Professor Su, As requested by other viewers, can you please offer this course on edX or Coursera--millions like me would benefit. These are the best series of lectures on Real Analysis I have come across.
At 25:00, what's interesting about order is that this also goes back to ancient civilizations. Human beings didn't just wake up one day and magically have the law of trichotomy and transitivity pop into their heads, they associated numerals to sense-perception and realized that if you had 1 candy on a table and added a another candy on a table then you have more candies than you started with, and thus 2 > 1 and thus an order is born. There's no way to use formal logic to deduce this without getting into the work of Frege and Bertrand Russell, and ultimately I think most mathematicians accept sense-perception as an intuitive understanding of the natural and rational numbers and their properties as more fundamental than straight up logicism. And so, in light of this, we *define* the order relation to *match* our sense-perception, *but this doesn't explain where it comes from or why there exists an order relation,* (and perhaps if our biology were different, so would our mathematics) in order to try and speak of things we can't perceive, like higher dimensions and other spaces and structures. It's up to you to decide philosophically if this is the correct approach, but this is the approach that mathematics has taken, and it is the one that anyone in science and engineering uses today, and any technology you are currently using to read this works using this philosophical approach. And perhaps even pure logic itself is also rooted in sense-perception. It would seem to me that even things such as the law of excluded middle rely on our perceptual experience to be valid. On the other hand, things like perfect geometries, like perfect triangles, can not be made in real life. We can think them, but they don't exist, so where does the definition of the triangle come from? Not sure. We can perceive line segments, and then abstract the idea to form triangles I'd suspect. Same goes for any shape in geometry, but the Greeks were rationalists; believed math was God Given and that there existed a mathematical universe separate from sense-perception. Ultimately I think the Greeks are wrong about this, but they are right to say we can imagine things that don't exist and use them. That's how the first record of irrational numbers came about, imagining a perfect unit square (which can never be built in real life) and noticing the sqrt(2) as the diagonal (which can never exist exactly in real life either, only in our heads). Even computers just approximate it, because there's no sqrt(2), there might be 100,000,000 digits of sqrt(2) in something, but never is there the whole thing. Same story for complex numbers. And yet complex numbers let us solve any equation, and give us fractal geometry, which quite ironically, describes natural biological structures better than anything else. So it's not right so say that we can trust only empiricism or rationalism, the truth is that I think we must use both.
@@fndTenorio Yeah, but there's a problem with geometry. If we remain literal with our sense perception it can work out fine, but if we start trying to make up things we end up with irrational numbers. If we used real world measurements, it's perfectly fine to have a 'right triangle' with side lengths '12"' '12"' '17"'. And I'm sure construction workers do this all the time when building tiles for floors and whatever else.
I wish my teacher was that clear, spoke that well, was that interactive with the students and was at least half as enthusiastic about teaching as this guy is!!!
Wow, I feel blessed to have access to this series of lectures in 2021. It is just amazing how me an applied math student from Florida International University is learning from this lectures. Thanks
@@berkeunal5773 I don't think so, just his notes on the website, which don't always have everything he puts on the board. He seems to follow the book on many occasions, so that helps.
17:30 It's hard to figure out at first (assuming I got this right) that a isn't the same exact value as a', but that the entire fraction of which a is a part is equivalent to the fraction involving a'. With so many new concepts flying around, it can be like trying to construct a house from a tornado.
12 years late but I recently read a bit of it, using "increment" is an interesting take on constructing addition, so far it's good but it'll probably take a while for me to get used to
great lecturer. as a freshman in college i decided to go computer science because I was afraid of math like this, but now i find it the more compelling subject between the two.
@HarveyMuddCollegeEDU. Thanks for the great lectures. Even MIT OCW has nothing on Analysis. Are there higher resolutions available, though ? Sometimes it is hard to read what is on the written on the board.
thank u so much for the lecture..i am not a pure maths student but i have to do research in financial maths and i cant even understand the workings, mathematical writing and all...this really help me.
Just wondering at 37:37 when you say m`/n` - m/n is positive do you have to define what it means to be positive for rations (in the same way you did for integers?)
Good content, nice pace and very clear. You´re a great teacher! Thank you prof. Su. What was the textbook for this course? Sadly I do agree with the critique regarding the low resolution.
Why does he assume the integer 5 is the rational number 5/1 at 20:45. OK, I think he justifies to some degree in the next minute by saying that an integer n and an integer m, expressed as n/1 and m/1 respectively will belong to equivalence classes in Q satisfying the usual multiplication and addition rules. I think that's what he does?
It is AN order; it is not THE order. He's talking about pairs of integers, which are not necessarily rational numbers. So, (1,2) < (2,4) is valid by the definition of dictionary order, but (1,2) does not have to be 1/2 in this case. The / symbol isn't defined for integers, so 1/2 is like saying 1@2, it has no meaning in this case.
1:01:07 I have a question? Why integers does not follow M5 they all have additive inverse I think that property should talk about multiplication inverse because is what all M properties are talking about maybe is just a writing mistake because then it would make totally sense,correct me if I am wrong
The multiplicative inverse of, for example, 5 is 1/5. 1/5 is not an integer. So Z doesn't fulfill M5. The multiplicative inverse of 5/1 is 1/5. Both of these are rationals. So Q *does* fulfill M5.
Nice lectures. Why is equivalence defined as having these three properties. What's the intuition behind choosing just these three things to describe this notion of equivalence
I am a bit confused about the definition of the dictionary ordering. As par the definition stated, (2,3) < (3,4) but also (2,3) < (3,2). Should not the definition be ((a < c) and (b < d or b = d)) or ((a = c) and (b < d))?
That's both correct and incorrect. This is still math. The difference is that assembly languages came before C: C was designed deliberately to be an abstraction of assemblies. But real analysis came after stuff like set theory and arithmetic. In one sense, this is actually Java or Erlang, not assembly, because it's adding a layer of abstraction, not removing one.
The addition for Q is not the same as the addition for Z. "Addition" isn't "all additions". It's specifically for a given set of entities; in this case, groups of numbers. You can say "a/b + c/d = ad+bc/bd" because a, b, c, and d are all integers.
I can work with the low resolution, but I hope you get better videographers for your videos. The nauseatingly abrupt zooms and camera swings gives one the impression that they were trained by Michael Bay and Zach Snyder.
Every video lecture I've watched is this way. If you were in class you'd turn your head and look at the lecturer and focus where he directed your attention. This camera work emulates this. A wide angle shot would include all kinds of useless information and be distracting.
I mean. It's probably some poor kid working a camera on a tripod. You can tell they're trying to catch up to where Professor Su is standing, trying to guess where he's going to stand ahead of time, etc.
I recall that at my university they had a pretty crappy remote controlled camera. I’m assuming it is the case here also. There is no real camera on a tripod, just someone with some kind of joystick. Unfortunately the market for good, cheap(-ish) camera equipment hadn’t exploded at the time of the recording.
Really wouldn't call this a good lecture, he "teaches" us what we already know and gives us what we don't know and can't do as a homework :D Are you kidding me?
It seems you've mistaken your lack of comprehension for a profound insight. While this lecture strives to enlighten even the dimmest minds, it appears to have missed its mark in your case.
Professor Su,
As requested by other viewers, can you please offer this course on edX or Coursera--millions like me would benefit. These are the best series of lectures on Real Analysis I have come across.
That would certainly be "Mathematics for Human Flourishing"! If would gladly give $100 to a "go fund me" if Professor Su chooses to do it.
At 25:00, what's interesting about order is that this also goes back to ancient civilizations. Human beings didn't just wake up one day and magically have the law of trichotomy and transitivity pop into their heads, they associated numerals to sense-perception and realized that if you had 1 candy on a table and added a another candy on a table then you have more candies than you started with, and thus 2 > 1 and thus an order is born. There's no way to use formal logic to deduce this without getting into the work of Frege and Bertrand Russell, and ultimately I think most mathematicians accept sense-perception as an intuitive understanding of the natural and rational numbers and their properties as more fundamental than straight up logicism.
And so, in light of this, we *define* the order relation to *match* our sense-perception, *but this doesn't explain where it comes from or why there exists an order relation,* (and perhaps if our biology were different, so would our mathematics) in order to try and speak of things we can't perceive, like higher dimensions and other spaces and structures. It's up to you to decide philosophically if this is the correct approach, but this is the approach that mathematics has taken, and it is the one that anyone in science and engineering uses today, and any technology you are currently using to read this works using this philosophical approach.
And perhaps even pure logic itself is also rooted in sense-perception. It would seem to me that even things such as the law of excluded middle rely on our perceptual experience to be valid.
On the other hand, things like perfect geometries, like perfect triangles, can not be made in real life. We can think them, but they don't exist, so where does the definition of the triangle come from? Not sure. We can perceive line segments, and then abstract the idea to form triangles I'd suspect. Same goes for any shape in geometry, but the Greeks were rationalists; believed math was God Given and that there existed a mathematical universe separate from sense-perception. Ultimately I think the Greeks are wrong about this, but they are right to say we can imagine things that don't exist and use them. That's how the first record of irrational numbers came about, imagining a perfect unit square (which can never be built in real life) and noticing the sqrt(2) as the diagonal (which can never exist exactly in real life either, only in our heads). Even computers just approximate it, because there's no sqrt(2), there might be 100,000,000 digits of sqrt(2) in something, but never is there the whole thing. Same story for complex numbers. And yet complex numbers let us solve any equation, and give us fractal geometry, which quite ironically, describes natural biological structures better than anything else. So it's not right so say that we can trust only empiricism or rationalism, the truth is that I think we must use both.
Good news, my friend. It appears to me that you have the working mind of a mathematical philosopher. I had fun reading your exposition 🙏🏽☺️
Bravo! Great comment, sir!
Nice. We can detect edges, hence we can "count" objects that our brain recognizes as being the "same". Everything follows.
@@fndTenorio Yeah, but there's a problem with geometry. If we remain literal with our sense perception it can work out fine, but if we start trying to make up things we end up with irrational numbers. If we used real world measurements, it's perfectly fine to have a 'right triangle' with side lengths '12"' '12"' '17"'. And I'm sure construction workers do this all the time when building tiles for floors and whatever else.
I disagree with the idea that our biology defines mathematics
I wish my teacher was that clear, spoke that well, was that interactive with the students and was at least half as enthusiastic about teaching as this guy is!!!
Wow, I feel blessed to have access to this series of lectures in 2021. It is just amazing how me an applied math student from Florida International University is learning from this lectures. Thanks
This man taught me how to add (classes) and I smiled so hard. I love it when people do things that make my mind giggle 🤓😊🙏🏽
Just wanted to say what an excellent teacher he is.. I understand everything he shows.
Up there with Gilbert W Strang's linear algebra classes at MIT.
This guy really motivates what's important in proofs
These videos are great, but needs more resolution.
I'm telling ya, your gonna want that resolution!
Did we get it in 5 years? If yes where can I find them?
@@berkeunal5773 I don't think so, just his notes on the website, which don't always have everything he puts on the board. He seems to follow the book on many occasions, so that helps.
17:30 It's hard to figure out at first (assuming I got this right) that a isn't the same exact value as a', but that the entire fraction of which a is a part is equivalent to the fraction involving a'. With so many new concepts flying around, it can be like trying to construct a house from a tornado.
Terence Tao's book "Analysis" give a good description of the construction of real numbers.
12 years late but I recently read a bit of it, using "increment" is an interesting take on constructing addition, so far it's good but it'll probably take a while for me to get used to
great lecturer. as a freshman in college i decided to go computer science because I was afraid of math like this, but now i find it the more compelling subject between the two.
@HarveyMuddCollegeEDU.
Thanks for the great lectures. Even MIT OCW has nothing on Analysis.
Are there higher resolutions available, though ? Sometimes it is hard to read what is on the written on the board.
Wonderful lecture! Thanks a lot Professor Francis.
thank u so much for the lecture..i am not a pure maths student but i have to do research in financial maths and i cant even understand the workings, mathematical writing and all...this really help me.
These lectures are absolutely world class.
Just wondering at 37:37 when you say m`/n` - m/n is positive do you have to define what it means to be positive for rations (in the same way you did for integers?)
1:05:20 what is the property in multiplication and addition that preserves that order.
Good content, nice pace and very clear. You´re a great teacher! Thank you prof. Su. What was the textbook for this course?
Sadly I do agree with the critique regarding the low resolution.
+kiwanoish The textbook is "Principles of Mathematical Analysis" by Walter Rudin.
Ain't no bitches in that class.
Why does he assume the integer 5 is the rational number 5/1 at 20:45.
OK, I think he justifies to some degree in the next minute by saying that an integer n and an integer m, expressed as n/1 and m/1 respectively will belong to equivalence classes in Q satisfying the usual multiplication and addition rules.
I think that's what he does?
thanks Professor! You are a superhero for me.
Your slide on the field axioms shows (M5) talking about additive rather than multiplicative inverses.
Ty, your lectures are helping me a lot with my course in real analysis.
It is AN order; it is not THE order. He's talking about pairs of integers, which are not necessarily rational numbers. So, (1,2) < (2,4) is valid by the definition of dictionary order, but (1,2) does not have to be 1/2 in this case. The / symbol isn't defined for integers, so 1/2 is like saying 1@2, it has no meaning in this case.
U define the construction of rational number......such a good way.this is fabulus.
1:01:07 I have a question? Why integers does not follow M5 they all have additive inverse I think that property should talk about multiplication inverse because is what all M properties are talking about maybe is just a writing mistake because then it would make totally sense,correct me if I am wrong
The multiplicative inverse of, for example, 5 is 1/5. 1/5 is not an integer. So Z doesn't fulfill M5.
The multiplicative inverse of 5/1 is 1/5. Both of these are rationals. So Q *does* fulfill M5.
Or if you just meant "it should say multiplicative inverse", yes, that is correct. The slide has an error there.
@@Duiker36 yeah that was,he corrected the problem a few minutes later
Is the "dictionary order" defined (around 30:00) valid? This would make 1/2 < 2/4 < 3/6
we don't use the dictionary order because as you pointed out it is not well defined.
Dictionary order is an order over ordered pairs, not rationals
beautiful topic, amazing prof, almost laughable camera qualtiy
Nice lectures. Why is equivalence defined as having these three properties.
What's the intuition behind choosing just these three things to describe this notion of equivalence
Can someone help me with the "homework" at the beginning of the video?
using cancelation properties.
I can't find the notes on the blogspot page and can't see the blackboard on youtube.
Thanks for sharing this wonderful session ♥️🐈
I still haven't figured out the first homework that you gave us (Addition on Q equivalence check) , pls help sensei Su ...
I am a bit confused about the definition of the dictionary ordering. As par the definition stated, (2,3) < (3,4) but also (2,3) < (3,2). Should not the definition be ((a < c) and (b < d or b = d)) or ((a = c) and (b < d))?
What's wrong with having (2, 3) < (3, 4) and (2, 3) < (3, 2)? As per dictionary ordering, that's how it's defined.
Excellent Excellent job.....You are Super Hero!!!
vector mathematics is woefully under represented with ALT codes.
“Want: notion that does not depend on representatives chosen!”
Every one knows about the first YT video about the elephants....is this the 2nd YT video?
OMG why am I only finding these a nite b4 my exams..... uhh!!!! Thanks alot tho these vids have cleared up a lot of stuff
“Some of you have seen this proof before, in discrete mathematics...”
Me: no, but I saw it in high school once, then a ton on UA-cam 😂😂
Has anyone done the HW problems or know where the solutions are posted?
Are all the videos this bad or does the camera man get it together later on?
Thank You so much for these videos! They help tremendously! Come teach at my school thanks lol.
Is a course in abstract algebra a prerequisite for these lectures?
I lost my glasses the other day so these are too blurry for me to see, but they sound like great lectures!
Yes, x [quantity] (p/q)
@mohammadaggan Principles of Mathematical Analysis by Walter Rudin
In the proof of root 2 is rational
Why don't he put (p)^2=2m?
Can any one tell me?
This guy is awesome!
18:08 in 240p he makes the word dissapears with his hand!
Listen to it and you will get it.
great lecture indeed!
Great lectures but poor video resolution
If math was like c, this is assembly
That's both correct and incorrect. This is still math. The difference is that assembly languages came before C: C was designed deliberately to be an abstraction of assemblies. But real analysis came after stuff like set theory and arithmetic. In one sense, this is actually Java or Erlang, not assembly, because it's adding a layer of abstraction, not removing one.
Great way to view real analysis! Very enlightment.
Thanks a lot teacher.
I dont umderstand why you are allowed to define addition in terms of itself
The addition for Q is not the same as the addition for Z. "Addition" isn't "all additions". It's specifically for a given set of entities; in this case, groups of numbers. You can say "a/b + c/d = ad+bc/bd" because a, b, c, and d are all integers.
We're assuming we know how addition is defined for integers
I can't see anything. Recorded on a potato
Super impressive
I can work with the low resolution, but I hope you get better videographers for your videos. The nauseatingly abrupt zooms and camera swings gives one the impression that they were trained by Michael Bay and Zach Snyder.
Every video lecture I've watched is this way. If you were in class you'd turn your head and look at the lecturer and focus where he directed your attention. This camera work emulates this. A wide angle shot would include all kinds of useless information and be distracting.
I mean. It's probably some poor kid working a camera on a tripod. You can tell they're trying to catch up to where Professor Su is standing, trying to guess where he's going to stand ahead of time, etc.
I recall that at my university they had a pretty crappy remote controlled camera. I’m assuming it is the case here also. There is no real camera on a tripod, just someone with some kind of joystick. Unfortunately the market for good, cheap(-ish) camera equipment hadn’t exploded at the time of the recording.
A very excellent videos. Very high math level. But the definition of the video is not good!.
Thank you
The word class is not defined before.
en.wikipedia.org/wiki/Class_(set_theory)
It is in the first lecture.
mmkay
Really wouldn't call this a good lecture, he "teaches" us what we already know and gives us what we don't know and can't do as a homework :D Are you kidding me?
It seems you've mistaken your lack of comprehension for a profound insight. While this lecture strives to enlighten even the dimmest minds, it appears to have missed its mark in your case.
the video is worse that useless, it's all blurry. might as well be just an audio lecture
These videos might as well be unwatchable.