Realizing that you can’t just define a fraction as a/b because that would mean we have the division operation (which is not a part of the integer group) is really illuminating. That’s why we construct an equivalence class of ordered pairs to represent a/b, and the set of all such equivalence classes in Q. I definitely haven’t thought about it like that before. Thank you, Professor Francis! You’re a legendary instructor, I can already tell 😭❤️🙏🏽🔥🤓
I see that Z is just a set of numbers, without "division" being part of that set, so we could not just say a/b and assume this was division. But a few minutes later he defines equivalence sets using multiplication (p*n = q*m). Why can you use multiplication in the second case when division was not permitted in the first. Can anyone offer any insight please? Is it because Z is closed under multiplication but not under division?
Just to write this out a little further, it's because if you have nothing except the set of all integers, you know how to add two integers, how to subtract two integers, and how to multiply two integers, because the result is another integer. BUT you cannot divide two integers, because the result *could* be an integer, but there exist pairs for which the result is not an integer. Concretely, I can do 1+2, 1-2, 1*2 and get 3, -1, and 2, but if I try 1/2, the result is not an integer. That's what "closed under multiplication" means in Paul's comment. Given an operation (multiplication), applying it to members of the set Z yields members of the set Z. Another operation (e.g. division or exponentiation) upon members of the set Z will sometimes yield members outside the set Z.
@@paulb75 I don't know how much abstract algebra you've had but the reason why certain (binary) operations are permitted on the integers makes sense if you talk about integers in the context of group theory. If you have the set of integers and you've defined the function f:Z x Z---->Z as the binary operation of integer multiplication on the set, then an inverse element doesn't exist for every a in Z which is why the integers under multiplication don't form a group (they're a monoid, which has less restrictive requirements than a group does). The multiplicative inverse (aka division) under the integers doesn't exist on the set in general, hence why the integers are closed under multiplication and not division.
At Harvey Mudd, classes numbered 0-99 are core (gen-ed) courses, or introductory/basic courses. All other undergraduate courses are numbered 100-199, and are organized by topic rather than difficulty. So Math 13x is the analysis sequence, Math 10x is the discrete math sequence, etc.
I wanted to make some joke about how the professor lied in the first minute of the video when he said, "Real Analysis is an exciting subject," but this guy taught one of the most engaging lectures I've ever seen on arguably the most difficult subject I know of in undergrad.
This is real teaching. I teach myself and learned so much how to teach better from these lectures in addition to better understanding of some of the (more advanced) material. Thank you.
@KillerNedDude f(x)=1 for all x in R satisfies the condition for a function: for a given input x, there is a unique output: 1. It does not have to be the other way around: a function has to turn an input into a unique output, not the other way around (if it does, it's a bijection). In a normal function plot, this means: on every vertical line (so a specific x), there can be at most one point y=f(x) on the plot (which f(x)=1 for all x satisfies).
@KillerNedDude That's a good question. Strictly speaking, you should also give the domain and codomain when giving a function. The arcsin function is normally defined to be from the input interval [-1,1] to the output interval [-pi/2,pi/2], in which case it has a unique output for every input (and is even a bijection). It's like the square root function, which goes from all non-negative real numbers to just the non-negative real numbers, in stead of giving 2 values for every function.
The homomorphism is definitely not onto from Z into Q, but it is into the subgroup of numbers in Q of the form n/1. If you can prove that this is a subgroup of Q and the function is an isomorphism (trivial), you have proven that Z is isomorphic with the whole numbers in Q, which is exactly what you would expect.
Antony Foster No, he says it's "not a set per se". As you show, it can be defined in terms of sets but the point he's making is that the ordered pair (a,b) shouldn't be confused with the set {a,b}, since {a,b}={b,a} but (a,b)!=(b,a).
@KillerNedDude He says that, given a specific input, a function has a unique output, which is correct. A function is surjective if every member of its codomain is the function value of at least one input, and a function is bijective if every member of its codomain can be traced back to exactly one input. So f(x)=x^2 (R to R) is a function, but not bijective because (2)^2=(-2)^2=4, and 4 cannot be traced back to one unique input. It's not even surjective, because no x satisfies f(x)= -1.
Awesome! Grateful that I found this set of lectures. So what, the video is poor - LISTEN and LEARN and hit pause if need be. Looking forward to the rest :)
Now here's a professor that knows how to teach! ...he does know the golden rule: if you can make it fun...why not? Some other guys just seem like they are teaching a wall...instead of groups of students; that, they can do at their homes! Nice class, even though I would've love to some more light...but I guess its ok...you know, its free!
Mind blown at 48:00... "Maybe we want to set up fractions defined in terms of equivalence relations." Holy crap! Lightbulb moment. Then each pair of equivalent ratios belong to some equivalence class, that might have lots of elements; call that equivalence class *the ratio, for instance, 1/3.* "Let *Q* be the set of *all such equivalence classes* (!!!!!!). ... You might also want to say how *Z* is embedded in *Q,* so that n/1 e *Q* [is an equivalence relation symmetric with] n e *Z.*" Wow! Only the first day and I'm already impressed. Just keep in mind, though, that without intuitive ideas about division already obtained from older civilizations, like Egypt, Babylon and Greece, we couldn't have ever come up with this logical construction. So, in this way we don't "explain why or how" division works, we just take division as a fact, and set up logical rules to make sure more complicated things that rely on division work. The only explanation for arithmetic like division is our intuition and sense-perception. It's important not to lose sight of this, and it plays a role in philosophy too. This is why we learn k-12 algebra before real analysis or abstract algebra, because these latter classes don't explain the stuff in k-12 math, they just build logic on top of it. This same argument could apply to calculus, but like the quote shown in the beginning of class, most people "draw the line" at the rational or natural numbers as far as intuition and sense-perception is concerned, and chalk up anything that uses real numbers as no longer sense-perceptible, and therefore some people believe calculus should be learned only with full rigour and logical construction saying that the real numbers have 0 intuitive properties, and others say that you still can use sense-perception to intuitively learn about real numbers. So, I'm not sure where I stand, but I think I lean more in the camp of "real numbers are not intuitive, and therefore should be learned only with full logical construction." This isn't to say that the uses of calculus are not intuitive, I think things like the Integral are incredibly intuitive/obvious, but the math used to write it down is not. Neither are infinitely non-repeating decimals. So it's a funny thing. Often the concepts are very intuitive, but the math to actually get an answer suddenly involves all this weird unintuitive crap. So, is calculus just the intuitive idea of adding up a bunch of rectangles, or is the weird unintuitive crap of the sum of an infinite series as a converging infinite sequence of partial sums, potentially never once repeating a value, forever? I don't think anyone knows the answer to this question. But I do think that Newton and Leibniz would say something like "don't worry about the details, just think of the idea and allow it to work" and it does give the right answer regardless of the details; so this view is basically valid. It's just like Zeno's Paradox and the whole idea that if an Archer shoots an arrow at a target then it could never reach the target because it always has to go half the distance and it'd never reach it; but on the contrary, if you shoot an arrow you better not be standing in front of that target, because if you are, not only will that arrow hit the target, it'll also go through your flesh on the way there. And so rational logic often contradicts empirical evidence. This is the crux of the philosophical debate of empiricism vs rationalism. I have to admit that I think this debate has less to do with the truth, and more to do with how our brains work. Because any human being can intuitively do calculus and classical physics; every student does it in gym class when they shoot a basketball, and every athlete is taking derivatives subconsciously constantly. And anyone who drives a car and figures out how much distance they need to stop does it. Yet, when we try to consciously sit down and do it fully aware, those same athletes who can make 90% of their baskets can't pass calculus class on paper in the classroom. That's odd right? Likewise, there are probably tons of mathematicians who can consciously write down and solve any calculus problem without breaking a sweat, but then they'd go and miss all their baskets. This relationship between subconscious math and fully conscious math is interesting to me. There's definitely a sense in which everyone is doing calculus and physics every day of their lives, but there is another sense in which it seems the relationship between doing it on paper, and *actually doing it in real life* is inversely proportional. Anyone can drive a car, but not everyone can ace calculus class. Any mathematician can pass calculus class, but not all mathematicians can shoot a basketball. Wierd.
I think there are basically two "answers" to this that are the same thing: 1. muscle memory helps encode the "subconscious math" you speak of, sort of in the same way we used look up trigonometric functions on tables. A basketball player can put the ball in the net because they've practiced it a lot, so their muscles told the brain what configuration they used to do the throw and the brain crunched those numbers. 2. first-person perspective is encoded into the "subconscious math". An athlete isn't generalizing their basketball throws into all possible basketball throws ever, let alone into ideal parabolic flight paths informed by gravitational constants corrected for air frictions. They're synthesizing basketball throws using their own body, standing on a flat surface, aiming at a specific point from inside a range of distances. I think that dodging specificity is part of what makes mathematics astounding. It extracts a simple essence out of everyday experience and shows how that essence can show up in lots of other places, too.
Thank you so much for these lectures! If I got a good grade in my analysis 1 course it's thanks to them. Now I'm reviewing the entire series of videos for analysis 2. It would really help us students if you could upload something similar for other topics too :-)
I though Kronecker meant that we had to assume that naturals are just there and build on that - as in, naturals are sort of like existing because they are very basic and man created the more complex math that somehow is based on the naturals, like how the rationals are an extension of the integers which themselves are an extension of the naturals and the naturals being the most basic - the first cause in some sense,
I think what Kronecker meant, or should've meant, is that bits are fundamental in the universe. Either something exists, or it doesn't (1 or 0). All other numbers are either derived, or are functions and not numbers in and of themselves. Try defining any other number without constructing it. This lecture is all about constructing numbers.
Forgive the poor production and just keep watching. This guy is the one. Sometimes I think he may be trying to tell us something with the fuzzy resolution and strange aspect ratio.
2:29 I think the stdudent does have a point there, In the real world everything is disrete and wheunting atomsn you zoom far enough you'll essentially be a collection of atoms. Real numbers and Continuity are just an approximation of the very dense collection of atoms as continous matter.
At 41:20, that's a subtle point! There can be relations that are not functions! We usually call those relations "curves" when talking about parametric representations of curves. Remember, a function is only a *relation* that has one unique output for each input, *by definition.* Therefore, not all relations are functions, but all functions are relations, and, functions are a special case of a relation! For instance, the graph of a Circle represents a relation that *is not a function* because for each input x, there exists two corresponding outputs, therefore the definition of a function is not satisfied, and the equation for a circle in euclidean space is *not a function,* but it *is a relation.*
yep, this is when you solve for y in the equation of a circle; you get two distinct functions, one corresponding to the upper half and the other to the lower half.
Thank you so much, these lectures are helping me prepare for my analysis course! I had some difficulty understanding compactness with just the book and these were so so helpful!
1:38, "God created the integers," what he meant was: we need to figure it out the rest, how we are going to use them, and applied them into the real world.
I think he meant other constructs like the complex number, which like it's cousins-vectors and quarternions-were harder to get your head around without context(polar co-ordinates, general rotations) and also were not that easy to apply in general (transformation matrices, jacobians) .It's funny that he has a tensor named after him.
At 53:58 he said that extending the Integers with the construction of the Rationals amounts to finding an isomorphism from Z into Q given by n --> n/1 (assuming we have been told how to operate on the rationals). Wouldn't that map be considered a homomorphism since it is not bijective (not onto)? I'm not trying to be picky. I just completed my first semester of modern algebra and want to make sure I understand this correctly.
Not exactly. An equivalence relation implies a set of equivalence classes and vice versa. So when you have one, you have the other, but the elements in an equivalence relation are ordered pairs in the Cartesian product of a set (with itself) but the elements in an equivalence class are elements in the original set.
Suppose we have a set X and a relation R. For every element of X we can define an equivalence class over them. Let's call a generic element of this set as "a". So an equivalence relation is defined as the set of all elements of X which are related to "a". We denote this as [a] = {x in X; xRa}
What Kronecker meant was -God gave us Integers, meaning Integers are the most basic intuition when it comes to math, i.e. counting. quantifying objects , differentiating between one and many, is what is the basic form of math itself, that existed in nature, long before man started deriving more complex relations and formulae that aided calculations of natural phenomena.
Thanks for a nice explanation. My lecture is using Mathematical Analysis I by Vladimir A. Zorich, which is considered extremely difficult to understand by freshman like me ;( Principle of Mathematical Analysis by Walter Rudin seems understandable and easy to catch up for me. Currently I am struggling with Mathematical Analysis and re-learn all the principles and theorems (Proving them is my everyday homework). Hope you can make more such videos and do remember to update the video quality. If I were not focusing on watching your video then I might not catch up what you had written on the board =)
@stijnisgoed but then what is unique about the outputs of say the inverse sine function which are the same for many different inputs i.e. asin(0) = 0, pi or npi (for all n in Z).
@stijnisgoed Cheers that makes sense, but what about say f(x) = 1 (for all x in R), is this not a function? Also are functions such as f(x) = { 0 if x is even, 1 if x is odd} (for all x in Z) not valid functions? (I'm guessing I probably have to study the course more to find out).
Prof made a mistake on the board when he wrote down the "definition" of the subset....he wrote A is a subset of B if x in A then x is in B. You need a universal quantifier right in front to indicate that you are talking about every element in A, is also in B, not just a particular x, or some x.
Not sure about his description of functions in this lecture, he say's that a function should have a unique output, surley this is only the case with a bijective function.
I am self-learning analysis by working through Spivak's Calculus. Should I wait until I finish the book to watch these lectures or do they complement the textbook?
The lecturer states that one should communicate math in whole sentences and not (e.g.) just in symbols. This is particularly true when the communication is done exclusively in writing. What the lecturer doesn't state explicitly is that in a lecture and/or oral exam, your communication is your narration plus your writing, and it is those two elements together which should fully and completely convey your ideas. That's why using shorthand notation on the blackboard without a written explanation (provided you give an oral one) is more appropriate than doing the same in your research papers (I assume) and your written homework.
We assume (1) pn=mq and (2) mb=an. The goal is pb=aq. If we multiply pn and mb, we should get mq*an, due to substitution. I.e. (3) pn*mb=mq*an. Rearranging that for clarity, we have (4) (mn)*(pb)=(mn)*(aq). Because we can cancel a term that multiplies the entirety of both sides (mn), we're left with (5) pb=aq. That's our goal, so we've proved transitivity. (I actually had to look up the answer twice before I realized what was happening.)
since pn = qm and mb = na, then (pn mb) must equal to (qm na), using cancelation law, you cancel m and n, it remains (pb=qa) which is what we want to show. QED
"using cancelation law, you cancel m and n", you can only use cancelation law for non-zero integers. n is guaranteed to be non-zero but for m you need a little bit more work.
in the case that m is not 0, the cancellation law would hold, but in the case that m is zero, the equation still holds true since 0 = 0. is this not sufficient?
How does that relate to division though? Like how do we get from 1/3 to 0.333.... 1/3 just seems like a symbolic representation without anything to do with division
![Путин и ПЕСКОВ ждут звонка от ТРАМПА 😁 [Пародия]](/img/n.gif)
Realizing that you can’t just define a fraction as a/b because that would mean we have the division operation (which is not a part of the integer group) is really illuminating. That’s why we construct an equivalence class of ordered pairs to represent a/b, and the set of all such equivalence classes in Q. I definitely haven’t thought about it like that before. Thank you, Professor Francis! You’re a legendary instructor, I can already tell 😭❤️🙏🏽🔥🤓
I see that Z is just a set of numbers, without "division" being part of that set, so we could not just say a/b and assume this was division. But a few minutes later he defines equivalence sets using multiplication (p*n = q*m). Why can you use multiplication in the second case when division was not permitted in the first. Can anyone offer any insight please? Is it because Z is closed under multiplication but not under division?
@@paulb75 That last part is exactly it.
Just to write this out a little further, it's because if you have nothing except the set of all integers, you know how to add two integers, how to subtract two integers, and how to multiply two integers, because the result is another integer. BUT you cannot divide two integers, because the result *could* be an integer, but there exist pairs for which the result is not an integer. Concretely, I can do 1+2, 1-2, 1*2 and get 3, -1, and 2, but if I try 1/2, the result is not an integer.
That's what "closed under multiplication" means in Paul's comment. Given an operation (multiplication), applying it to members of the set Z yields members of the set Z. Another operation (e.g. division or exponentiation) upon members of the set Z will sometimes yield members outside the set Z.
@@paulb75 I don't know how much abstract algebra you've had but the reason why certain (binary) operations are permitted on the integers makes sense if you talk about integers in the context of group theory. If you have the set of integers and you've defined the function f:Z x Z---->Z as the binary operation of integer multiplication on the set, then an inverse element doesn't exist for every a in Z which is why the integers under multiplication don't form a group (they're a monoid, which has less restrictive requirements than a group does). The multiplicative inverse (aka division) under the integers doesn't exist on the set in general, hence why the integers are closed under multiplication and not division.
@@jacobflores8666 nice answer
At Harvey Mudd, classes numbered 0-99 are core (gen-ed) courses, or introductory/basic courses. All other undergraduate courses are numbered 100-199, and are organized by topic rather than difficulty. So Math 13x is the analysis sequence, Math 10x is the discrete math sequence, etc.
I wanted to make some joke about how the professor lied in the first minute of the video when he said, "Real Analysis is an exciting subject," but this guy taught one of the most engaging lectures I've ever seen on arguably the most difficult subject I know of in undergrad.
This guy is awesome. Explained everything like a hero.
This is real teaching. I teach myself and learned so much how to teach better from these lectures in addition to better understanding of some of the (more advanced) material. Thank you.
What else did you expect? Complex teaching?
@@Krispio666 lmao
This dude literally made real analysis fun. Wow, such lucky students.
very good teaching, just wish the videos were better quality.
Agreed. He’s a wonderful lecturer but I really can’t make out a lot of what he’s writing as the video quality is so poor.
@KillerNedDude f(x)=1 for all x in R satisfies the condition for a function: for a given input x, there is a unique output: 1. It does not have to be the other way around: a function has to turn an input into a unique output, not the other way around (if it does, it's a bijection). In a normal function plot, this means: on every vertical line (so a specific x), there can be at most one point y=f(x) on the plot (which f(x)=1 for all x satisfies).
Yea that's right
Wish the video quality was better though, but none the less the lecture is awesome. Thank you!!
@KillerNedDude That's a good question. Strictly speaking, you should also give the domain and codomain when giving a function. The arcsin function is normally defined to be from the input interval [-1,1] to the output interval [-pi/2,pi/2], in which case it has a unique output for every input (and is even a bijection). It's like the square root function, which goes from all non-negative real numbers to just the non-negative real numbers, in stead of giving 2 values for every function.
I'm fortunate to find these lectures.
The professor knew the name of all the students in the first class!
The homomorphism is definitely not onto from Z into Q, but it is into the subgroup of numbers in Q of the form n/1. If you can prove that this is a subgroup of Q and the function is an isomorphism (trivial), you have proven that Z is isomorphic with the whole numbers in Q, which is exactly what you would expect.
Antony Foster No, he says it's "not a set per se". As you show, it can be defined in terms of sets but the point he's making is that the ordered pair (a,b) shouldn't be confused with the set {a,b}, since {a,b}={b,a} but (a,b)!=(b,a).
@KillerNedDude He says that, given a specific input, a function has a unique output, which is correct. A function is surjective if every member of its codomain is the function value of at least one input, and a function is bijective if every member of its codomain can be traced back to exactly one input. So f(x)=x^2 (R to R) is a function, but not bijective because (2)^2=(-2)^2=4, and 4 cannot be traced back to one unique input. It's not even surjective, because no x satisfies f(x)= -1.
Awesome! Grateful that I found this set of lectures. So what, the video is poor - LISTEN and LEARN and hit pause if need be. Looking forward to the rest :)
Very good comment ❤
Construction of Q begins 41:55
mad props. One of the best "yo dawg" lines ive come across
Hey wassup
man this guy is great , I hope he continues on with other courses ,hint hint
Now here's a professor that knows how to teach! ...he does know the golden rule: if you can make it fun...why not? Some other guys just seem like they are teaching a wall...instead of groups of students; that, they can do at their homes! Nice class, even though I would've love to some more light...but I guess its ok...you know, its free!
"Little" X cited 'red, corvette, red corvette' when listing his properties of conveyance and conveniences.
Mind blown at 48:00... "Maybe we want to set up fractions defined in terms of equivalence relations." Holy crap! Lightbulb moment. Then each pair of equivalent ratios belong to some equivalence class, that might have lots of elements; call that equivalence class *the ratio, for instance, 1/3.* "Let *Q* be the set of *all such equivalence classes* (!!!!!!). ... You might also want to say how *Z* is embedded in *Q,* so that n/1 e *Q* [is an equivalence relation symmetric with] n e *Z.*" Wow!
Only the first day and I'm already impressed. Just keep in mind, though, that without intuitive ideas about division already obtained from older civilizations, like Egypt, Babylon and Greece, we couldn't have ever come up with this logical construction. So, in this way we don't "explain why or how" division works, we just take division as a fact, and set up logical rules to make sure more complicated things that rely on division work. The only explanation for arithmetic like division is our intuition and sense-perception. It's important not to lose sight of this, and it plays a role in philosophy too. This is why we learn k-12 algebra before real analysis or abstract algebra, because these latter classes don't explain the stuff in k-12 math, they just build logic on top of it.
This same argument could apply to calculus, but like the quote shown in the beginning of class, most people "draw the line" at the rational or natural numbers as far as intuition and sense-perception is concerned, and chalk up anything that uses real numbers as no longer sense-perceptible, and therefore some people believe calculus should be learned only with full rigour and logical construction saying that the real numbers have 0 intuitive properties, and others say that you still can use sense-perception to intuitively learn about real numbers.
So, I'm not sure where I stand, but I think I lean more in the camp of "real numbers are not intuitive, and therefore should be learned only with full logical construction." This isn't to say that the uses of calculus are not intuitive, I think things like the Integral are incredibly intuitive/obvious, but the math used to write it down is not. Neither are infinitely non-repeating decimals. So it's a funny thing. Often the concepts are very intuitive, but the math to actually get an answer suddenly involves all this weird unintuitive crap.
So, is calculus just the intuitive idea of adding up a bunch of rectangles, or is the weird unintuitive crap of the sum of an infinite series as a converging infinite sequence of partial sums, potentially never once repeating a value, forever? I don't think anyone knows the answer to this question. But I do think that Newton and Leibniz would say something like "don't worry about the details, just think of the idea and allow it to work" and it does give the right answer regardless of the details; so this view is basically valid. It's just like Zeno's Paradox and the whole idea that if an Archer shoots an arrow at a target then it could never reach the target because it always has to go half the distance and it'd never reach it; but on the contrary, if you shoot an arrow you better not be standing in front of that target, because if you are, not only will that arrow hit the target, it'll also go through your flesh on the way there. And so rational logic often contradicts empirical evidence. This is the crux of the philosophical debate of empiricism vs rationalism.
I have to admit that I think this debate has less to do with the truth, and more to do with how our brains work. Because any human being can intuitively do calculus and classical physics; every student does it in gym class when they shoot a basketball, and every athlete is taking derivatives subconsciously constantly. And anyone who drives a car and figures out how much distance they need to stop does it. Yet, when we try to consciously sit down and do it fully aware, those same athletes who can make 90% of their baskets can't pass calculus class on paper in the classroom. That's odd right? Likewise, there are probably tons of mathematicians who can consciously write down and solve any calculus problem without breaking a sweat, but then they'd go and miss all their baskets. This relationship between subconscious math and fully conscious math is interesting to me.
There's definitely a sense in which everyone is doing calculus and physics every day of their lives, but there is another sense in which it seems the relationship between doing it on paper, and *actually doing it in real life* is inversely proportional.
Anyone can drive a car, but not everyone can ace calculus class. Any mathematician can pass calculus class, but not all mathematicians can shoot a basketball. Wierd.
I think there are basically two "answers" to this that are the same thing:
1. muscle memory helps encode the "subconscious math" you speak of, sort of in the same way we used look up trigonometric functions on tables. A basketball player can put the ball in the net because they've practiced it a lot, so their muscles told the brain what configuration they used to do the throw and the brain crunched those numbers.
2. first-person perspective is encoded into the "subconscious math". An athlete isn't generalizing their basketball throws into all possible basketball throws ever, let alone into ideal parabolic flight paths informed by gravitational constants corrected for air frictions. They're synthesizing basketball throws using their own body, standing on a flat surface, aiming at a specific point from inside a range of distances.
I think that dodging specificity is part of what makes mathematics astounding. It extracts a simple essence out of everyday experience and shows how that essence can show up in lots of other places, too.
What a wonderful professor and lecture! Thank you Harvey Mudd College
Thank you so much for these lectures! If I got a good grade in my analysis 1 course it's thanks to them. Now I'm reviewing the entire series of videos for analysis 2. It would really help us students if you could upload something similar for other topics too :-)
There is a series of videos for analysis 2? where can I find them?
@@limyeewee3518 Did you find it? If that’s the case, please put that link here
I though Kronecker meant that we had to assume that naturals are just there and build on that - as in, naturals are sort of like existing because they are very basic and man created the more complex math that somehow is based on the naturals, like how the rationals are an extension of the integers which themselves are an extension of the naturals and the naturals being the most basic - the first cause in some sense,
I think what Kronecker meant, or should've meant, is that bits are fundamental in the universe. Either something exists, or it doesn't (1 or 0). All other numbers are either derived, or are functions and not numbers in and of themselves. Try defining any other number without constructing it. This lecture is all about constructing numbers.
Forgive the poor production and just keep watching. This guy is the one. Sometimes I think he may be trying to tell us something with the fuzzy resolution and strange aspect ratio.
2:29 I think the stdudent does have a point there,
In the real world everything is disrete and wheunting atomsn you zoom far enough you'll essentially be a collection of atoms. Real numbers and Continuity are just an approximation of the very dense collection of atoms as continous matter.
Did not know the word parallelogon before this!
Thanks a lot Mr. Francis Su ❤
@DarthCormac
At HMC, the courses are numbered in odd ways. This course is taken by sophmores, or advanced freshmen in some cases.
At 41:20, that's a subtle point! There can be relations that are not functions! We usually call those relations "curves" when talking about parametric representations of curves. Remember, a function is only a *relation* that has one unique output for each input, *by definition.* Therefore, not all relations are functions, but all functions are relations, and, functions are a special case of a relation!
For instance, the graph of a Circle represents a relation that *is not a function* because for each input x, there exists two corresponding outputs, therefore the definition of a function is not satisfied, and the equation for a circle in euclidean space is *not a function,* but it *is a relation.*
yep, this is when you solve for y in the equation of a circle; you get two distinct functions, one corresponding to the upper half and the other to the lower half.
if you're familiar with sets and relations concept thoroughly directly jump to 42:00.
Love how he says "little".
Mr. Su is a great lecturer!
great lectures but terrible video quality...would be nice if you updated this with new videos
Jess Schwartz I agree with your insights. You did real analysis on the quality of this video.
@@TheLililitu Gotta love Real Analysis puns!
mmkay
Thank you the honoured and venerable Francis Edward Su for this introductory lecture on real analysis, namely on the construction of the real numbers.
Is there any exercise problems corresponding to this series of lectures?
Thank you so much, these lectures are helping me prepare for my analysis course! I had some difficulty understanding compactness with just the book and these were so so helpful!
1:38, "God created the integers," what he meant was: we need to figure it out the rest, how we are going to use them, and applied them into the real world.
Awesome to watch it During Covid😅
Chup hutiya
I think he meant other constructs like the complex number, which like it's cousins-vectors and quarternions-were harder to get your head around without context(polar co-ordinates, general rotations) and also were not that easy to apply in general (transformation matrices, jacobians) .It's funny that he has a tensor named after him.
I don't know much about uni class numbers but isn't real analysis usually way more than 131?
6:58, 58 i had the same issue with how obvious the stuff is
At 53:58 he said that extending the Integers with the construction of the Rationals amounts to finding an isomorphism from Z into Q given by n --> n/1 (assuming we have been told how to operate on the rationals). Wouldn't that map be considered a homomorphism since it is not bijective (not onto)? I'm not trying to be picky. I just completed my first semester of modern algebra and want to make sure I understand this correctly.
Z is not isomorphic to Q but it is isomorphic to the subset containing equil classes of the form n/1
Yes Josh, you are correct.
Very good lectures on the core concepts of real analysis, great videos
Thanks for making these videos available. You should do this for Coursera.
If you're taking real analysis during your freshman or sophmore year, you are pretty badass
thank you so much for this, it saved me one whole module in sch
Is an equivalence class the same as a equivalence relation?
Not exactly. An equivalence relation implies a set of equivalence classes and vice versa. So when you have one, you have the other, but the elements in an equivalence relation are ordered pairs in the Cartesian product of a set (with itself) but the elements in an equivalence class are elements in the original set.
Suppose we have a set X and a relation R. For every element of X we can define an equivalence class over them. Let's call a generic element of this set as "a". So an equivalence relation is defined as the set of all elements of X which are related to "a". We denote this as [a] = {x in X; xRa}
Go to Keith Devlin's youtube account. He's the Professor of Math at Stanford, and he has tons of Real Analysis videos up.
What exactly does he write about the order of integers at 43:08 ? Can anyone tell me?
He just wrote "their arithmetic and order."
OK, thanks.
np
What Kronecker meant was -God gave us Integers, meaning Integers are the most basic intuition when it comes to math, i.e. counting. quantifying objects , differentiating between one and many, is what is the basic form of math itself, that existed in nature, long before man started deriving more complex relations and formulae that aided calculations of natural phenomena.
@stijnisgoed Correction: I meant 2 values for every input (accept zero).
Thanks for a nice explanation. My lecture is using Mathematical Analysis I by Vladimir A. Zorich, which is considered extremely difficult to understand by freshman like me ;( Principle of Mathematical Analysis by Walter Rudin seems understandable and easy to catch up for me. Currently I am struggling with Mathematical Analysis and re-learn all the principles and theorems (Proving them is my everyday homework). Hope you can make more such videos and do remember to update the video quality. If I were not focusing on watching your video then I might not catch up what you had written on the board =)
How’s it going now? :)
@stijnisgoed but then what is unique about the outputs of say the inverse sine function which are the same for many different inputs i.e. asin(0) = 0, pi or npi (for all n in Z).
the image's quality is not good , it is hard for me to see clearly , but this is a good video and a good professor
@stijnisgoed Cheers that makes sense, but what about say f(x) = 1 (for all x in R), is this not a function? Also are functions such as f(x) = { 0 if x is even, 1 if x is odd} (for all x in Z) not valid functions? (I'm guessing I probably have to study the course more to find out).
Any group theory lecture might be helpful.. please..
Prof made a mistake on the board when he wrote down the "definition" of the subset....he wrote A is a subset of B if x in A then x is in B. You need a universal quantifier right in front to indicate that you are talking about every element in A, is also in B, not just a particular x, or some x.
Not sure about his description of functions in this lecture, he say's that a function should have a unique output, surley this is only the case with a bijective function.
Anyone have link for Real Analysis 2 course ??
Bundle of thanks
I am self-learning analysis by working through Spivak's Calculus. Should I wait until I finish the book to watch these lectures or do they complement the textbook?
syggelekokIe hey man so I’m trying to start Spivak calculus but it’s just difficult...I was wondering which path u took?
@@viccctv9106 before doing Spivak do a proof based calculus textbook. Jerrold Marsden or Peter lax books
Thx for the videos but where are the notes? i don't see any.
26:13 bookmark
Princeton math majors take this freshman fall (same book and class essentially). It's called MAT 215, but it's first year
JacobShulman which text is used???
@WorldCollections Thanks for recommending it!:)
@Evan Zamir Thank you too!:)
The lecturer states that one should communicate math in whole sentences and not (e.g.) just in symbols.
This is particularly true when the communication is done exclusively in writing. What the lecturer doesn't state explicitly is that in a lecture and/or oral exam, your communication is your narration plus your writing, and it is those two elements together which should fully and completely convey your ideas.
That's why using shorthand notation on the blackboard without a written explanation (provided you give an oral one) is more appropriate than doing the same in your research papers (I assume) and your written homework.
how would you do that last exercise: show that (p,q) ~ (m,n) and (m,n) ~ (a,b) => (p,q) ~ (a,b)?
We assume (1) pn=mq and (2) mb=an. The goal is pb=aq.
If we multiply pn and mb, we should get mq*an, due to substitution. I.e. (3) pn*mb=mq*an.
Rearranging that for clarity, we have (4) (mn)*(pb)=(mn)*(aq).
Because we can cancel a term that multiplies the entirety of both sides (mn), we're left with (5) pb=aq.
That's our goal, so we've proved transitivity.
(I actually had to look up the answer twice before I realized what was happening.)
Wow Professor Su knows most of the names of students...
if anyone is new here and is complaining about the video quality. I suggest you to go to his website and see the notes and follow along like I did.
Best lecturer ever
could anyone tell me how to solve the homework question at the end of the lecture?
with simple algebric calculation...
since pn = qm and mb = na, then (pn mb) must equal to (qm na), using cancelation law, you cancel m and n, it remains (pb=qa) which is what we want to show. QED
"using cancelation law, you cancel m and n", you can only use cancelation law for non-zero integers. n is guaranteed to be non-zero but for m you need a little bit more work.
can you elaborate on the work necessary for m?
in the case that m is not 0, the cancellation law would hold, but in the case that m is zero, the equation still holds true since 0 = 0. is this not sufficient?
You are really gorgeous. Best explanation, this lecture still the gem for now.
@50:03 -BODY ONCE TOLD ME
Where are the handouts and notes mentioned in the blogspot link?
Edit video, so we can actually read what the professor writes.
what is this "process of doing mathematics" that he is referring to?
Is there any textbook for free for this course?
There is a recent free e-book but it it is unrelated to the class outside of subject
Does anyone know where to get notes. The website dosn't seem to work.
Parallelogon. Yes.
the link for the notes and study tools is bunk. whats up with that? is there another link?
very nice lecture
homomorphism of Z into Q (trivial kernel）
If sets A and B both contain the number 5, does that mean A U B contains 2 copies of the number 5 or just one?
Duplicates are discarded in sets, so it'd be just 1. However, if you wrote two copies, the set wouldn't change
I wan't able to find notes..! Any link to notes ?
Ajay G write your own
Awesome audio lecture dude!
This guy is a hero
i actually searched for a comment referring to this aspect before liking it.
@cpowel2 I think you mean extraneous and not erroneous but we get each other. :) Good luck on computer science!
He's my uncle
He's our hero
no one asked
He's my father
Math majors- our uncle
I’d smash
is this a beginner graduate course? cause i'm kinda confused with the '131'
Is it me or Prof. Su sounds a bit like Mr.Mackey! Great course btw.
you are from russia?
you are from russia?
P.S. The actual construction begins at 42:00
8 years ago, but thank you lmao
Skipped over the construction of the integers as N X N.
Do they have a complex analysis course?
26:55
39:00
Thank you professor.
How does that relate to division though? Like how do we get from 1/3 to 0.333....
1/3 just seems like a symbolic representation without anything to do with division
wait til he gets to defining real numbers in general from the rationals.
We can't really talk about decimals (infinite decimals) yet before we've defined the real numbers, which we construct form the rationals.
@@x0cx102 ohhhh thenkss