2:20 Good Place To See That 14:47 Michael’s « homework » ? 15:58 0 is already in N in my book 25:22 Michael’s homeworks (great places to start) 27:04 Good Place To Stop Who asked, who cares but I’m going into a 3 week vacation on Wednesday. I should be able to post timestamps still but not everyday.
I'm currently working on a problem of the distribution of primes in certain intervals. I am astonished time and again at the amount of structure that arises from these basic facts about divisibility. In particular, I'm finishing preparations for a lecture in a couple of weeks on the Sieve of Eratosthenes, which I hope you will enjoy on my channel when I publish it. It is clear to me that there are somewhat more modern and sophisticated perspectives, not far off the beaten path, using this ancient technique coupled with other accessible facts, many of which are presented in this video.
Great video! Following the lead of this one, can you make also a video about the tricks to understand if a prime divides a generic integer? Thank you in advance
Do we have to prove the uniqueness of r and q for the division algorithm? Isn’t min of a subset of natural number is always unique and q being a one to one mapping of r hence also unique?
Trying to find where to practice these problems more. It's very interesting and not as complicated as one would think but, I like to put in the work. Does anyone know, could suggest, a good book or site to practice this more in depth?
20:45 it seems wrong to me that you're using the Archimedean principle (which is theorem of the real numbers provable only using the completeness of the reals) to prove a basic result in the theory of integers / natural numbers.
When Michael says "Archimedean principle", it's referring to the fact that any nonempty subset of the natural numbers has a minimum. (BTW if you're referring to the property that says the natural numbers are unbounded, then you can prove it without the completeness of R, because it's also true for the rationals.)
@@integralboi2900 The Archimedean principle is that given two positive numbers x and y, there is an integer n so that nx > y. It shows that there are no "infinitesimal" numbers. The fact that any nonempty subset of the natural numbers has a minimum is the well ordering principle and can be proven using induction.
@@normanstevens4924 From what I understand, Michael made the same mistake in the first video of the series, calling the well-ordering principle the Archimedean principle.
@@normanstevens4924 I think you're referring to the archimedean property, Michael calls the property he used "The Archimedean Principle" in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-IaLUBNw_We4.html
This obviates the need to define N +0 as N has 0 W does not Z has N and -W and is the equivalence class of pairs of integers defined with a1+b2=a2+b1 means a1,a2 ~ b1,b2 etc.
I don't know if you are still looking for the solution or not but this is how I tried to figure this out:- drive.google.com/file/d/1QV6l9o1_qrW-93uiDUxirzWQgXHuK719/view?usp=drivesdk
@@debtanaysarkar9744 yeah I guess its enough to just change the union to include integers bigger than -b/2 and the rest of the proof holds up, ps you have very nice handwritting!
I did something very similar to the general divisibility problem note that is min(s) =< b/2, the proof remains identical, I just added that if min(s) > b/2 to use min(k)-b and added one to the quotient (basically using the first number and index that would be negative), you can prove that this has to work on the second case because the difference would have to be greater than b for it not to work, and, given that we are substracting b that would be impossible
Indeed, there are several comments but here's the most detailed : ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-IaLUBNw_We4.html&lc=UgzHphYr2JcJs4qFsoh4AaABAg
(Note: In Vietnamese, we use the "divides notation" oppositely because it makes sense) For example, you can write: 40 | 8 (because 40/8 = 5, 5 is an integer) But you cannot write: 8 | 40 (like the one that has been introduced in the video) because 8/40 = 1/5, the result is NOT an integer.)
@@timobrien6957 I pressed shift but my phone had other ideas. But if you're going to be so rudely technical, I too might as well be. "Right." Is not a proper sentence. And you failed to specify what you are speaking in reference to in your second sentence.
@@PubicGore Unlucky. Also, since this is the context of a dialogue (and I am clearly the one responding to a statement), one word sentences are grammatically correct. And I made no failure to specify anything.
@@timobrien6957 Where are you pulling the rule that being in "context of a dialogue" makes it grammatically correct to use a single word sentence? That's not a rule. Also, you did fail to specify the subject precisely.
I tend to distinguish whole numbers and natural numbers - whole numbers are 1,2,3 natural are those produced from set theory so emptyset - 0 set containing empty set 1 and the sucessor set contains all the priors in sets
0:18 "Suppose that a and b are integers" Me: But how do you knoooooowwwwwwww? "And a is not equal to zero" Me: Well, if I can just make stuff up. There's always someone like this in the class. Usually it's me.
