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!
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
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 !!
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
@@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
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 :)
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 UA-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.
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."
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.
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.
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
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?
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.
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.
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!
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. ;-)
These proofs literally left me speechless.
Still taking my breath away…
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!
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.
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 !!
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 A TEACHER THIS IS? Beautiful
The best of the best description of proof of Euclidean Division, I like more than VedTutor
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?
so nice to see this channel growing!
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 :)
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 !!!
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 UA-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.
Anytime I am stuck in any proof
You come as a saviour
Thank you so much Sir 🙏
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."
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.
Please countinue this amazing videos
Excellent explanation. This most certainly helps. Thank you.
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.
thank you sir, love from india
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.
your teaching method are perfect
i really prefer chalks over blackboards, the teachings are so smooth thanks!
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
Thanks sir!
❤️ From India.
What does _unique_ mean in the definition and in general?
What are the common prerequisite courses for this class?
A normal college math curriculum up through Linear Algebra or Differential Equations.
Thanks again for another wonderful video! Division? More like di-vision, because now I can see things clearly!
Thank you Sir! You made this proof undarstandable
U r amazing teacher love from india👍❤️
Being a student , i am saying you are the best teacher sir . You made it soo easy for me to understand . Just awesome 🔥 .
You sir are incredible.
Is this playlist (113 videos in total) in order??
a perfect pure mathematician.
Can you show another example division algorithm with proof?
A lot of existence proofs in number theory depends on well ordering principle.
What an amazing video! Thank you.
Why we know that r-b is the element of the S - after as we subtract b of both sides of equation ?
great video
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.
Thank you sir you are great 👍
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
Hello Sir ,. Please upload videos on Combinatorics also plzzz
Can any1 help me..
Why we have to consider two cases.... 1) b>= 0 2) b
Where are you?
Amazing
thanks
Thank You So Much Sir
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
Sir , the only multiple between 0 and b is 0.How?
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
Thank you.... from Sri Lanka
A great help for me
what would happen if b < a?
Bravo!
Mind blowing.
Damn man what did the board do to you why you hitting it so hard
Thank you from india
Thank u from India...
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.
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!
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.
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. ;-)
Why you ppl forget to write a>b
Wanna watch more
Good
Awsome
Nice
thanks man 😁
Thanks a lot from india
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
❤❤❤
❤️
6:04 wait what? How?
Oh nvm. I forgot that b∈ℤ
Still confused 😭
👍👍👍👍👍
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