Archimedean Property
ฝัง
- เผยแพร่เมื่อ 9 พ.ย. 2024
- 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
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!
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.
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
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.
Ok for the pedagogy of the LUB. Thank you very much.
I was just reading about the Archimedean property in my analysis text book. Great timing π-m!
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
I wanna see Oreo!!! I think she (or he) is also learning analysis from you. :))
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?
Every Dedekind-complete ordered field is isomorphic to the reals, so no. A good, insightful question.
@@tomkerruish2982 But if complete means Cauchy complete, there are the surreal numbers
It looks like super trivial but it's actually pretty cool ! It says that there is no infinitely small or large real numbers.
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.
best of luck
which textbook you used in video ?
btw Excellent video ! thanks , helped a lot
Ross
This was "discovered" before Archimedes, by Eudoxos
The history of mathematics is kind of counfusing
@@mepoor761 If you're curious, there's an interesting "History of Science" series on some TH-cam channel that you should check out.
@@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
Wow.. . Great sir 🙏🙏🙏🙏 I am from India
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!
sites.uci.edu/ptabrizi/math140asp20/
@@drpeyam yeaaay thank u !! Also ur bunny is super cuute!
Good teature
Hi Dr. Peyam, Love your videos. Can you please make one on Singular Value Decomposition (SVD)?
Very good
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.
I got a mic, you’ll see the difference in maybe 3 months or so
It is equivalent to n>b/a and the theorem follows because N ha sup = infty
This is ridiculously intuitive but mathematicians wants to minimize the number of axioms, so they do this? OMG! Save me...
You should do real analysis course for beginners
That’s what this is!
@@drpeyam I mean the complete course
The whole course is on my playlists
Sodium is bigger than b. :P
Does this property not also exist for the rationals?
Of course, but the whole point is to prove it in R.
@@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?
What motivation did Archimedes use? (If it is known).... = = = >>>>>what kind of problem did he develop this to use in solving it?
He tried to fill a bathtub with a teaspoon hahaha
Dr Peyam silly but funny!
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?
It does, but can you prove it for real numbers?
@@drpeyam i guess it depends on what definition of addition im allowed to use haha
Hey Doctor Peyam!!
Seeing u after a long time!! how r u??