I swear to god, you are so much better than my professor. Unfortunately, I have to do it his way instead of yours. We are all over the place and he isn't tying things together or building up like you are doing. I hate taking hard classes with awful teachers, especially ones like mine who think they are god's gift to math. Thank you for these videos. I may not pass my class, but at least I'll know it wasn't entirely my fault. Your lectures are outstanding.
I had a similar experience. Lots of guys were seemingly hired for their ability to contribute to the body of mathematics and not for their ability to teach.
Thank you so much for your wonderful lectures. Thanks also to your student who encouraged you to have these lectures taped and posted on UA-cam. And thanks to UA-cam. I am a 39-year-old who is considering a new career in statistics, and need to make up for all the mathematics I never learned. I have been trying to learn Real Analysis by studying Baby Ross and Strichartz's The Way of Analysis on my own, and your lectures make the topic so much more accessible!
If the lub, L = a/b, satisfies L^2 > 2, then consider the set of rationals greater than 2 and less than L^2; it suffices to find a square. Consider (an-1)^2 / (bn)^2 for large n. the difference between this and L^2 = (an)^2 / (bn)^2 is equal to ( 2an - 1 ) / (bn)^2 < (2a / b^2) / n, which goes to zero as n grows large since a and b are fixed. Thus, we simply pick sufficiently large n such that this difference is less than L^2 - 2, ie. any integer n > (2a / b^2) / (L^2 - 2) works; And this rational square will be in (2, L^2), contradicting minimality. And of course, if L^2 = 2, we know that there's no solution to this in Q.
40:50 if you don't have to check the condition thst the empty set is bounded by 2, why does that mean it is true that it is bounded by 2 (vacuously true).
He 29:15 he was talking about the whole set of rationals. If you are talking about the set of negative rationals then the sup is, in fact, 0; remember that the sup(E) does not have to be an element of E, it just needs to be the least upper bound of E.
I didn't know symbols were disallowed in proofs. I thought they were a universal way to communicate truth across languages. Hopefully that isn't a hard/fast rule.
contradiction is unnecessary for the x^2 < 2, 3/2 is an ub example. We just note that if x in A, then x^2 < 2 < (3/2)^2, so x < 3/2; hence 3/2 is an ub. Though since square roots don't exist you'd have to just use other axioms (like positivity) to show that if x and a are positive rational numbers such that x^2 < a^2, then x < a.
comment on 30:20 show that Q has no supremum in Q. It is sufficient to show that Q has no upper bound, since supremum must be an upper bound (in addition to being the least upper bound). proof (by contradiction). Assume Q does have an upper bound in Q, call it b. This means that for all x in Q , x
“Let’s take a break and when we get back, we’ll define Dedekind cuts” Me: sweet! *proceeds to get up* Editor: “Not the people at home. We cut out your break. Sit down” 💀😭😂
This lecture could be a little better if it proved that in any interval there is no largest or smallest rational number, so therefore there is no max or min in that interval; the difference between max and upper bound is that max has to be in the set, and upper bound does not. Therefore, there is no max, and therefore the lub or sup can not be the max, and therefore it can not be a rational number. And therefore the real numbers are *defined* to be the lim of a sequence of rationals, and therefore a real number is the lub, and therefore they fill in the holes in the rationals on a "number line." I also wish we didn't take the completeness of R to be an axiom. I'd rather construct that as a theorem from Q; I'm pretty sure that cauchy sequences are the only way to do that. This alternative construction is the one put forth by Cantor; he defines real numbers to literally be equivalence classes of entire infinite cauchy sequences, where each decimal digit in an infite never repeating decimal represents each element of the infite never repeating cauchy sequence. Real numbers don't have to be never repeating infinite cauchy sequences. This definition also is what allows us to say that 0.99999... = 1.0000... and that's true because each corresponding cauchy sequence both converge to 1. Therefore the two decimal representations are equivalent. The point being, decimals are unnecessary in mathematics once you understand sequences and series. It turns out that every real number has 1 or 2 decimal representations, 2 if the decimal representation is terminal and non-repeating and 1 if it never repeats. It also turns out that every real number is the limit of a bounded cauchy sequence of rationals, and if some sequence is divergent, then it is not in R, thus actual infinity is not a number in R, so any function that is unbounded for some neighborhood of x values has no limit in R itself, like lim x->0 1/x^2, even if it appears to be infinity. Also, any two cauchy sequences that have the same limit are the same real number, even if the terms in the sequences are slightly different, as long as they approach the same limit, they are equivalent; though, I honestly couldn't tell you why this number would then only have one decimal expansion given that the two sequences might have different terms -- it's just beyond me right now, but I'll work on figuring it out and update this comment when I do. You can find this explanation in Terence Tao's "Analysis 1" third edition. If I'm honest, Tao combined with Rudin is a pretty damn solid Analysis education. Add in Johnsonbaugh's "Foundations of Mathematical Analysis" on top of that for better treatment of the integral and some other cool things not in the other two books and you're ready to kick some serious analysis ass. No joke. I call these three books the "Perfect Trinity" of analysis, well, undergraduate analysis. You can get all three books on Amazon for less than $60 if you buy the international paperback version of "Principles of Mathematical Analysis" by Rudin. Happy learning!
Robert Wilson III Thanks for the lovely answer,it's obvious that you took time and effort into crafting it.Im currently self studying analysis,do you recommend "The Holy Trinity" or do you have another particular book in mind?I have Tao's and Rudin's book with me,the concept by itself isn't that hard,but I find Rudin's book sometimes to be a little too terse and I have trouble watching the lectures because of my wifi(I come from a not-so-well-to-do background)
Incorrect cut example? Video 45:30 says that set A from before described in video 9:25 as A = {x an element of Q such that x^2 < 2} is NOT a cut becuase it fails to satisfy property 2 of being closed downward. Video 48:45 says that it is a cut which I believe to be correct. Did I miss something? Is A something else?
Not sure if this will help but I used this counter-example to explain to myself why I think A = {x an element of Q such that x^2 < 2} is not a cut: Take x to be -2, you can verify that x is an element of Q and it does not belong to A. Now consider another element, call it r, that we know will be in A - How about we set r = 1? Then you have r in A, x in Q and x < r, but x is NOT in A. Thus A fails to be closed downward and thus not a cut.
(for revision, and as a check list if i forget what dedekind cuts are) ->just understand these terms: # non trivial: not empty and quite literally not the whole set of rationals. # closed downwards: we are talking about the closure property (like in equivalence relations) # no largest number: you can always have a number larger than what you currently have that belongs to 'alpha', this property specifically wouldn't work in closed intervals since you can define a 'largest number' in that interval.
also, frick this lecture, although so many others are giving shit ton of praises to it, i spent 3 hours to still not understand it fully, i appreciate u sir francis, but damn u gotta work on that video quality (i don't quite understand additive and multiplicative identities still now, i am gonna be very thankful if anyone bothers to explain me those with a reply)
+Ste Rose According to this definition, every real number is a dedekind cut. So if you have a transcendental number, then you have a dedekind cut written down in front of you, which you call a transcendental number. So your question probably only makes much sense if you use a different definition of real numbers. For example you can ask: How can I construct a dedekind cut out of an infinite non-repeating decimal (because many people think of irrational numbers as infinite non-repeating decimals) or you can ask, how to construct a dedekind cut out of a cauchy sequence (because some people also define real numbers as equivalence classes of cauchy sequences). So if you specify more precisely, in which format you have the transcendetal number, then your question can be answered.
The question posed is a very good one and lies at the core of the problem with Dedekind cuts. If you look at this a bit futher you find the problem is in the CONSTRUCTION of the Dedekind cut. It seems to work for algebraic numbers, e.g. the rather overworked example of sgrt 2. But it is somewhat unclear for the Transcendentals. I made this comment at the beginning of the video. I hope when I have seen the whole thing, I am shown a demonstration of a Transcendental Dedekind cut. The other methods you mention Myren Eario are alternatives to the Dedekind cut. You don't need to construct a number twice.
A dedekind cut is just a partition of the rationals into two sets. Notice that I am not describing a cut using "the rationals less than a real number and the rationals greater than it" as that would require the existence of a real number that we haven't even constructed yet. Instead, just think of a dedekind cut as a partition of the rationals into two sets. Now, think of the set of all dedekind cuts, so all of the partitions of Q. That set is what we call the real numbers R where each cut corresponds to a unique element in R. Once we've created this set can we only talk about the properties of each element (being algebraic, transcendental, squares to give two etc...). For example, the set of all rationals x such that x*x less than 2 is indeed one way to describe a cut so it represents a real number. But also note that all we have done is describe an element of R, we haven't proved that it squares to two and only once we've done that can we call this element in R "sqrt(2)". That is how we go about doing things we "name" the elements of R based on the properties they might have like "pi" or "e" or even their decimal expansion based on the rationals in each cut they correspond to.The elements already existed from construction we just go about discovering their properties. You don't need the existence of an element to define a cut In fact what you do is first define all possible cuts of Q (which are all of its possible partitions) to create this set R and give it the operations addition and multiplication and then only you go about "naming" the elements of the set from their properties. The question on describing the cut of a transcendental number well I assume that that number is an element of the reals because a number being transcendental only makes sense if it is an element of some field like the reals and we check if there exists a polynomial with coefficients in say the integers in which it is a root so if it is in the reals we just look for the cut that represented it in the first place because from construction, every element is essentially just a cut from the set of all possible cuts
@@revooshnoj4078 Ok then, so how would you construct say "Pi" using the cut method? By that I mean: can you show how to construct a partition of the rationals such that this partition behaves in a certain sense just like pi? I guess this is what OG meant, atleast thats how I get it
@@TheDummbob asked "can you show how to construct a partition of the rationals such that this partition behaves in a certain sense just like pi? " We can use Leibniz formula for pi. Let a(k) = (-1)^k * (1 / (2 * k + 1)). Let s(n) = sum(from k = 0 to n)a(k). Let L = { x in Q | (En)(n in N and x < s(n)) }. Then, L is a (lower) Dedekind cut for pi. You can take U = Q - L for the upper Dedekind cut for pi. Note that the "for some n" gets around the need to specify a specific upper bound for pi.
At 28:57 why should gamma be in the set if 0 isn't? Seems like (not that I know anything) that it would need to be shown that for any arbitrarily small negative irrational there is a larger negative rational. Not correcting, jsut trying to understand.
Question about Dedekind cuts? I believe all sets of the form X = {x an element of Q such that x < 2} are cuts and those of the form Y = {y an element of Q such that y
At 28:57 why should gamma be in the set if 0 isn't? Seems like (not that I know anything) that it would need to be shown that for any arbitrarily small negative irrational there is a larger negative rational. Not correcting, just trying to understand.
A great class. This might be only me, but I am VERY uncomfortable with constructing R using Dedekind cuts in the sense that cuts create only an isomorphism of the real world R instead of the real world R, whatever "real world R" means. I am saying this because it is meaningless to say that a real number is a proper subset of another real number whereas here we define a binary relation (order) by means of another (being a proper subset of) and say that this cut (real number) is a proper subset of that cut.
How can an additive identity be a set of negative rationals? Additive identity is something that does not change the element that we are adding it to. So if q is a rational and we add to it the additive identity, we get back q. Whereas here the additive identity is a set of negative rationals.
Also, for the definition of a product, wouldn't you need it to be the union of the set described and the negative rationals? Otherwise the set isn't closed downwards and isn't a cut.
@mappingtheshit Once again: there is no dumb questions in Math. For example in the set of rationals there is no convergence of series like lim n->inf (1+1/n)^n ; because the limit (e number) is not in the set of rationals. Thats why I am asking, you see?.
No offense to other great professors but Prof Su kills it !!! I think more school should upload their video lecture for foundational class in Math, especially those bridge lower and upper level
couldn't you also show that there's no lub in Q by noting that the lub, L, must satisfy L^2 = 2? If we have L^2 > 2, it suffices to show that there is a And we already know that x^2 = 2 has no solution in Q. Done.
@@jonalderson5571 Ok, thanks for the replay, it makes some sense. But anyway, I think it's just a matter of a habbit. You get used to it in time like with alphabet of a new language. And you also read math in a different way than you would read some ordinary text. So, I would say that it could be helpfull in some way if you are new to it but after some time the idea that you wanna get through is much more important than the way you write it down. Better use that extra intelectual energy to figure out the math concepts better, than get lost in the way you "need" to write it down. Making it a strickt class rule doesn't make sense to me and has nothing to do with the main subject. That would be my oppinion.
I was talking about the previous example on 27:50. True the Supremum does not have to be an element of the set, but it has to be an element of the Metric Space where the set resides. Notice that he had define S ﬤ E as the set of rationals. (So my previous comment is indeed false as I failed to notice this fact due to the poor quality of the video) Had we chosen Q_
Actually in my comment below I meant to claim you can't construct IR by trying to complete "real" operations under Q. By "real" I mean excluding sqr. negative number constructs). Looking further in the video to the actual construction of IR, the idea of satisfying the need for completeness under Q is abandoned in favour of creating Dedekind cuts based directly on the rational numbers. But can there really be a justification for doing this? But even if it can be justified can you really have a construction which is non ennumerable? Surely you can only do this if each cut contains a non enumerable set (of irrational numbers). But the (potential) cuts cannot contain any irrational numbers because they have not yet been created. So again I don't see how the Dedekind cut construction can be valid.
It seems all Dedekind cut explanations in these courses use square root 2. But you can extend the idea and using algebraic numbers you can create an infinite set of non rationals. You can then throw in the transcendental numbers. So you have constructed an infinte set of non rationals spread over the number line. But surely this infinite set is ennumerable. Now Cantor claimed that IR is non ennumerable although he could not prove how non ennumerable (the so called continuum problem is still unsolved). So even if an infinite set of irrationals can be constructed this still leaves 100% of the irrationals unconstructed. So from the above argument it follows you cannot construct the irrationals (from Dedekind cuts). Or is there a flaw in my argument? The alternative is to define IR axiomatically which is less than satisfactory.
This series of videos is hands-down, the supremum of the set of real analysis lectures on UA-cam
wouldnt that make it not real analysis videos the the definition? =D
@@ayadevin2413 but the set of all real analysis videos is large but finite. So the sup exists in the set. lol
I normally don't comment on videos, but this is single handedly the best Real Analysis lecture I've seen. Thanks for uploading it!
he is a talented teacher
I swear to god, you are so much better than my professor. Unfortunately, I have to do it his way instead of yours. We are all over the place and he isn't tying things together or building up like you are doing. I hate taking hard classes with awful teachers, especially ones like mine who think they are god's gift to math. Thank you for these videos. I may not pass my class, but at least I'll know it wasn't entirely my fault. Your lectures are outstanding.
I had a similar experience. Lots of guys were seemingly hired for their ability to contribute to the body of mathematics and not for their ability to teach.
Thank you so much for your wonderful lectures. Thanks also to your student who encouraged you to have these lectures taped and posted on UA-cam. And thanks to UA-cam. I am a 39-year-old who is considering a new career in statistics, and need to make up for all the mathematics I never learned. I have been trying to learn Real Analysis by studying Baby Ross and Strichartz's The Way of Analysis on my own, and your lectures make the topic so much more accessible!
Wonderful! It is never too late to begin again. Good luck sir
What does the Rational Field have in common with a well done Still Life? - Ordered Pears!
Thank you for being the best real analysis teacher!!!!!!
Maaaaan, I so wish the video quality was higher!
If the lub, L = a/b, satisfies L^2 > 2, then consider the set of rationals greater than 2 and less than L^2; it suffices to find a square.
Consider (an-1)^2 / (bn)^2 for large n. the difference between this and L^2 = (an)^2 / (bn)^2 is equal to ( 2an - 1 ) / (bn)^2 < (2a / b^2) / n, which goes to zero as n grows large since a and b are fixed. Thus, we simply pick sufficiently large n such that this difference is less than L^2 - 2, ie. any integer n > (2a / b^2) / (L^2 - 2) works; And this rational square will be in (2, L^2), contradicting minimality.
And of course, if L^2 = 2, we know that there's no solution to this in Q.
More exactly, n=2a is always large enough, so we don’t even have to prove that the expression goes to 0 for large n
comment on 26:30
if g < 2 (g for gamma), then it is false that 2
40:50 if you don't have to check the condition thst the empty set is bounded by 2, why does that mean it is true that it is bounded by 2 (vacuously true).
He 29:15 he was talking about the whole set of rationals. If you are talking about the set of negative rationals then the sup is, in fact, 0; remember that the sup(E) does not have to be an element of E, it just needs to be the least upper bound of E.
I didn't know symbols were disallowed in proofs. I thought they were a universal way to communicate truth across languages. Hopefully that isn't a hard/fast rule.
I did get counted off for it in a homework, so best to follow his advice on this. Maybe more about it in books on proof?
contradiction is unnecessary for the x^2 < 2, 3/2 is an ub example.
We just note that if x in A, then x^2 < 2 < (3/2)^2, so x < 3/2; hence 3/2 is an ub.
Though since square roots don't exist you'd have to just use other axioms (like positivity) to show that if x and a are positive rational numbers such that x^2 < a^2, then x < a.
comment on 30:20
show that Q has no supremum in Q. It is sufficient to show that Q has no upper bound, since supremum must be an upper bound (in addition to being the least upper bound).
proof (by contradiction). Assume Q does have an upper bound in Q, call it b. This means that for all x in Q , x
Yep. A set that is not bounded above cannot have a least upper bound.
“Let’s take a break and when we get back, we’ll define Dedekind cuts”
Me: sweet! *proceeds to get up*
Editor: “Not the people at home. We cut out your break. Sit down” 💀😭😂
This lecture could be a little better if it proved that in any interval there is no largest or smallest rational number, so therefore there is no max or min in that interval; the difference between max and upper bound is that max has to be in the set, and upper bound does not. Therefore, there is no max, and therefore the lub or sup can not be the max, and therefore it can not be a rational number.
And therefore the real numbers are *defined* to be the lim of a sequence of rationals, and therefore a real number is the lub, and therefore they fill in the holes in the rationals on a "number line." I also wish we didn't take the completeness of R to be an axiom. I'd rather construct that as a theorem from Q; I'm pretty sure that cauchy sequences are the only way to do that.
This alternative construction is the one put forth by Cantor; he defines real numbers to literally be equivalence classes of entire infinite cauchy sequences, where each decimal digit in an infite never repeating decimal represents each element of the infite never repeating cauchy sequence. Real numbers don't have to be never repeating infinite cauchy sequences. This definition also is what allows us to say that 0.99999... = 1.0000... and that's true because each corresponding cauchy sequence both converge to 1. Therefore the two decimal representations are equivalent. The point being, decimals are unnecessary in mathematics once you understand sequences and series. It turns out that every real number has 1 or 2 decimal representations, 2 if the decimal representation is terminal and non-repeating and 1 if it never repeats. It also turns out that every real number is the limit of a bounded cauchy sequence of rationals, and if some sequence is divergent, then it is not in R, thus actual infinity is not a number in R, so any function that is unbounded for some neighborhood of x values has no limit in R itself, like lim x->0 1/x^2, even if it appears to be infinity. Also, any two cauchy sequences that have the same limit are the same real number, even if the terms in the sequences are slightly different, as long as they approach the same limit, they are equivalent; though, I honestly couldn't tell you why this number would then only have one decimal expansion given that the two sequences might have different terms -- it's just beyond me right now, but I'll work on figuring it out and update this comment when I do.
You can find this explanation in Terence Tao's "Analysis 1" third edition. If I'm honest, Tao combined with Rudin is a pretty damn solid Analysis education. Add in Johnsonbaugh's "Foundations of Mathematical Analysis" on top of that for better treatment of the integral and some other cool things not in the other two books and you're ready to kick some serious analysis ass. No joke.
I call these three books the "Perfect Trinity" of analysis, well, undergraduate analysis. You can get all three books on Amazon for less than $60 if you buy the international paperback version of "Principles of Mathematical Analysis" by Rudin.
Happy learning!
Robert Wilson III Thanks for the lovely answer,it's obvious that you took time and effort into crafting it.Im currently self studying analysis,do you recommend "The Holy Trinity" or do you have another particular book in mind?I have Tao's and Rudin's book with me,the concept by itself isn't that hard,but I find Rudin's book sometimes to be a little too terse and I have trouble watching the lectures because of my wifi(I come from a not-so-well-to-do background)
Incorrect cut example? Video 45:30 says that set A from before described in video 9:25 as A = {x an element of Q such that x^2 < 2} is NOT a cut becuase it fails to satisfy property 2 of being closed downward. Video 48:45 says that it is a cut which I believe to be correct. Did I miss something? Is A something else?
Not sure if this will help but I used this counter-example to explain to myself why I think A = {x an element of Q such that x^2 < 2} is not a cut: Take x to be -2, you can verify that x is an element of Q and it does not belong to A. Now consider another element, call it r, that we know will be in A - How about we set r = 1? Then you have r in A, x in Q and x < r, but x is NOT in A. Thus A fails to be closed downward and thus not a cut.
Beautiful lecture, prof. Mackey, m’kay.
(for revision, and as a check list if i forget what dedekind cuts are)
->just understand these terms:
# non trivial: not empty and quite literally not the whole set of rationals.
# closed downwards: we are talking about the closure property (like in equivalence relations)
# no largest number: you can always have a number larger than what you currently have that belongs to 'alpha', this property specifically wouldn't work in closed intervals since you can define a 'largest number' in that interval.
also, frick this lecture, although so many others are giving shit ton of praises to it, i spent 3 hours to still not understand it fully, i appreciate u sir francis, but damn u gotta work on that video quality (i don't quite understand additive and multiplicative identities still now, i am gonna be very thankful if anyone bothers to explain me those with a reply)
Would the set B, of x in Q s.t. x < 2 be a cut since there is always some r in B s.t. x
yes Ethan, that is requirement 3. Specifically if we define B = { x : x in Q and x
@@maxpercer7119 dang this is an old comment. Brings back memories of first getting into math haha.
I just understand the Dedekind sets today. Thanks to u prof
I wish the video had better quality >.
How would you describe a Dedekind cut for a transcendental number?
+Ste Rose
According to this definition, every real number is a dedekind cut. So if you have a transcendental number, then you have a dedekind cut written down in front of you, which you call a transcendental number.
So your question probably only makes much sense if you use a different definition of real numbers. For example you can ask: How can I construct a dedekind cut out of an infinite non-repeating decimal (because many people think of irrational numbers as infinite non-repeating decimals) or you can ask, how to construct a dedekind cut out of a cauchy sequence (because some people also define real numbers as equivalence classes of cauchy sequences). So if you specify more precisely, in which format you have the transcendetal number, then your question can be answered.
The question posed is a very good one and lies at the core of the problem with Dedekind cuts.
If you look at this a bit futher you find the problem is in the CONSTRUCTION of the Dedekind cut. It seems to work for algebraic numbers, e.g. the rather overworked example of sgrt 2. But it is somewhat unclear for the Transcendentals. I made this comment at the beginning of the video. I hope when I have seen the whole thing, I am shown a demonstration of a Transcendental Dedekind cut. The other methods you mention Myren Eario are alternatives to the Dedekind cut. You don't need to construct a number twice.
A dedekind cut is just a partition of the rationals into two sets. Notice that I am not describing a cut using "the rationals less than a real number and the rationals greater than it" as that would require the existence of a real number that we haven't even constructed yet. Instead, just think of a dedekind cut as a partition of the rationals into two sets. Now, think of the set of all dedekind cuts, so all of the partitions of Q. That set is what we call the real numbers R where each cut corresponds to a unique element in R. Once we've created this set can we only talk about the properties of each element (being algebraic, transcendental, squares to give two etc...). For example, the set of all rationals x such that x*x less than 2 is indeed one way to describe a cut so it represents a real number. But also note that all we have done is describe an element of R, we haven't proved that it squares to two and only once we've done that can we call this element in R "sqrt(2)". That is how we go about doing things we "name" the elements of R based on the properties they might have like "pi" or "e" or even their decimal expansion based on the rationals in each cut they correspond to.The elements already existed from construction we just go about discovering their properties. You don't need the existence of an element to define a cut In fact what you do is first define all possible cuts of Q (which are all of its possible partitions) to create this set R and give it the operations addition and multiplication and then only you go about "naming" the elements of the set from their properties. The question on describing the cut of a transcendental number well I assume that that number is an element of the reals because a number being transcendental only makes sense if it is an element of some field like the reals and we check if there exists a polynomial with coefficients in say the integers in which it is a root so if it is in the reals we just look for the cut that represented it in the first place because from construction, every element is essentially just a cut from the set of all possible cuts
@@revooshnoj4078 Ok then, so how would you construct say "Pi" using the cut method?
By that I mean: can you show how to construct a partition of the rationals such that this partition behaves in a certain sense just like pi?
I guess this is what OG meant, atleast thats how I get it
@@TheDummbob asked "can you show how to construct a partition of the rationals such that this partition behaves in a certain sense just like pi? "
We can use Leibniz formula for pi.
Let a(k) = (-1)^k * (1 / (2 * k + 1)).
Let s(n) = sum(from k = 0 to n)a(k).
Let L = { x in Q | (En)(n in N and x < s(n)) }.
Then, L is a (lower) Dedekind cut for pi.
You can take U = Q - L for the upper Dedekind cut for pi.
Note that the "for some n" gets around the need to specify a specific upper bound for pi.
dedekind cuts are open rays that point to the left, with some structure added
Great lecture, thanks for your work!
Hate it when professors start to take leaps through the material of the course and then tell you to just check things out on your own.
That's why it's "non-positive" and not "negative". "Non-positive" means negative +0. "Non-negative" is positive +0.
Will you ever upload Real Analysis II?
Can anyone explain what is the Dedekind cuts for me, I'm totally lost.
First class material bro.
On the example of minute 29:15 he defined the set of non-positive rationals. Zero is NOT a negative number. The set he had on mind has no Supremum.
At 29:15, the set E does not have to contain supremum. Because "0" is in the Q, "0" is the supremum of the subset E.
At 28:57 why should gamma be in the set if 0 isn't? Seems like (not that I know anything) that it would need to be shown that for any arbitrarily small negative irrational there is a larger negative rational. Not correcting, jsut trying to understand.
Question about Dedekind cuts? I believe all sets of the form X = {x an element of Q such that x < 2} are cuts and those of the form Y = {y an element of Q such that y
At 28:57 why should gamma be in the set if 0 isn't? Seems like (not that I know anything) that it would need to be shown that for any arbitrarily small negative irrational there is a larger negative rational. Not correcting, just trying to understand.
I wish the volume was louder
One question: 0 is not included in the set: Q- (negative rationals); does the supremum must be included in the set in order to be a supremum?
A great class. This might be only me, but I am VERY uncomfortable with constructing R using Dedekind cuts in the sense that cuts create only an isomorphism of the real world R instead of the real world R, whatever "real world R" means. I am saying this because it is meaningless to say that a real number is a proper subset of another real number whereas here we define a binary relation (order) by means of another (being a proper subset of) and say that this cut (real number) is a proper subset of that cut.
Quantifier symbols are most certainly used in official mathematical publications...
Is there a way to make the video better so I can read what's being written on the board?
You may need to watch using your mobile device.
How can an additive identity be a set of negative rationals? Additive identity is something that does not change the element that we are adding it to. So if q is a rational and we add to it the additive identity, we get back q. Whereas here the additive identity is a set of negative rationals.
Because of the way dedekind cuts and addition between cuts are defined.
How do you define a cut for say pi or e? You cannot write an inequality for those....
The professor keeps saying ''see the book''. Can one of you guys tell me the name of the book? thank you in advance.
pma rudin
Also, for the definition of a product, wouldn't you need it to be the union of the set described and the negative rationals? Otherwise the set isn't closed downwards and isn't a cut.
I believe the law of trichotomy is a hybrid of the property of reflexivity and the property of anti-symmetry, is this the case?
Anybody please suggest me a book where Dedekind cut is explained well and can be found as an e-book.
Are yall also here for fun?:)
I've never seen anyone be so conservative with a chalk board.
what book are they using
amazing lectures! Although, the video quality is very bad :( This lecture specifically seemed kind of quiet too.
The video only captures pixels with rational coordinates because the reals haven't been defined yet. The small gaps make the video blurry.🙂
😂
What book this teacher use?
I couldn't understand from 19.53 to 19.58 😰 pls someone explain !
Principles Of Math. Analysis. Walter Rudin
@mappingtheshit Once again: there is no dumb questions in Math. For example in the set of rationals there is no convergence of series like lim n->inf (1+1/n)^n ; because the limit (e number) is not in the set of rationals. Thats why I am asking, you see?.
Why is it so damn quiet though
I watched this whole video waiting for an explanation of the proof of the LUB property of R, only to have the instructor go “welp, next time!” 😭😭😭
which text HMC use for real analysis? anyone
Which book are they using? Thanks
This 🥺💝💝
Sir beautiful lecture...thank you so much...
No offense to other great professors but Prof Su kills it !!! I think more school should upload their video lecture for foundational class in Math, especially those bridge lower and upper level
couldn't you also show that there's no lub in Q by noting that the lub, L, must satisfy L^2 = 2? If we have L^2 > 2, it suffices to show that there is a
And we already know that x^2 = 2 has no solution in Q. Done.
Why would you say that symbols for "exist", "for all" and "implies" are not appropriate for math notations?!
@@jonalderson5571
Ok, thanks for the replay, it makes some sense.
But anyway, I think it's just a matter of a habbit. You get used to it in time like with alphabet of a new language. And you also read math in a different way than you would read some ordinary text. So, I would say that it could be helpfull in some way if you are new to it but after some time the idea that you wanna get through is much more important than the way you write it down. Better use that extra intelectual energy to figure out the math concepts better, than get lost in the way you "need" to write it down. Making it a strickt class rule doesn't make sense to me and has nothing to do with the main subject. That would be my oppinion.
wait he's actually using baby rudin for this
thank you very much teacher
Really good at explaining! Thank you:)
@MegaBushmann "Principles of Mathematical Analysis" by Walter Rudin
Bravo!
whats the name of the book he is using
+xoppa09 Rudin's Principles of Mathematical Analysis
dedekind cuts are so cool! WOAH OWO OWOWOWOWOWOWWO
What do you mean by "the book"?
I thin its principles of analysis by rudin
I'm just taking real analysis in my junior year, and I'm on track for what my college requires...
Thank you!
Francis Su
Francis Supreme.
For me watching this is like see math for the first time
*cut* to the chase HAHAHAHAHAAH LOLS
I was talking about the previous example on 27:50. True the Supremum does not have to be an element of the set, but it has to be an element of the Metric Space where the set resides. Notice that he had define S ﬤ E as the set of rationals. (So my previous comment is indeed false as I failed to notice this fact due to the poor quality of the video) Had we chosen Q_
SPEAK UP, STEVE.
Actually in my comment below I meant to claim you can't construct IR by trying to complete "real" operations under Q. By "real" I mean excluding sqr. negative number constructs).
Looking further in the video to the actual construction of IR, the idea of satisfying the need for completeness under Q is abandoned in favour of creating Dedekind cuts based directly on the rational numbers. But can there really be a justification for doing this? But even if it can be justified can you really have a construction which is non ennumerable? Surely you can only do this if each cut contains a non enumerable set (of irrational numbers). But the (potential) cuts cannot contain any irrational numbers because they have not yet been created. So again I don't see how the Dedekind cut construction can be valid.
@mappingtheshit Yes, it's something like that...thank you.
LIttLE P
I wish he repeated the doubts his students asked him.
anyone wants to learn together by the way? shoot me an email.
Hooboy. I would never read a proof that didn't make smart and economical use of quantifiers/shorthand.
It seems all Dedekind cut explanations in these courses use square root 2. But you can extend the idea and using algebraic numbers you can create an infinite set of non rationals. You can then throw in the transcendental numbers. So you have constructed an infinte set of non rationals spread over the number line. But surely this infinite set is ennumerable. Now Cantor claimed that IR is non ennumerable although he could not prove how non ennumerable (the so called continuum problem is still unsolved). So even if an infinite set of irrationals can be constructed this still leaves 100% of the irrationals unconstructed. So from the above argument it follows you cannot construct the irrationals (from Dedekind cuts). Or is there a flaw in my argument?
The alternative is to define IR axiomatically which is less than satisfactory.
...listening to Pangaea by Michael Cassette in the background... perfect.
NJ is not impressed 😂
LITTLE P
dope
This "little p" shenanigans annoys me.
Sup