@@geovanygrant5298 i'm not preparing for olympiad. but from any english number theory sources i've given attempt on youtube so far, this is notably the best for pure beginners. If intermediate enough, you could check out richard e boecherds
@@henocksherlock3340 I would actually only recommend it for advanced students, or math undegrads. It definitely requires abstract algebra to fully understand what he says.
@@chrisjuravich3398 But since this is a video for beginners THAT is an unforgivable mistake. Some will stop watching right there because they don't understand what is going on. If a mathematician doesn't master simple multiplication then maybe he shouldn't make videos. THE LEAST he could do is to put a TEXT TAG over the video saying he fk'ed up, but maybe his ego won't allow him to do that?
Definition: For integers A, B , A | B if and only if B = A*k for some integer k. That is, A | B iff B is a multiple of A." As an alternative to the word 'divides' , A | B can mean equivalently A is a factor of B . Note: The notation A | B is actually more nuanced that simply saying A is a factor of B. To be consistent with the definition , we say A | B when B can be factored (split up) in such a way "B = x * y" , such that x = A and y is some integer. So though B | B is true because B is a factor of B since B = B, it is true that b | b using the definition since B = B * 1. Why go through all this trouble , instead of just using the definition? I think saying a is a factor of b is a nice mental shortcut. Also the word factor has the connotation of being split up. Single factorizations of numbers are done for the sake of making statements more concise. In any case, when i see a number like B, then i think B = x * y * z * w , then x, y, z, w are factors of B. Also for any number B, B = 1 * B , so 1 is a factor of B, and B = B * 1 , so B is a factor of B. The proof that a | 1 then a = {-1, 1} is quite short. Since 1 = 1 * 1 and 1 = (-1) * (-1) , so a = { 1, -1} . ok, to prove that there are no other factors might involve some heavy proof methods.
Yeah but the whole thing with number theory is using proofs rather than words like that. We're learning exactly what those words mean. Ie: think like a robot.
I notice that your play list closely follows the chapters in "Elementary Number Theory" by Gareth A Jones and J May Jones by Springer. Is that the textbook you're using?
I wonder if my proof of 2 and 3 are correct (I mean, if it respects all the integers axioms, I need to practice my ability to write a proof). Proof of 2: If a│b then exists an integer k such that a.k=b. Multiplying both sides by an integer c we get c.(a.k)=c.b By associative: a.(c.k)=c.b. Denoting c.k=k' we have a.k'=c.b and it implies that a│c.b. # Proof of 3: If *a│b* then exists a integer k such that a.k=b. If *a│b* then exists a integer l such that a.l=c. By 2, there is a integer k' such that a.k'=b.x for some integer x. (i) By 2, there is a integer l' such that a.l'=c.y" for some integer y. (ii) Adding (i) and (ii): a.k'+a.l'=b.x+c.y a.(k'+l')=b.x+c.y If (k'+l')=m, then a.m=b.x+c.y and it follows that a│b.x+c.y #
Thank you for making these wonderful videos; you explanations are very clear and help me to understand. Divisibility? More like divis-easy-ability now!
6:10 The absolute value of any integer is greater than equal to 1" that is false, the absolute value of an integer, |a|, can be zero. however, since we assume a | 1 , then we reject the case a = 0 since 0 | 1 is impossible.
I know this may descend into a bottomless pit. Given this is about number theory, how far do we go with various "operations". For example, do we need to define/prove addition, multiplilcation; associativitiy (which was assumed in video); communitivity (needed in proves for the other 2 propositiions); distribution of multiplication over addition etc. There is Peano, but where does that fit in? Should it be learnt before everything else?
I personally feel like the absolute value proof to not be satisfying, using more intuition than rigor. My constructive approach : First realize that a and k must be of same sign. For the positive case, if k > 1, then a < 1. As a is an integer and there exists no integer between two consecutive integers (in this case 0 and 1), then k can't be > 1. As k is positive, and cannot be greater than 1, than k = 1. The negative case is similar. QED.
I am from India and it's Soo helpful I have learnt number theory like this for olympiads u should see some questions of JEE ADVANCED exam it's one of the most roughest exam with impossible integrals
The best thing on divisibility is the fact that the definition uses an non constructiv existence to avoid division and make "divisibility" a multiplication. This is so nice and powerful.
@mohamed benabbou The definition is: For a, b integers we say: a divides b, notation: a | b : it exist k integer, s. t. b = ka So there is the word "exist". The definition say: a divides b if you can find k, s.t. b = ka. But it doesn´t tell you, how to find this k. This is "non constructive existence".
Mark Zuckerberg hitting the gym!
lol
gordon ramsay
😂😂😂
Do you like being touch?
Finally a course on Number Theory to help me in self study and Olympiad preparation. Thanks a lot sir!
Are you still preparing? Did you get it right?
Did it help?
@@geovanygrant5298 i'm not preparing for olympiad. but from any english number theory sources i've given attempt on youtube so far, this is notably the best for pure beginners. If intermediate enough, you could check out richard e boecherds
@@henocksherlock3340 Thanks
@@henocksherlock3340 I would actually only recommend it for advanced students, or math undegrads. It definitely requires abstract algebra to fully understand what he says.
Your channel is gold, man. Much respect.
MR. Penn, thank for a fantastic introduction to Number Theory
Love the videos :), but isn’t 2 times 4 equal to 8 and not 16?
That was a nasty mistake by him😁
I think his brain switched to powers of 2 lol
2 times 4 is 16 for sure.. Yes it is
Thank you for the correction. Cut him some slack! These are awesome videos, and they’re free for everyone.
@@chrisjuravich3398 But since this is a video for beginners THAT is an unforgivable mistake. Some will stop watching right there because they don't understand what is going on. If a mathematician doesn't master simple multiplication then maybe he shouldn't make videos. THE LEAST he could do is to put a TEXT TAG over the video saying he fk'ed up, but maybe his ego won't allow him to do that?
Thanks for your efforts, i really liked this video and honestly one of the reason to like this video was blackboard and chalk .
It is clear exposition, and you are making proof, step by step, thank you very much, I like your explanation very much.
Hi, I wanted to ask whether all of the videos in your number theory Playlist (113 videos in total) are properly ordered?
Definition: For integers A, B , A | B if and only if B = A*k for some integer k. That is, A | B iff B is a multiple of A."
As an alternative to the word 'divides' , A | B can mean equivalently A is a factor of B .
Note: The notation A | B is actually more nuanced that simply saying A is a factor of B. To be consistent with the definition , we say A | B when B can be factored (split up) in such a way "B = x * y" , such that x = A and y is some integer.
So though B | B is true because B is a factor of B since B = B, it is true that b | b using the definition since B = B * 1. Why go through all this trouble , instead of just using the definition? I think saying a is a factor of b is a nice mental shortcut.
Also the word factor has the connotation of being split up. Single factorizations of numbers are done for the sake of making statements more concise.
In any case, when i see a number like B, then i think B = x * y * z * w , then x, y, z, w are factors of B.
Also for any number B, B = 1 * B , so 1 is a factor of B, and B = B * 1 , so B is a factor of B.
The proof that a | 1 then a = {-1, 1} is quite short.
Since 1 = 1 * 1 and 1 = (-1) * (-1) , so a = { 1, -1} .
ok, to prove that there are no other factors might involve some heavy proof methods.
equivalently,
a | b iff a has a multiple that is b,
or even briefer , a | b iff a has multiple b.
Yeah but the whole thing with number theory is using proofs rather than words like that. We're learning exactly what those words mean. Ie: think like a robot.
I notice that your play list closely follows the chapters in "Elementary Number Theory" by Gareth A Jones and J May Jones by Springer. Is that the textbook you're using?
You pick up any textbook for elementary numbet theory it follows more or less the same pattern
Great Chanel, the best that I have seen talking about Number Theory. You are helping me a lot. Congratulations from Spain.
Tengo ke vi larco n tigo oi
Despacito
Your videos are realy helping me lot. Thank so much Sir
Sir, you must have made a slight mistake.... 2 * 4 = 8 and not 16... did you just mean 2^4 = 16?
typo maybe 2*8 = 16
thank you very much sir .really helped me a lot .now i can clear my doubts through your videos
.and also i am preparing for isi examination
I wonder if my proof of 2 and 3 are correct (I mean, if it respects all the integers axioms, I need to practice my ability to write a proof).
Proof of 2: If a│b then exists an integer k such that a.k=b.
Multiplying both sides by an integer c we get c.(a.k)=c.b
By associative: a.(c.k)=c.b. Denoting c.k=k' we have a.k'=c.b and it implies that a│c.b. #
Proof of 3:
If *a│b* then exists a integer k such that a.k=b.
If *a│b* then exists a integer l such that a.l=c.
By 2, there is a integer k' such that a.k'=b.x for some integer x. (i)
By 2, there is a integer l' such that a.l'=c.y" for some integer y. (ii)
Adding (i) and (ii):
a.k'+a.l'=b.x+c.y
a.(k'+l')=b.x+c.y
If (k'+l')=m, then a.m=b.x+c.y and it follows that a│b.x+c.y #
1:06
16=2×8
i saw that too lol
Yes
I must say, that was for a mathmatician like him
16=2*4?
Thank you for making these wonderful videos; you explanations are very clear and help me to understand. Divisibility? More like divis-easy-ability now!
Great efforts , thanks a lot
6:10 The absolute value of any integer is greater than equal to 1" that is false, the absolute value of an integer, |a|, can be zero.
however, since we assume a | 1 , then we reject the case a = 0 since 0 | 1 is impossible.
He has also defines this in equation that where a is not equal to 0
firstly no integer times zero equal 1
Very clear and efficient. Love this!
I know this may descend into a bottomless pit. Given this is about number theory, how far do we go with various "operations".
For example, do we need to define/prove addition, multiplilcation; associativitiy (which was assumed in video); communitivity (needed in proves for the other 2 propositiions); distribution of multiplication over addition etc.
There is Peano, but where does that fit in? Should it be learnt before everything else?
Your channel is great 💛
I personally feel like the absolute value proof to not be satisfying, using more intuition than rigor. My constructive approach :
First realize that a and k must be of same sign. For the positive case, if k > 1, then a < 1. As a is an integer and there exists no integer between two consecutive integers (in this case 0 and 1), then k can't be > 1. As k is positive, and cannot be greater than 1, than k = 1. The negative case is similar. QED.
Valeu!
Pra que tu tá estudando isso, doidão? O bagulho é pegar mulher e assistir o framengo! Nerd.
how does the ineqaulity work
Love from 🇮🇳India
I really thought he would notice the mistakes in 2|16. Anyway, thanks for this video!
thank you very much ❤
you are great ❤
and can you talk about polynomials pleas😊?
for preparing to its hard problem
great effort love from india
I felt like i am in my actual math class again.
Your videos are great!
Ok. Just in case someone doesn’t know… a|b means a divides b NOT a/b . This looks like b/a in lower math common usage.
16= 2*8
I didn't understand part when he says c = a(kl) = ak' ....I mean what is k' ?
nvm I got it
Cool lecture my fellow chimpin-puppydog
great thanks. Haven't done that in a while and needed this🙂
Bro, you really cleaned that board fast
Thank you sir ❤
16 = 2 \times 8 ... when are you going to fix that ?
I’m not sure why we should exclude a=0 for divisibility
good job thank you
wow penn great lecture
I am from India and it's Soo helpful I have learnt number theory like this for olympiads u should see some questions of JEE ADVANCED exam it's one of the most roughest exam with impossible integrals
Why can a be 0?
Great video....! One question..if a devides b, b=a×k, for some k€Z ...does k need to be greater than 0 always, or can it be negative also..?
It can be negative also.
i like the way u wrote a euro sign for ‘is an element of’
thank you
Interesting, I have never seen it proved ;o
You need to place a quantifier on k.
You have a mistake 2|16 because
16 = 2 . 8 not 2.4
UKW, 2 multiplied by 4 is 8 and not 16
The best thing on divisibility is the fact that the definition uses an non constructiv existence to avoid division and make "divisibility" a multiplication. This is so nice and powerful.
This definition is very neat because it holds for other rings where division is not always well defined but multiplication is
@mohamed benabbou The definition is:
For a, b integers we say:
a divides b, notation: a | b
:
it exist k integer, s. t. b = ka
So there is the word "exist". The definition say: a divides b if you can find k, s.t. b = ka.
But it doesn´t tell you, how to find this k.
This is "non constructive existence".
Good
💙
16 does not equal 2 times 4 :/
2|16 16=2*8😊
ok, great.
And that's a good place to stop
But you have to add that k can't be 0
He says theres additional stuff like that that can be added before he does into it.
"Good!"
We are barely even a minute in and we've already forgot that 2*4=8. Otherwise, nice channel.
16=2.8
Why does this all make sense… what is going on with my brain…
“Good”!
bros jacked
pove that : 123456789123456789.......123456789 is neither a perfect square or a cube
Because yes proved
how in the world is 2*4=16! have we forgotten that 2*4=8 !!!!!!!!!!!!!!!!!!!!
Easy to make a mistake, if you are also a genius and have your mind on other things and putting out videos every few hours.
LOLOLOLOLOLOLOLOLOOLOLOLOLOL OLOL
Great class!! I'm motivated to wath the rest of the course.
Who else here cuz the number theory professor can't explain this stuff well?!
i cant read ur hand writing
16=2×8?
1:05 16= 2•8