Good timing... right now, we’re about to define the real numbers in my Analysis course! Though now that I think about it, that’s probably the case in most universities in the Northern Hemisphere... so not too much of a coincidence.
It should've started with redefining Cauchy sequences, since the epsilon there can no longer be 'real' that we don't know about as yet. And to redefine, we can just take the epsilon to be any positive non zero rational, as the crux lies in the fact that it still gets arbitrarily close to 0 in the usual metric on Q.
In the beginning, we need to emphasize that epsilon is rational because we only know about rationals as of that time and that cauchy criterion is different from the cauchy criterion for reals
14 seconds in.. what did I notice? Dr. P. is wearing a Gaussian sweet Integral blouse, he is super happy to introduce us to the Real Numbers, and........... I mean, isn't that just a hot looking "R" on the top-left corner of the board.
12:00 Im not entirely sure I understand how the order relation is defined. Is this true for all epsilon > 0? If it were, if we chose two rational numbers r and s then we could choose epsilon > |r-s| and this would surely fail, since both sequences are constant?
No it's true for some epsilon, i.e. there is such a positive rational epsilon and such a natural number N that for all natural M, P >= N the equation p_M + epsilon < q_P holds. It may be a bit confusing, because most definitions in calculus start with "for all epsilon...", but here it's "there is such an epsilon...".
why not have the definition of q_n>p_n be that there exists some number N and some number 𝜖>0 such that when n⩾N q_{n}-p_{n}>𝜖? why does the statement after there exists a number N depend on 2 variables?
How do we know that R contains the irrationals? I see no proof that this definition of R contains anything more than Q already does. Pi might be irrational but youre defining it as a sequence of rationals... and I just dont see why this gives rise to an R with something more than rationals in it.
ou are right, it is weird, but the thing is that a real/irrational number is itself rather the SET of all the rational numbers being less than that supposed real number. Now you might ask isn‘t that stupid to call the sets the reals numbers thenselves? Well maybe not so stupid after all if you can show that you can add and multiply with these sets like rational or integer numbers as usual.
It's kinda confusing for me when you are talking about the "smallest integer upper bound," you say that the existence of the smallest one is ensured by the fact that there is finite of them to choose from. I was so confused because that I'm sure that there's infinite upper bounds once there's one. Later, I see that you're basically saying a non-empty set of integer that is bounded both above and below can have a well-defined smallest element. The thing kinda bothers me is not you're not mentioning the well-ordering principle (in natural numbers, in this case) while using it. (Well, to be precise, it's that of countable bounded-below set, but it's equivalent to that of natural numbers.) I get it you can check inductively one by one and end up somewhere, but for that to be able to be carried out in a rigorous fashion you also need the well-ordering principle, I believe. That being said, not stating the principle didn't really disqualify the proof. Also, I believe you can elaborate more on if order relation on the equivalence class is well-defined. All in all, this is still a wonderful video. I never got to see the construction myself. Thanks for making a video on this.
No for finite sets you don’t have to worry about the well ordering principle since you can literally compare each element one by one. It’s for infinitely many elements that you have to think about the well ordering principle.
@@drpeyam Yes I agree However, the "finite set" you wanna find a smallest number on is not clearly specified in the video. (Sorry if you found me annoying.)
I think it is specified. You first pick one integer upper bound and then there are only finitely many integers between 0 and that integer, so you just find the one that is smallest, and you can do so by writing a recursive procedure that only involves finitely many steps. And no, not annoying at all
Sorry, I don't understand. How can a number n be defined as a series of other numbers that eventually converges to n but could be no closer than 1dp from m by the trillionth term, which presumably goes on forever but never includes n? That is a rubbish idea. Imagine the scene. Nice car, how much was it? £12,457 £8,400 £19,850 5p .......
Good timing... right now, we’re about to define the real numbers in my Analysis course! Though now that I think about it, that’s probably the case in most universities in the Northern Hemisphere... so not too much of a coincidence.
"let's the problem become the solution"
alright I'll go dissolve myself in acid then :D
Wow got to love this comment
Makes me happy every time i see it. Thanks
Oh man i need to learn Analysis asap to understand this channel!!!
Dr Peyam is the coolest mathematician 😎😎😎😎
Based 😎
It should've started with redefining Cauchy sequences, since the epsilon there can no longer be 'real' that we don't know about as yet. And to redefine, we can just take the epsilon to be any positive non zero rational, as the crux lies in the fact that it still gets arbitrarily close to 0 in the usual metric on Q.
when I defined f(x) to be the special function that solves the ode in my differential equations class, I got points off my test :(
@7:09 This was too much fun!!!
In the beginning, we need to emphasize that epsilon is rational because we only know about rationals as of that time and that cauchy criterion is different from the cauchy criterion for reals
The cool way... Really ? Whoa !!! Thank you very much.
14 seconds in.. what did I notice?
Dr. P. is wearing a Gaussian sweet Integral blouse, he is super happy to introduce us to the Real Numbers, and........... I mean, isn't that just a hot looking "R" on the top-left corner of the board.
You should definitely consider making more videos
Sir, your videos ar really amazing. IA am from India.
That was a journey!
whoa, you uploaded this 1 day bfore my 18th birthday.
Metric completion is the way to go! Isn't that the standard to learn about real numbers?
No there’s also dedekind cuts
but that's not so cool. and you don't get p-adic numbers via cuts, or do you?
Dedekind cuts make proving the least upper bound property super elegantly, check out my real numbers playlist
Cool way, indeed!
I watched the other videos and thought "these aren't the cool way, it's cauchy sequences"
:) :) :)
How to show the relation of order does not depend on the representants of the class?
Happiest man on earth.
Cool!
Amazing video!!
Such a great video! Helped me a lot :))
12:00 Im not entirely sure I understand how the order relation is defined. Is this true for all epsilon > 0? If it were, if we chose two rational numbers r and s then we could choose epsilon > |r-s| and this would surely fail, since both sequences are constant?
No it's true for some epsilon, i.e. there is such a positive rational epsilon and such a natural number N that for all natural M, P >= N the equation p_M + epsilon < q_P holds.
It may be a bit confusing, because most definitions in calculus start with "for all epsilon...", but here it's "there is such an epsilon...".
Does LUB imply completeness?
For R it does
this is nice
why not have the definition of q_n>p_n be that
there exists some number N and some number 𝜖>0 such that when n⩾N q_{n}-p_{n}>𝜖? why does the statement after there exists a number N depend on 2 variables?
ok never mind i think 1 can prove that they are in fact equal. assuming p_n and q_n are both convergent.
@@aneeshsrinivas9088 being cauchy implies being convergent
@@sebastianwidua2055 then i guess they are equivilent
Unreal (U) are neat aswell
What’s with the equivilence classes?
?
Why are real numbers equivilence classes of cauchy sequences over just regular cauchy sequences?
Beautiful, isn’t it?
It would be if i understood this better.
Ok I'm starting to understand this better and can see the beauty of this.
Sir ,,, please give some lacture series on a particular chapter.
?
It’s called a playlist
As you showed that it is ordered you should have shown the axioms i.e. that always [(p_n)] < or = or > [(q_n)]
?
Dr Peyam that would be the law of trichotomy
K
Why is this unlisted?
Because
@@drpeyamb... e... c... a... u... s... e...
nani
whats an equivilence class. i know what an equivilence relation is but not an equivilence class?
How do we know that R contains the irrationals? I see no proof that this definition of R contains anything more than Q already does. Pi might be irrational but youre defining it as a sequence of rationals... and I just dont see why this gives rise to an R with something more than rationals in it.
Q is not complete. Since we are showing R has the LUB property, R is complete.
ou are right, it is weird, but the thing is that a real/irrational number is itself rather the SET of all the rational numbers being less than that supposed real number. Now you might ask isn‘t that stupid to call the sets the reals numbers thenselves? Well maybe not so stupid after all if you can show that you can add and multiply with these sets like rational or integer numbers as usual.
How is this the cool way of defining ℝ?!
It’s really cool
I fail to see why this is better than dedekind cuts.
I think you have it backwards, the dedekind cuts is the cool way of defining ℝ
Why would we want this construction of ℝ over the dedekind cuts way
No, this is cooler than cuts
How are there comments from 2 months ago
I was wondering that, too. But it is less disturbing than comments from the future.
Magic
It's kinda confusing for me when you are talking about the "smallest integer upper bound," you say that the existence of the smallest one is ensured by the fact that there is finite of them to choose from. I was so confused because that I'm sure that there's infinite upper bounds once there's one. Later, I see that you're basically saying a non-empty set of integer that is bounded both above and below can have a well-defined smallest element. The thing kinda bothers me is not you're not mentioning the well-ordering principle (in natural numbers, in this case) while using it. (Well, to be precise, it's that of countable bounded-below set, but it's equivalent to that of natural numbers.) I get it you can check inductively one by one and end up somewhere, but for that to be able to be carried out in a rigorous fashion you also need the well-ordering principle, I believe. That being said, not stating the principle didn't really disqualify the proof. Also, I believe you can elaborate more on if order relation on the equivalence class is well-defined.
All in all, this is still a wonderful video. I never got to see the construction myself. Thanks for making a video on this.
No for finite sets you don’t have to worry about the well ordering principle since you can literally compare each element one by one. It’s for infinitely many elements that you have to think about the well ordering principle.
@@drpeyam Yes I agree However, the "finite set" you wanna find a smallest number on is not clearly specified in the video. (Sorry if you found me annoying.)
I think it is specified. You first pick one integer upper bound and then there are only finitely many integers between 0 and that integer, so you just find the one that is smallest, and you can do so by writing a recursive procedure that only involves finitely many steps. And no, not annoying at all
Hmm, I felt like I can understand more now. Thanks fore replying. :)
Sorry, I don't understand. How can a number n be defined as a series of other numbers that eventually converges to n but could be no closer than 1dp from m by the trillionth term, which presumably goes on forever but never includes n? That is a rubbish idea.
Imagine the scene. Nice car, how much was it?
£12,457 £8,400 £19,850 5p .......
This video had 69 comments and I destroyed it by commenting, now it has 70