The golden ratio appears everywhere in art... if you take "golden ratio" to mean any number between 1/2 and 2/3. The artists who use the number intentionally only do so because they've heard the same baloney as the rest of us (and this is fine; lots of great art is inspired by fiction). I once read an entertaining blog post by someone trying to prove that nautilus spirals have golden proportions. The author had a golden spiral overlayed on a nautilus shell and it wasn't lining up, so they kept tweaking parameters trying to make it fit. Finally, they centered the spiral somewhere _other_ _than_ the center of the shell and it lined up OK in some places, so they declared victory. You could see how the insanity had taken hold because, in their mind, the harder they had to work to find a golden proportion the deeper and more sophisticated the connection must be.
23:00 Euclid Algorithm is based on the following property: For any number x < y and y = xq + r, r < x, we get gcd(x,y) = gcd(x, xq + r) = gcd(x,r). ( I did not know this until I watch this video) In fact, set d := (x,y) and d' := (x,r). Because left hand side of y = xq + r is divided by d', we get d'| y. And obviously, d'|x. Therefore by the definition of d, we get d'
Hi hausdorff I dont understand what you mean for your first point ? choose k elements in a set of N elements= N!/(k!*(N-k)!) if k=3 and N=n+2=> ((n+2)!/(3!*(n+2-3)!))=1/6*(n + 2)*(n + 1)*n=1/6*n^3 + 1/2*n^2 + 1/3*n =1/6*(n^3 + 3*n^2 + 2*n)
40:24 Fibonacci numbers in biology is not total nonsense. For example, if you look at seeds at the center of a sunflower, they seem to be arranged in a spiral-like pattern. If you count the number of spirals going one way and the number of spirals going the other way, most likely they are going to be consecutive Fibonacci numbers like 21 and 34 or 34 and 55. You can't really call 34 "small"; certainly not a lot of numbers near 34 are Fibonacci. Turns out, that these kind of spiral-y patterns are efficient in their use of space - you can fit a lot of seeds in a small circle this way. Suppose you have put a bunch of seeds in a circle and want add another. You add it to the least crowded place on the boundary. If you repeat this process many many times, you will end up with a Fibonacci spiral pattern. Vi Hart did a video series on this topic several years ago, which I found very interesting: Part 1: ua-cam.com/video/ahXIMUkSXX0/v-deo.html Part 2: ua-cam.com/video/lOIP_Z_-0Hs/v-deo.html Part 3: ua-cam.com/video/14-NdQwKz9w/v-deo.html
And that has to do with the fact that phi is the "least rational" in some sense. Basically, phi is the irrational number worst approximated by rational numbers. Well, there are a few more of these badly approximated irrational numbers, but you can generate all of them by taking phi and applying integer addition or multiplication. Relevant Mathologer video: ua-cam.com/video/CaasbfdJdJg/v-deo.html
how is the binomial expression when n=n+2 and k=3 equal to the statement "number of ways to choose 3 elements from 6 element set?" wouldn't that expression equal to the number of ways to choose 3 elements from an n+2 element set? 3:40
Why is the worst case when q=1? With the numbers 100 and 101, we have 101 = 100*1 + 1. Q is 1, but we've already reduced the number down to 1 in just one step.
I was wondering the same thing. He said "we want r to be small" but he means "we want r to be small if we want the algorithm to be as fast as possible". We want it to be as slow as possible, so we want r to be as big as possible. To make r as big as possible we need to make b as big as possible in relation to a, so we take q=1.
Is this taught in college? Learned some of these things in the 6th-7th grade. Of course, not about Fibonacci sequences which the lecturer mixes in an incoherent and sloppy way with elementary-school topics. A shabby course overall!
Well... you certainly didn't learn EVERYTHING covered in this video in the 8th grade, did you? If so, good for you. Did you complete elementary studies in France then? I hear that five-year-olds can hold fairly intelligent conversations about rings and fields there. I never verified whether that's fact or myth (I think it's most likely a friendly exaggeration. They may be explicitly told that addition and multiplication over integers are closed and commutative. Rings and fields? I don't think so, not for an average five-year-old at least). As someone who didn't formally study these things (of course I heard about them in high school) until the first year of college, I was very intimidated and sometimes apologetic because of what I learned and hadn't learned. Tao in his blog says (and I paraphrase) "always relearn mathematics". If you haven't experienced moments like "holy *** (insert God's name), why haven't I seen it this way before?", you are truly, fundamentally missing out on the joy of mathematics.
Well you learn only the basic... not the whole course (part 20++). The point of this in college is to get everyone on the same car, so there is no one lost in the middle of the course. Yes you may had learnt this in highschool, but you can’t treat everyone the same, some of them maybe didn’t understand math in early days but want to take this course, maybe they are’nt as good as you but they want to learn too! (Sorry for my english btw, have a nice day)
If you think back to the course where you learned this in eighth grade, I bet the first month of that eighth month math course, you could have complained, "Is this taught in eighth grade? I learned such things in fourth grade." Every properly taught course usually starts by reviewing what the student should have already learned. This particular lesson was probably the first week of the college course, so yes, a professor would expect that everyone would have been exposed to this. However, for purposes of argument, I took all the advanced math courses in high school one year ahead (did five years high school math grade 8 to 12, 1965-1969) then took all the Advanced Math courses at a regional technical college in retirement (2014-2020), yet there's things he's said in the first two lessons that are new to me, but much is a review.
Not a big deal (you can understand everything fine) but there's a bit more echo than usual in the sound
27:06 Physicists definitely approve this approximation
That took a laugh out of me!
Loved the lecture.
0:00 Divisibility
7:08 Euclid's Division Algorithm
10:54 Ideals
13:36 Greatest Common Divisor
19:42 Euclid's Algorithm
25:48 Fibonacci Numbers (worst case for Euclid's algorithm)
31:43 Formula for fibonacci numbers
"Everything about the Fibonacci numbers in popular perception is non-sense, except in Witchcraft" xD
man had me rolling laughing
The golden ratio appears everywhere in art... if you take "golden ratio" to mean any number between 1/2 and 2/3. The artists who use the number intentionally only do so because they've heard the same baloney as the rest of us (and this is fine; lots of great art is inspired by fiction).
I once read an entertaining blog post by someone trying to prove that nautilus spirals have golden proportions. The author had a golden spiral overlayed on a nautilus shell and it wasn't lining up, so they kept tweaking parameters trying to make it fit. Finally, they centered the spiral somewhere _other_ _than_ the center of the shell and it lined up OK in some places, so they declared victory. You could see how the insanity had taken hold because, in their mind, the harder they had to work to find a golden proportion the deeper and more sophisticated the connection must be.
23:00
Euclid Algorithm is based on the following property:
For any number x < y and y = xq + r, r < x, we get
gcd(x,y) = gcd(x, xq + r) = gcd(x,r). ( I did not know this until I watch this video)
In fact, set d := (x,y) and d' := (x,r).
Because left hand side of y = xq + r is divided by d', we get d'| y. And obviously, d'|x.
Therefore by the definition of d, we get d'
Hi hausdorff
I dont understand what you mean for your first point ?
choose k elements in a set of N elements= N!/(k!*(N-k)!)
if k=3 and N=n+2=> ((n+2)!/(3!*(n+2-3)!))=1/6*(n + 2)*(n + 1)*n=1/6*n^3 + 1/2*n^2 + 1/3*n
=1/6*(n^3 + 3*n^2 + 2*n)
Where does the 6 elements set come from?
@@yunjiangjiang6146 Hi, I forget about it. So, I remove the sentence. I regret that I wrote something boring....
Minor correction at 3:26: (n + 2)* element set
Thank you for the great lecture!
40:24 Fibonacci numbers in biology is not total nonsense.
For example, if you look at seeds at the center of a sunflower, they seem to be arranged in a spiral-like pattern. If you count the number of spirals going one way and the number of spirals going the other way, most likely they are going to be consecutive Fibonacci numbers like 21 and 34 or 34 and 55. You can't really call 34 "small"; certainly not a lot of numbers near 34 are Fibonacci.
Turns out, that these kind of spiral-y patterns are efficient in their use of space - you can fit a lot of seeds in a small circle this way.
Suppose you have put a bunch of seeds in a circle and want add another. You add it to the least crowded place on the boundary. If you repeat this process many many times, you will end up with a Fibonacci spiral pattern.
Vi Hart did a video series on this topic several years ago, which I found very interesting:
Part 1: ua-cam.com/video/ahXIMUkSXX0/v-deo.html
Part 2: ua-cam.com/video/lOIP_Z_-0Hs/v-deo.html
Part 3: ua-cam.com/video/14-NdQwKz9w/v-deo.html
And that has to do with the fact that phi is the "least rational" in some sense. Basically, phi is the irrational number worst approximated by rational numbers. Well, there are a few more of these badly approximated irrational numbers, but you can generate all of them by taking phi and applying integer addition or multiplication.
Relevant Mathologer video: ua-cam.com/video/CaasbfdJdJg/v-deo.html
@@blairkilszombies fun fact: you can as well call it "almost rational", because rational numbers are even worse approximated by other rationals
I like his bluntness.
Thank you! Very good stuff
@3:27 How come ((n+2) choose 3) equals choosing 3 elements from a 6 element set.
How did we come to n+2 = 6? So n = 4?
nPr is n!/r!... (n+2)!/3! must be a whole number, so (n+2)! must be divisible by 3!=6.
how is the binomial expression when n=n+2 and k=3 equal to the statement "number of ways to choose 3 elements from 6 element set?" wouldn't that expression equal to the number of ways to choose 3 elements from an n+2 element set? 3:40
Great lecture thank you very much
Professor:
Its fairly ez to proof.
*Me looking at my 10 wasted papers that don't give my shit*: Thats ez?
I'll try solving em later today... hopefully it'll infact be easy.
Why is the worst case when q=1? With the numbers 100 and 101, we have 101 = 100*1 + 1. Q is 1, but we've already reduced the number down to 1 in just one step.
Worst case is when we dont know the number yet, just the steps
I was wondering the same thing. He said "we want r to be small" but he means "we want r to be small if we want the algorithm to be as fast as possible". We want it to be as slow as possible, so we want r to be as big as possible. To make r as big as possible we need to make b as big as possible in relation to a, so we take q=1.
4:11 he says "8 divides n squared minus n ..." when (as immediately shown in the examples) he means n squared minus one.
I like how he slipped and said "ideals". What are those, lol?
Lol the golden ratio is useful in witchcraft. very funny
Do you notice the humour in him?
👏
31:22 🇮🇳😍
yeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
Is this taught in college? Learned some of these things in the 6th-7th grade. Of course, not about Fibonacci sequences which the lecturer mixes in an incoherent and sloppy way with elementary-school topics. A shabby course overall!
Well... you certainly didn't learn EVERYTHING covered in this video in the 8th grade, did you? If so, good for you. Did you complete elementary studies in France then? I hear that five-year-olds can hold fairly intelligent conversations about rings and fields there. I never verified whether that's fact or myth (I think it's most likely a friendly exaggeration. They may be explicitly told that addition and multiplication over integers are closed and commutative. Rings and fields? I don't think so, not for an average five-year-old at least). As someone who didn't formally study these things (of course I heard about them in high school) until the first year of college, I was very intimidated and sometimes apologetic because of what I learned and hadn't learned. Tao in his blog says (and I paraphrase) "always relearn mathematics". If you haven't experienced moments like "holy *** (insert God's name), why haven't I seen it this way before?", you are truly, fundamentally missing out on the joy of mathematics.
@@SaveSoilSaveSoil It is 100% a lie, not even an exaggeration
Well you learn only the basic... not the whole course (part 20++). The point of this in college is to get everyone on the same car, so there is no one lost in the middle of the course. Yes you may had learnt this in highschool, but you can’t treat everyone the same, some of them maybe didn’t understand math in early days but want to take this course, maybe they are’nt as good as you but they want to learn too! (Sorry for my english btw, have a nice day)
If you think back to the course where you learned this in eighth grade, I bet the first month of that eighth month math course, you could have complained, "Is this taught in eighth grade? I learned such things in fourth grade."
Every properly taught course usually starts by reviewing what the student should have already learned. This particular lesson was probably the first week of the college course, so yes, a professor would expect that everyone would have been exposed to this.
However, for purposes of argument, I took all the advanced math courses in high school one year ahead (did five years high school math grade 8 to 12, 1965-1969) then took all the Advanced Math courses at a regional technical college in retirement (2014-2020), yet there's things he's said in the first two lessons that are new to me, but much is a review.
this is lecture 3...