In this video, I prove the Archimedean property of real numbers, which says that for every real numbers and b positive 0, there is an integer n such that na is greater than b Check out my Real Numbers Playlist: • Real Numbers
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@starter4974 жыл бұрын
This is an extremely powerful theorem. It took me a while to understand the importance of it but its probably the most useful theorem when doing any sort of crazy delta epsilon proofs through not obvious direct or indirect proofs. For instance because of this property, you can always assume that for every epsilon > 0 there exists an N in naturals such that N(epsilon) is bigger than any fixed number. So when doing proofs by contradiction, generally you fix a value ( 1 is probably the easiest) and try to get a contradiction. It turns out if you fix 1 to be that value, you get there exists N such that N(epsilon) > 1. After some calculations you get, 1/ N < epsilon. Then for certain problems, you can see how to go about the proof accordingly. Sorry Dr. Peyam! I am studying for a math analysis qualifier exam and these videos are really helpful for me to refresh my memory on the fundamentals. Anyways, keep up the videos!
@ai_serf9 ай бұрын
The more I do math the more I realize how powerful these simple theorems are, regardless of how seemingly arbitrary or obvious they are. When we grok them deeply, we can use them to make rigorous arguments, that otherwise, would be filled with hidden assumptions.
@andreutormos72103 жыл бұрын
I very much appreciate analogies to understand what a given proposition is fundamentally saying. In this case, thinking it in terms of a given total and a given currency really helped me visualize the Archimedean Property :D
@albertodelaraza44754 жыл бұрын
I have always found your videos both entertaining and informative. Thank you. May I make one suggestion, however? Since you're left-handed, may I suggest that you angle the camera from the left side of the board instead of the right side, or even straight on? This will give us a much better view of what you are writing as you verbalize the lesson. Often, all we see is your back during some critical steps. I've always found that writing on a board while teaching any concept superior to the "PowerPoint" method. But, if we can't see what is being written, then we lose a little something when we have to back read what was already discussed and fall out of sync with your discussion. Just a friendly suggestion.
@Kdd1604 жыл бұрын
I wanna see Oreo!!! I think she (or he) is also learning analysis from you. :))
@BootesVoidPointer4 жыл бұрын
I was just reading about the Archimedean property in my analysis text book. Great timing π-m!
@dgrandlapinblanc2 жыл бұрын
Ok for the pedagogy of the LUB. Thank you very much.
@s00s774 жыл бұрын
reminds me of good old baby rudin, but better written (or said) i remember i could understand his proof, but it took me a while to seem this natural
@xavierplatiau46354 жыл бұрын
It looks like super trivial but it's actually pretty cool ! It says that there is no infinitely small or large real numbers.
@cerwe88614 жыл бұрын
This was "discovered" before Archimedes, by Eudoxos
@mepoor7613 жыл бұрын
The history of mathematics is kind of counfusing
@Natalija3793 жыл бұрын
@@mepoor761 If you're curious, there's an interesting "History of Science" series on some KZbin channel that you should check out.
@mepoor7613 жыл бұрын
@@Natalija379 idon't like this kind of course to be honest altough im curious about mathemtics and its history I praper for my own resreach because i want my ideas and my view about the subject to be deep rather then whatching few epaiods But think you any way and i think this could be useful for me for seek of explration
@hassanalihusseini17174 жыл бұрын
Thank you for the video. When I was introduced to the real numbers I was taught the Archimedian property as an axiom. So I was always wandering how would a complete ordered field be without the Archimedian axiom. Can there be even something like that?
@tomkerruish29824 жыл бұрын
Every Dedekind-complete ordered field is isomorphic to the reals, so no. A good, insightful question.
@orlandomoreno61684 жыл бұрын
@@tomkerruish2982 But if complete means Cauchy complete, there are the surreal numbers
@muskannm13423 жыл бұрын
which textbook you used in video ? btw Excellent video ! thanks , helped a lot
@drpeyam3 жыл бұрын
Ross
@nailabenali74884 жыл бұрын
You make maths so much fun and even beautifull!! I just wanna know if there is some pdf when u keep notes of every video?Because I need to watch 3 of your playlists and it's really hard and time consuming to write it all :D!! Thanks a loooot!
@drpeyam4 жыл бұрын
sites.uci.edu/ptabrizi/math140asp20/
@nailabenali74884 жыл бұрын
@@drpeyam yeaaay thank u !! Also ur bunny is super cuute!
@shashwatbajpai69112 жыл бұрын
Very good
@user-td6pl6wk6s Жыл бұрын
Good teature
@SartajKhan-jg3nz4 жыл бұрын
I never thought analysis could be this fun albeit being so difficult. Damn Dr Payem YOU ARE SICK. P.S i am learning all of this for the first time and idk how i am understanding all of it.
@igorfritz29733 жыл бұрын
best of luck
@xhem10613 жыл бұрын
Wow.. . Great sir 🙏🙏🙏🙏 I am from India
@knivesoutcatchdamouse21374 жыл бұрын
I appreciate your videos, but on my cell phone I have trouble hearing you on some of them (like this one) even with the volume at max. Please try to speak a bit louder, maybe? Thank you.
@drpeyam4 жыл бұрын
I got a mic, you’ll see the difference in maybe 3 months or so
@ramitsurana4 жыл бұрын
Hi Dr. Peyam, Love your videos. Can you please make one on Singular Value Decomposition (SVD)?
@xriccardo18314 жыл бұрын
It is equivalent to n>b/a and the theorem follows because N ha sup = infty
@TechToppers3 жыл бұрын
This is ridiculously intuitive but mathematicians wants to minimize the number of axioms, so they do this? OMG! Save me...
@andreapaps3 жыл бұрын
Sodium is bigger than b. :P
@nathanisbored4 жыл бұрын
to me this feels more like a property of addition than a property of real numbers.... doesnt this also apply to positive rationals and integers for example?
@drpeyam4 жыл бұрын
It does, but can you prove it for real numbers?
@nathanisbored4 жыл бұрын
@@drpeyam i guess it depends on what definition of addition im allowed to use haha
@gisopolis774 жыл бұрын
Does this property not also exist for the rationals?
@drpeyam4 жыл бұрын
Of course, but the whole point is to prove it in R.
@BLCAAN2 жыл бұрын
@@drpeyam In R we use the l.u.b. property in order to prove the A.P., how would you go about to prove the A.P. in Q?
@dougr.23984 жыл бұрын
What motivation did Archimedes use? (If it is known).... = = = >>>>>what kind of problem did he develop this to use in solving it?
@drpeyam4 жыл бұрын
He tried to fill a bathtub with a teaspoon hahaha
@dougr.23984 жыл бұрын
Dr Peyam silly but funny!
@ritesharora21234 жыл бұрын
You should do real analysis course for beginners
@drpeyam4 жыл бұрын
That’s what this is!
@ritesharora21234 жыл бұрын
@@drpeyam I mean the complete course
@drpeyam4 жыл бұрын
The whole course is on my playlists
@ShivanshTrisal4 жыл бұрын
Hey Doctor Peyam!! Seeing u after a long time!! how r u??