Thank you.

Thank you professor. You really clarified the line of thoughts with the discipline of geometry. Is there a connection between Diophantus appraoch and algebraic geometry as suggested by Papus and Desargue.

Thales was a Pythagoras teacher. Thank you for your information.

Thnkx.professor u r assential.asert for us stay blessd live long

thank you so much prof. 😄😄😄

:O! mind-blown!

15/8/2023. Magistrale conférence. On pourrait y ajouter ma découverte du 5 juin 2022 à savoir: De Z² = X² + Y² ,on peut désormais écrire : Zpuissance(N)= X² + Y² ,avec +1

That is good to know, because I was not following that presentation at all. I came here looking for some math history, and yes with lots of math, but I was about ready to ditch the series because that whole discussion of the circle was too confusing. I'm now hoping that the rest of the series will be easier to follow. Thanks for the heads up!

If you add any two unit fractions with denominators n and n+2, then the resulting fraction, say m/n, will be the first two numbers of a pythagorean triple. I think this was a way used to generate triples before Euclid's method - but I don't know for sure. Perhaps you could then parameterise a circle using just one parameter - n

Very good.

i learn something, thx. i need 1 week to understand all of this teach. it speak of trigonometry who come from reductio ad absurdum

Hello profesor Wildberger! Thanks for all of your video. I am very happy that in youtube I can find and learn so good and usefull things about Maths. Only today I found your canal and I watched the first part of your collection of Math History (first two videos). I am not very good at English because it is not my native language. And I am not very good at Maths too because I am still studying and I am only in begining on Maths. I realy like your way of teaching. Your explanation of everything is very good. Even I am able to understand everything very clearly. But there was only one thing that I didn't understand. Probably I have a mistake, but atleast at the moment I think that you have one mistake in your proof. I am thinking about this video time after 8:30. At this moment you started to devide the equation with 'm' in power 2. In my understanding you should devide both sides with 'm' in power 2. not only the left side. In video you devided only left side of equation, but in right side you didn't do anything with number 1. In my opinion that is incorrect, but probable I had made mistake somewhere. I only want to understand where is my mistake. Hope to hear answer from you soon. Thanks®ards Marta.

@jangwlee

Жыл бұрын

When there is a rational number (e.g. 4/8), what we can do is to divide the numerator by its common factor (e.g. 4) and also divide the denominator by 4 .. so it ends up having 1/2. So, the professor was using the fact: for a given rational number, if we divide the same number to both numerator and denominator will result in the same number. (not about applying the same operation for both right-hand side and left-hand side)

my god... diophantus is the one who's really responsible for the "Weierstrass substitution" thanks for the great videos! I'll be watching them all :D I've gone up through DiffEQ, LinAlg, and Vector Calc (Green's, Stokes', & Divergence Thms.), but hearing the history behind all the math I've learned is awesome.

@njwildberger thank you, good sir.

That cool with mentioning how the calculus derivation where relevant.

good,i like it.

Thank you

Insightful video. Thank you very much for the tutorial. Worthwhile formula: (m^2 - n^2)^2 + (2mn)^2 = (m^2 + n^2)^2 I am writing that into my old formula booklet. (m^4 - 2m^2.n^2 + n^4) + 4m^2.n^2 = m^4 + 2m^2.n^2 + n^4 m^4 + 2m^2.n^2 + n^4 = m^4 + 2m^2.n^2 + n^4

17:17 it’s the Vieta’s theorem. Product and sum of roots of a quadratic polynomial.

Hi MrDesignMagic Yes you are right.

wow....very helpful and easy to understand. you re great mathematician..what grade students are there in the class sir?

This is insane holy.

Hello Norman, going through this again, this time with pen and paper(long story actually). I've actually never heard of parameterization in school; i remember some sort of parameterization when I self studied calculus with Morris Kline's Calculus book. I'm thinking that parameterization has to do with the opposite of polarization of equations? It's like you're re-expressing equations in terms of cartesian coordinates?

So the dashed line from -1 to point y becomes tangent to the circle when infinite. Cool

Appreciate you doing these videos, if you can could you please do some Number theory videos :) Thank youuuuuuuuu

So if you have a linear function, its zero A, and another random point B there is a relationship to a quadratic function with zeroes A and B?

I've watched the whole history of math series, but now I'm going over it it and taking detailed notes. I breezed through 1A and then came to "parameterization of the unit circle." I think this may be the most dense lecture in the whole series. The calculations are all basic algebra with a little bit of trig, but there's a lot of calculation and the underlying concepts are far reaching. I think I'm going to be reviewing this lecture many times.

interesting that the random number theory theorem to generate pythagorean triples is related to ... parametrization of the circle

whats the song from? I quite cant grasp where i heard this from.

Do you have the numer theory proof for this?

Cool lecture. Does anybody know what song the music at the beginning and ending credits is from?

@jordanmarloy9748

7 жыл бұрын

Very cool indeed, and to answer your question, you're hearing Haydn's string quartet No. 61 = )

Hi, can you please explain (or refer to an explanation of) the transformation from the euclidian identity of right triangles to the circle? I didn't follow at all... Thanks 😁

@jangwlee

Жыл бұрын

Here is some story which I interpret this lecture: 1. First, Euclidean identity is introduced using m and n 2. Unit circle equation is introduced.. every point in unit circle (x,y) satisfies x^2 + y^2 = 1 equation -> actually this equation also came from Pythagoras’ theorem since (0,0), (0,x) (x,y) points consists of the right triangle and x and y are legs’ length and 1 is hypotenus’s length 3. Now, we are curious about how parameterization of unit circle is possible (btw, parameterization is a way to express line using only a single parameter) 4. The professor went to back to Euclidean equation and was able to come up with a parameterization form using a single parameter “t” So, at this moment, we can realize that there are two ways of parameterization of unit circle: the one the professor derived during #4 step using a single parameter “t” and another “famous” one using a angle (theta), i.e. (cos theta, sin theta) 5. The professor went on how Diophantus came up w/ unit circle’s parameterization, which is the third way.. (i.e. intercept between the line and the unit circle) but it ends up w/ the same result like #4 6. But, interestingly Diophantus’s approach provides us with geometric meaning of “t” (btw note that Euclidean’s derivation does not provide this geometric meaning), this “t” is actually slope of the line (which passes (-1,0) and a point in the circle) 7. Note that “t” is also interpreted as tan(theta/2) when (x,y) = (cos theta, sin theta) 8. Finally, we compare the well known parameterized formula (cos theta, sin theta) and the one we derived using “t” and showed the relationship between cos theta, sin theta and tan(theta/2) see here math.stackexchange.com/questions/1367487/prove-that-sin-x-2t-1t2-and-cos-x-1-t2-1t2-t-tanx-2 .. how boring to show those relationship.. but, here the professor proved those relationships in a different manner.. which looks amazing

Hi proffessor Cn u plz answer me ... Why we use tan to find argumnt in cmplex analysis Also why we use parametric equation to find cmplex plain in a prpndiculr line

I think these ancient guys has some problem. How on earth they came up with this. It is remarkable, it is unbelievable. Human being's at their best.

Can you put an annotation in the video pointing this out?

It goes to show what you can learn if you will but study original Euclid. It is a pity it is not taught in schools anymore. Knowing Euclid makes the modern material much easier to understand. I don't believe many modern mathematicians really appreciate what is in those books.

Hi AZGG7, It's Haydn.

6 trigonometric ratios where they are in circle

you saved me!

very nice , it's a good break from KhanAcademy videos

0:22 teach me about Pythagorean triples Mr. Wildberger

I always judged book three of Euclid's Elements as the most boring book of the Elements. Knowing how to find the center of a cirlce is kind of cool, but all the rest seemed not so interesting; well, your showing the relation between angle theta and angle/2 certainly gives added importance! On the other hand, I've often found myself saying "no piece of mathematics proven by logic can ever be considered 'unimportant".

That's what pause is for. :D (I had to pause it, go back, rewatch a section until I got it.) I stopped at Algebra... I wish real-life math classes had pause! :D

@MWTan-ho2to

3 жыл бұрын

8 years later you have your wish haha

cool

trying to do this as a playlist is backwards

@9:09 into the video, it seems (2mn)^2 /m^2 = (2n)^2 (m^2 /m^2) =(2n)^2 & not= (2n/m)^2 ?Could someone point out my or Dr Wildberger's error? Thks

@Anakim416

5 жыл бұрын

Tom B you’re dividing 2mn by m^2 not (2mn)^2. The m on the top is canceled out by one of the m’s in m^2 so you’re left with one m on the bottom and one n on the top.

What they don't tell you anywhere is that there is a different Pythagorean theorem for any number of dimensions greater than or equal to two. In three dimensions, A^2 + B^2 + C^2 = S^2, where A, B, and C are the areas of the right triangle sides of a right tetrahedron, and S is the area of the remaining side. I discovered this in jail.

Uhhh this guy like knows stuff.

a little too fast-paced for my neurons (I stopped my math education at calculus 2)... very enjoyable nevertheless! Thanks for posting!

4:27 Unexpected Borat reference

But, Diophantus get coordinates of intersection point, [2t/(1+t^2), (t^2-1)/(t^2+1)], not [2t/(1+t^2), (1-t^2)/(t^2+1)]. Where is the trick?

5:04 I don't get why 8 and not 10 (2mn)2 M=4 N=1 So it's 10

@jangwlee

Жыл бұрын

Please note that the formula is (2mn)^2 not (2mn)*2 so it is 8^2

You talk about existence of square root of 2 in the previous lecture Mr Wildberger. Writing the square root 2 symbol doesn't mean it exists, right? What about drawing? Drawing a triangle on the board doesn't mean it exists, right? How do you know a right-angled triangle with the side length 1 exists? Specially... what is the meaning of the word existence? Answer this one.... please. PLEASE.

@njwildberger

10 ай бұрын

The crucial question here is not about existence but rather about definition.

@ZiroOne-hw7iw

10 ай бұрын

@@njwildberger you used the word existence. So I'm requesting its definition.

Hold on you get an undefined if you reduce parameter.

Sir, how Euclid got that identity 5:07

@jangwlee

Жыл бұрын

Because I guess he was smart and was interested in it😊 The equation is identity as the professor pointed out and you pointed out, which means anyone may come up w/ it.

The worst thing about watching on KZread I that I can't ask a question.

@jangwlee

Жыл бұрын

Yeah.. that could be true but you can ask a question here in the comment so that others may answer it .. prob. not instantly but sometime later 😊

I will be honest. You lost me. Math is very hard for me. I was in special ed my entire life, so I have to work my way up. I am still going to go through the videos though. I am hoping that building a historical framework will help to clarify math for me. I have one question. I don't know if you know this, but why did some greeks think the soul was a moving number? Aristotle mentions this in his metaphysics.