This was extemtly useful. Spent 10 minutes trying to understand my textbook explanation of this. I immediately understood what I was missing within 2 minutes of your video. Thank you
I discovered your channel a few days ago. What a pity you haven't more subscribers ! I want to learn number theory and go back to the fundamental WITH PROOFS seems an ideal way. Thanks!
Even though it's obvious and doesn't really require it, I want to fill in the gaps of the last step of the uniqueness proof. We have b(q-q')=r'-r, where 0
Good algorithm to divide. Can be called "Division by successive aproximation". It's only necesary to generate a seed big enough to reduce the amount of iterations. Good video
What an amazing channel for learning mathematics. I wonder how it has so less subscribers. Simply mind blowing playlist on number theory. Keep growing mate !!
@@antoniojg-b8284 what are you studying specifically as an applied math student. I only ask because I will soon be starting this degree as a freshmen in college :D
Does anyone know that this means? Its the first sentence of this cryptography class and I think it means I should drop it: "We say that a nonzero b divides > a if a = mb for some > m, where > a,b, and > m are integers. That is, b divides > a if there is no remainder on division. The notation > b|a is commonly used to mean b divides > a. Also if > b|a , we say that b is a divisor of > a. The positive divisors of 24 are 1,2,3,4,6,8,12 and 24."
Such a great proof. The way you explain things using examples is a lot clear than those literal "bookish" proofs. Thanks a lot. Keeping making things like these :)
Nice one (I'm aware that the length of the videos doesn't allow intuition, but for some, the steps may seem like magic). For the intuition about the set 'S', we can think about all 'b'-steps to the left and right (at the left is how many 'b' can we fit in 'a'). We don't allow it to go less than zero. It becomes apparent that S includes 'a' itself. If 'a' is negative, we simply extend the logic, and go 'b'-steps to the right until we reach something greater than zero, then start adding to the set 'S'. We then want to find 'r'. Of course it's the minimum of 'S', otherwise, we can subtract out another 'b'. For the final part, as we know 'r' and 'r'' are between 0 and 'b', then their difference must be less than 'b'. Now it's obvious that there is only one 'q' to reach between 0 and 'b', as each leap is at least 'b'-length, so anything else would escape.
Great proof, love your content, I can barely understand while looking at my lecture note, now I feel better when someone explains, keep up the good work, Jesus bless.
Oh boy time to binge number theory! I always feel so stupid that I can’t solve the number theory questions on this channel and everyone in the comments are talking about how trivial they are :/ Hopefully this series helps, it’s the only one I can find in number theory on TH-cam, so thank you for making it, Michael Penn!
same!! you probably got better (and I really hope so). I have a test on Thursday and I'm freaking out lol plus: English is not my native language, so it's even harder.
My God :))) I'm not an English speaker and I'm not a high level. But think about it man! How you teach and explain that even I could understand it well 🥺 thank youuuuuu
My godd....just speechless..this proof is just so easy to understand....I wish the same I had in the book. 😔...the way it is given in the book is literally going above my head😂
hi Michael I liked your video about euclidean algorithm but I know an another form to proof, suppose that q e q' are consecutive integers, then exist a rational between them, then we have. q0
Thank you for your videos! 😍 But I got lost at the contradiction on 8:21 .. Could someone please help me understand why did he do it and where did he take r >= b from?
what book do you think is perfect for learning number theory for the math olympics, because I was thinking about hardy, but this book has no exercises, so for me who am a neophyte I think it should be accompanied by an exercise book, or at least replaced with a book of theory and exercises, so what do you recommend?
What would've happened if r >= r' instead of vice versa (10:37)? Can the uniqueness proof be applied also to the generalization of b € Z-{0} instead of b>=0? Thank you
If we considered the other case that r >= r', you'd just rearrange the equation bq+r=bq'+r' to get r-r'=b(q'-q) and the same logic would apply. WLOG" just means that the other cases have the exact same steps and logic but with symbols swapped, and that we don't want to bother writing out something that's essentially identical.
@@salad7389 Since this is from a while ago I forget the details, but in logic, you can assume anything you want at any time. However, whatever consequences you get from an assumption are always chained to that assumption (at first), meaning you need to remember that they were a result of that assumption. If you want to forget about the WLOG business, you can do each case separately. You can first assume r'>=r and work out the consequence, and then assume r>=r' and work out the other consequence, and since both of these are the only possible options, then if their consequences are the same then their consequence has escaped the chains of the assumptions and is true regardless of the assumption. I hope my explanation provides some clarity. You can assume whatever you want at any time basically. We then used this logic theorem: P->Q. R->Q. P or R. Therefore Q.
Great video, but at 6:07 I did not understand how you went from b is greater than zero to 'it is bigger than or equal to one'. By 'it' are you referring to b here or not?
Sometimes I adjust the speed to 1.5 on certain channels, I find it disrespectful, and I am ashamed, even though they don't know. - But with you, Michael, I have to adjust it to 0.75. At that speed you sound more human, I suppose. Sorry. Your math is excellent. ;-)
Note that r' - r is a non-negative integer which is a multiple of b. Also note that since r is non-negative r' - r is a non-negative integer which is strictly less than b and hence r' - r is a non-negative integer less than or equal to b - 1. Since the only divisor of b between 0 and b - 1 is 0 itself the desired result follows. Hope it helps!
Not that good actually. The example are contrived and do not explore the realm of possibilities: a being negative, b being larger than a for instance. And because the result is proven over Z this could have been nice. The problem is that the intuition is not that of a problem: you want to measure a with b but the beauty reside in the fact that it is possible even with the case above.
These proofs literally left me speechless.
Still taking my breath away…
This was extemtly useful. Spent 10 minutes trying to understand my textbook explanation of this. I immediately understood what I was missing within 2 minutes of your video. Thank you
Both times are good. Hahahaha No one needs to understand a subjec int less than 10 minutes.
I discovered your channel a few days ago. What a pity you haven't more subscribers !
I want to learn number theory and go back to the fundamental WITH PROOFS seems an ideal way.
Thanks!
Even though it's obvious and doesn't really require it, I want to fill in the gaps of the last step of the uniqueness proof. We have b(q-q')=r'-r, where 0
WHAT A TEACHER THIS IS? Beautiful
Good algorithm to divide.
Can be called "Division by successive aproximation".
It's only necesary to generate a seed big enough to reduce the amount of iterations.
Good video
What an amazing channel for learning mathematics. I wonder how it has so less subscribers. Simply mind blowing playlist on number theory. Keep growing mate !!
Soo helpful!!I'm a Bsc mathematics final year student
I am 3rd year applied mathematics student. I just want to comment my status
@@antoniojg-b8284 what are you studying specifically as an applied math student. I only ask because I will soon be starting this degree as a freshmen in college :D
I comment coz u guys r funny I m phd student btw 🤣
Did you finish your degree?
The best of the best description of proof of Euclidean Division, I like more than VedTutor
Anytime I am stuck in any proof
You come as a saviour
Thank you so much Sir 🙏
Professor Penn ,thank you for an awesome lecture on The Division Algorithm.
Your channel really save my life. When my professor taught this session, he made it really difficult to understand. Thanks a lot !!!
Does anyone know that this means? Its the first sentence of this cryptography class and I think it means I should drop it: "We say that a nonzero b divides > a if a = mb for some > m, where > a,b, and > m are integers. That is, b divides > a if there is no remainder on division. The notation > b|a is commonly used to mean b divides > a. Also if > b|a , we say that b is a divisor of > a. The positive divisors of 24 are 1,2,3,4,6,8,12 and 24."
so nice to see this channel growing!
Please countinue this amazing videos
Such a great proof. The way you explain things using examples is a lot clear than those literal "bookish" proofs. Thanks a lot. Keeping making things like these :)
Nice one (I'm aware that the length of the videos doesn't allow intuition, but for some, the steps may seem like magic).
For the intuition about the set 'S', we can think about all 'b'-steps to the left and right (at the left is how many 'b' can we fit in 'a'). We don't allow it to go less than zero. It becomes apparent that S includes 'a' itself. If 'a' is negative, we simply extend the logic, and go 'b'-steps to the right until we reach something greater than zero, then start adding to the set 'S'.
We then want to find 'r'. Of course it's the minimum of 'S', otherwise, we can subtract out another 'b'.
For the final part, as we know 'r' and 'r'' are between 0 and 'b', then their difference must be less than 'b'. Now it's obvious that there is only one 'q' to reach between 0 and 'b', as each leap is at least 'b'-length, so anything else would escape.
Great proof, love your content, I can barely understand while looking at my lecture note, now I feel better when someone explains, keep up the good work, Jesus bless.
thank you sir, love from india
Excellent explanation. This most certainly helps. Thank you.
your teaching method are perfect
Oh boy time to binge number theory! I always feel so stupid that I can’t solve the number theory questions on this channel and everyone in the comments are talking about how trivial they are :/
Hopefully this series helps, it’s the only one I can find in number theory on TH-cam, so thank you for making it, Michael Penn!
same!! you probably got better (and I really hope so). I have a test on Thursday and I'm freaking out lol
plus: English is not my native language, so it's even harder.
Thank you from New Zealand. :)
Thank you from Singapore!
My God :)))
I'm not an English speaker and I'm not a high level. But think about it man! How you teach and explain that even I could understand it well 🥺 thank youuuuuu
My godd....just speechless..this proof is just so easy to understand....I wish the same I had in the book. 😔...the way it is given in the book is literally going above my head😂
Same here 😢
Thank you from Italy
fra ciao mi sai spiegare una cosa?
@@IPear Ciao, se posso volentieri.
@@fedepan947 Ti ringrazio, ho risolto.
i really prefer chalks over blackboards, the teachings are so smooth thanks!
Thanks sir!
❤️ From India.
hi Michael I liked your video about euclidean algorithm but I know an another form to proof, suppose that q e q' are consecutive integers, then exist a rational between them, then we have. q0
Being a student , i am saying you are the best teacher sir . You made it soo easy for me to understand . Just awesome 🔥 .
Thank you Sir! You made this proof undarstandable
You sir are incredible.
What an amazing video! Thank you.
Can you show another example division algorithm with proof?
U r amazing teacher love from india👍❤️
a perfect pure mathematician.
Thanks again for another wonderful video! Division? More like di-vision, because now I can see things clearly!
A lot of existence proofs in number theory depends on well ordering principle.
What does _unique_ mean in the definition and in general?
Thank you for your videos! 😍
But I got lost at the contradiction on 8:21 .. Could someone please help me understand why did he do it and where did he take r >= b from?
Because it is a contradiction r can't be greater are equal to b r is greater or equal to zero but less than b
What are the common prerequisite courses for this class?
A normal college math curriculum up through Linear Algebra or Differential Equations.
great video
Thank You So Much Sir
Why we know that r-b is the element of the S - after as we subtract b of both sides of equation ?
Thank you sir you are great 👍
what would happen if b < a?
Is this playlist (113 videos in total) in order??
Can any1 help me..
Why we have to consider two cases.... 1) b>= 0 2) b
Where are you?
how is the RHS been established can you clarify?
Thank you.... from Sri Lanka
Hello Sir ,. Please upload videos on Combinatorics also plzzz
what book do you think is perfect for learning number theory for the math olympics, because I was thinking about hardy, but this book has no exercises, so for me who am a neophyte I think it should be accompanied by an exercise book, or at least replaced with a book of theory and exercises, so what do you recommend?
NUMBER THEORY: CONCEPTS AND PROBLEMS by tutu, what do you think about this , is it too complex? or the exercises are not for beginners?
Thank you very much TT
Amazing
A great help for me
Thank u from India...
Thank you from india
How can we say that r-b
I’m also confused regarding this, it seemed like a jump in logic
Edit: Never mind, it was initially stipulated that b is greater than 0
What would've happened if r >= r' instead of vice versa (10:37)?
Can the uniqueness proof be applied also to the generalization of b € Z-{0} instead of b>=0?
Thank you
If we considered the other case that r >= r', you'd just rearrange the equation bq+r=bq'+r' to get r-r'=b(q'-q) and the same logic would apply. WLOG" just means that the other cases have the exact same steps and logic but with symbols swapped, and that we don't want to bother writing out something that's essentially identical.
@@wiggles7976 why are we allowed to assume r'>=r in the firstplace
@@salad7389 Since this is from a while ago I forget the details, but in logic, you can assume anything you want at any time. However, whatever consequences you get from an assumption are always chained to that assumption (at first), meaning you need to remember that they were a result of that assumption.
If you want to forget about the WLOG business, you can do each case separately. You can first assume r'>=r and work out the consequence, and then assume r>=r' and work out the other consequence, and since both of these are the only possible options, then if their consequences are the same then their consequence has escaped the chains of the assumptions and is true regardless of the assumption.
I hope my explanation provides some clarity. You can assume whatever you want at any time basically. We then used this logic theorem:
P->Q. R->Q. P or R. Therefore Q.
@@wiggles7976 that makes sense! Thanks for replying to my 5 month late question lol
Sir , the only multiple between 0 and b is 0.How?
Why do we have to prove the uniqueness of r ? Didn't we assume that r is the min(S) which makes it unique?
He _defines_ r to be min(S) and then shows that _this particular_ r satisfies a=bq+r (such an integer q must exist by the definition of S) and 0≤r
Mind blowing.
Great video, but at 6:07 I did not understand how you went from b is greater than zero to 'it is bigger than or equal to one'. By 'it' are you referring to b here or not?
b is an integer greater than zero. That means it is either 1 or greater than 1.
@@souverain1er Thanks, forgot we were dealing with integers, oops.
Thanks
Wanna watch more
I feel stupid. Hardly understood anything. I'm a regular viewer, so I'm sure the explanation was great, but it didn't work for me.
thanks man 😁
Thanks a lot from india
Good
Nice
Bravo!
Damn man what did the board do to you why you hitting it so hard
Awsome
Why you ppl forget to write a>b
Sometimes I adjust the speed to 1.5 on certain channels, I find it disrespectful, and I am ashamed, even though they don't know. - But with you, Michael, I have to adjust it to 0.75. At that speed you sound more human, I suppose. Sorry. Your math is excellent. ;-)
you are great❤❤❤❤❤❤
Sir in 12:49, why LHS=RHS=0?
Note that r' - r is a non-negative integer which is a multiple of b. Also note that since r is non-negative r' - r is a non-negative integer which is strictly less than b and hence r' - r is a non-negative integer less than or equal to b - 1. Since the only divisor of b between 0 and b - 1 is 0 itself the desired result follows.
Hope it helps!
Not that good actually. The example are contrived and do not explore the realm of possibilities: a being negative, b being larger than a for instance. And because the result is proven over Z this could have been nice. The problem is that the intuition is not that of a problem: you want to measure a with b but the beauty reside in the fact that it is possible even with the case above.
why a-bx? My textbook says a+bx
It isn't correct the rest is the result of a-bx for example 41/5 we have 40-5×8 = 1
Still confused 😭
6:04 wait what? How?
Oh nvm. I forgot that b∈ℤ
❤❤❤
❤️
Why is it important to show there is a minimum element?
👍👍👍👍👍
Binod😆
The glistening glorious physician coincidingly rain because accountant intraspecifically burn given a light square. nervous, fluffy beautician
I can clearly understandable thank you sir.
Join Jesus and spread the gosepl
thanks