63 and -7/4 are special - Numberphile
HTML-код
- Опубликовано: 8 фев 2025
- Dynamical sequences, prime divisors and special exceptions.
More links & stuff in full description below ↓↓↓
Featuring Dr Holly Krieger from MIT.
More primes: bit.ly/primevids
More videos with Holly Krieger: bit.ly/HollyKri...
Since this was filmed, Holly has become a mathematics Lecturer at the University of Cambridge and the Corfield Fellow at Murray Edwards College.
Support us on Patreon: / numberphile
NUMBERPHILE
Website: www.numberphile...
Numberphile on Facebook: / numberphile
Numberphile tweets: / numberphile
Subscribe: bit.ly/Numberph...
Numberphile is supported by the Mathematical Sciences Research Institute (MSRI): bit.ly/MSRINumb...
Videos by Brady Haran
Brady's videos subreddit: / bradyharan
Brady's latest videos across all channels: www.bradyharanb...
Sign up for (occasional) emails: eepurl.com/YdjL9
Numberphile T-Shirts: teespring.com/...
Other merchandise: store.dftba.co...
best handwriting on numberphile
Really pretty.
Not just the handwiting.
best hands!
hands down
Bottoms up
"So it's all Mandelbrot?"
"Always has been."
Is that "Never Back Down"?
That's Numberwang
Seeing someone just freehand the general shape of a fractal like that is quite impressive.
Had a random bout of obsession with the Mandelbrot once, and learnt the dimensions of the fractal as a consequence. It's a fun shape to doodle when bored, and instantly recognisable by any passing math enthusiast.
it wasnt even drawn right but ok
Every time Ms Holly laughs, a new kitten is born.
Ser
@@ECSSANJAY-lr2hu was that a joke referencing "white knight"? If so, well done.
Statistically speaking, probably yeah.
Haha. I'm a believer.
My thoughts exactly.
How to get rich: live in the UK, sell brown paper and sharpies
Anglcaangelics
Mbta problems
Numberphile: *i'll take ur enitre stock boi*
like David Brent?
@@EHMM the more British version would be "we'll take the lot"
Numberphile has become my favorite channel now, I watch it every day and can't stop. Thank you all for those awesome videos.
We meet again, Mandelbrot set, and you never stop surprising me.
4:28 Well that escalated quickly.
+ILikeWafflz Exponentially even.
Yep
I saw that. Brick killed a guy.
hahahahahhahahahahha lol
Bravo
I've got the feeling that, all of a sudden, a lot of people are going to become very interested in maths.
Glad someone else noticed :D
I know, right? An American for a change.
what university is this I need to apply, wanna study math.
Yeah, how did this channel go about cornering the market on brainy, beautiful, freckly redheads and strawberry blondes?
Math has always been sexy. It's the language of the universe. When I was getting my Masters at Columbia at the business department, the ladies were getting it done.
I have no idea what I just watched. But that lady’s enthusiasm and intelligence is amazing
Watching Numberphile is always awesome but sometimes also surprisingly relaxing too!
Brown paper from this video: cgi.ebay.co.uk/ws/eBayISAPI.dll?ViewItem&item=380872542159
Please sell more brown papers ^^
OrphanPaper xD
Should have gone for £63. ;)
Numberphile::: you missed to present f(x)=1/2{6x-(-1)^x+3}. This gives more prime numbers.
mayank acharya Really ? Where did you see that ?
"It might be a harder question, depending on how specific you want to get." A truer statement never told. :)
7/4 is a pretty rad time signature! I suppose -7/4 would mean the song has to be played in reverse.
+Scott Lee absolutely rad. let's see if people know the (arguably) most famous song in 7/4...
+Alexander Konczal Probably something by Dream Theater, I'm sure lol. Oh wait!! they play in 9.2/5. thats right...
what abouuuut... MONEY by pink floyd! ha! such a well known song, but people don't think about it.
ahh. Haven't heard much my Pink Floyd so that's probably why I didn't recognize it
I have to correct myself - I originally wrote time, I meant money. I guess in my mind, the saying "time is money" is true.
This was fascinating. Any more footage explaining what exactly the Mandelbrot set is would be great.
There is a video (not Numberphile) linking the M-set to Feigenbauw constant and the behaviour of the Xn+1= r.Xn.(1-Xn) eq.
I love how f(x) = x^2 -2 gives you a square root sign when you plot the results
underrated
True, didn’t think of that!
Man, if someone would've shown me this channel when I was in high school I would've liked math so much more. Maybe I'd be somewhere better than community college :(
Oh well, at least I've learned to love it now.
I think people don't realize how much work goes into your videos. Thanks Bradypus!
It seems like every number is special. What number is _not_ special? Because _that's_ the special one.
Lincoln Lopes RUclips is on the conspiracy!
If all numbers are Special, what makes an individual number special? All of them are special so it must be "Common" really (to be special).
this is known as the smallest boring number paradox
For sure all positive integers are special! Assume not. Then there exist a set of non-special numbers that has a minimal element. But hey that smallest element is special because it's the smallest! Contradiction. Thus all positive integers are special 🙃
A famously known proof by contradiction that every number is, indeed, special!
Is there anything Amy Adams doesn’t know how to do?
Raise babies?
B.D what??
Winning an Oscar.
@@luciano53688 oooof
I think that's Isla Fisher
It's moments like these that I'm proud to be studying math. This sounds a lot like the useless "tinkering" I did in highschool to get through the boring hours. Nice to know that there are very intelligent adult people who, instead of going "why would that ever be usefull", say "hey, this is peculiar. Let's see what happens when I do this".
At the beginning of this video, it was stated how the series generated by f(x) = 2x + 1 was always equal to 2^n - 1. If you perform the function in binary, rather than decimal, it becomes clear why. The function f(x) = 10x + 1 [or f(x) = 10x + 9] would have a visually similar effect in decimal.
I guess f(x)=10x+1 is the function that truly would have the same visual effect. Namely, the terms would be
1, 11, 111, 1111, ...
for each corresponding bases.
How to heck did she draw that graph without grid paper? She must be a wizard.
Dustin Boyd 🤣🤣🤣
I just tried _f(x) = x² - 2_ and when I went to calculate the fourth element in the sequence I actually lold. Well played, ma'am.
Every time I start being a bit afraid Numberphile will burn out after all, it suprises me with something really new to me.
Thank you to all creating and participating in this exciting series - and its siblings as well, of course!
Best handwriting in the series so far...
Seems like women have prettier handwriting than men.
I have noticed same. But I remember one dude with handwriting I thought was of female origin. I did a double take, mentally, heh
Moruk turksen turkce konus
Hayrican Durmuş Türklere mi konuşuyo
Moruk o zaman yazmayacak bir sey
Dr. Krieger seems like a good fit in Numberphile! Hopefully she does more videos in the future :)
Thank you sir
65535 is interesting because all the prime factors are one more than a power of two (3, 5, 17, 257)
I know this comment is ancient but I want to point out that all numbers of the form 2^2^n - 1 have this property (except that these divisors may not be prime). They will be divisible only by all numbers 2^2^m + 1 for m < n. It's not super hard to prove either :)
However, it's an open question whether 2^2^n + 1 numbers are prime for infinitely many n, or even whether they're composite for infinitely many n. Heuristics strongly suggest 2^2^4 + 1 = 65537 is the last one that's prime.
It's always great to see someone new in Numberphile! And this is a fascinating topic as well! :)
Your username is Moon Rabbit? Not slagging. Just think it's cool.
Technically, it is "The Hare of the Moon", rabbit is "cuniculus"
You should definitely do more vids with Dr. Holly Krieger!
And for those wondering but too lazy to do the figuring, x^2 - 2 ends up being -2, 2, 2, 2, 2, 2, 2, etc.
Am I the only person who heard a chicken at 6:46?
I've watched this several times and never heard that before. But it is definitely there.
its a squeak in the desk that the paper is on
That does seem more logical than a random chicken.
+Civil Engineering Philosophy taga um ka Noah?
lol
She's great. Hope we can see some more numberphile feat. Dr kreiger
Intelligence is beautiful. I hope we get to see more videos with Holly.
what are your thoughts on Grigori Perelman?
perelman gigachad
I could listen to her all day. Love the maths and the explanations!
She has a really lovely voice. A joy to listen to while I was driving home from work today.
That giggle after "if your rational number is very, very close to that special point in a technical way that's hard to formulate." is incredibly cute - largely in part to the intelligence that proceeds it.
Oh, and fun fact. Mandelbrot literally means "almond-bread" in German.
If this is indeed true, I appreciate this trivia. :)
With the fractions, you could also multiply by the denominator and then ignore it as a common factor (and not a new prime). By that you also encompass ax²+c, where a and c can be rational
I did maths at London Uni (Kings College ) in late 1960s and still find this stuff fascinating- thanks
Amazing handwriting.
her circles are amazing
I'd like to see more about this sequence.
It's actually completely mind-blowing that -7/4 is an exception is to this rule considering that even if there were an infinite number of exceptions within the Mandelbrot set, the fact that even one of them is a rational number is unbelievable, since the Mandelbrot set only contains a countably infinite number of rational numbers whereas there are an uncountably infinite number of irrationals within that set. Meaning even with a countably infinite number of exceptions there's effectively a 0% chance that any of them would be rational...
Whole numbers are also rational.
Yes they are, what's your point?
Whole numbers are pretty
Wouldn't any set containing an interval of the real line contain countable rationals and uncountable irrationals?
I understand your amazement, but keep in mind that irrational numbers, as interesting as they are, are completely irrelevant in this discussion - there's nothing like prime factorization of an irrational number, or even a denominator to speak about...
Has James Grime done something different with his hair? He looks different in this video...
It's the shirt, I think.
I love going along with my own math while watching these videos. Makes for a fun time.
It'd be cool if you showed the proof that those sequences will always have new prime divisors if its not too complicated.
I still really want to know why it matters that 63 was the 6th element of the sequence...
+Michael Edenfield All multiples of 3... ILLUMINATI !
+Michael Edenfield Zsigmondy's theorem
Me too! I do know that 63 is special because it is the smallest whole number that can be divided by all whole numbers between 1 and 9 without producing a remainder. No clue what the significance of it being the 6th element in the sequence is about though :(
Kyle Eggemeyer
I don't think that I understand your comment... There's something wrong here "63 is special because it is the smallest whole number that can be divided by all whole numbers between 1 and 9 without producing a remainder" right ? Or am I just too tired to understand what you meant...
+Kyle eggmeyer I second the previous thought about your statement 63 =9*7 it doesn't divide by any numbers other than 1,3,7,9,21, and 63 ?
Brill as usual. But eagerly awaiting a SIXTYSYMBOLS on the recent gravity wave discovery confirming inflation ?
Never have I scrolled past this video and not click it
I wish she was my math teacher.
I don't. I wouldn't learn anything.
Conner Trieskey why?
Conner Trieskey 🧐😆😆😆😆 I got that
Helvecio Sniper wait lol Ive just realized 😂😂
@@rothgang me too and it's quite pathetic really...
Probably the cutest mathematician / teacher i've ever seen. kudos!
+jolichja The profile pic though :D
jolichja Clearly you've never seen Dr. Hannah Fry.
i cant tell sometimes if primes are some weird thing humans have this fascination with or some massive universal truth we have only scratched the surface of
At least in some areas, primes are useful tools. I'm sure they also represent some universal truth that nobody knows yet, but for now they can make our encryption keys.
They are a universal truth about a particular complementary concept. They define what is NOT, rather than what IS. (That is, they are defined by not having a factor other than trivial factors.)
I wonder if there are other concepts that are defined by "nots"?
@@brendanh8193 I mean you could define a prime as having only 2 factors
Mandelbrot sets are connected to logarithmic spirals so yes you're not far off with that statement about special universal truth.
@raglanheuser
The fact that prime numbers give a unique decomposition of any number gives a clue about why they are fundamental. But of course, there are other far more advanced or fundamental things in number theory that involve prime numbers that I know absolutely nothing about. But, yeah, they are not just weird random artifacts.
If you zoom into the Mandelbrot set at a point on the real axis which corresponds to -7/4 (i.e -1.75) you come to a needle-thin point at the very innermost tip of the split in the bulb on that mini Mandelbrot. You can keep zooming in to that tip but you can never "arrive", no matter how many times you increase the iterations, it's like you have reached an infinity point. I dare say that there are an infinite number of these points along the Mandelbrot set's "real" axis which appear at the same point at the very tip of the split in the bulb of each minibrot, of which there are an infinite number.
Nice speaking voice, sounds like a professional broadcaster. Then again lecturing is good practice.
The numbers that add together in sequence that is your phone number, is the only numbers I think are special. The best part about infinity and multiple world theory is in one of them , I'm taking you to dinner right now.
It's been 3 months since you posted this comment. i hope you already getting the square root of 4761 in that timeline.
So interesting! I'd love to see more about fractals and the Mandelbrot set!
I first read that as "freckles". 🤣
Wow, Dr Holly Krieger is a stunner:)
You should do a video about the different fractal sets. I would like to understand what exactly they are.
what a calming voice
*looks on the chalkboard in the background*
wut
more mondelbrot math vids please.. and Julia..
well now I want an extra video about why 6 is special :)
also she has a great voice
brains and beauty
7:19 I keep hearing it as Bill Cosby ''You can ask this as a fraction, see? Instead of a whole number, see?''
Petar Simic 0:46 "This sequence is special..."
***** too soon... (but secretly I laughed)
I watch a wholesome mix of math videos, skateboarding, and prison gangster documentaries
So impressive...and people are sharing videos of kittens. This is youtube gold and has me love the internet again. Thank you.
Richard Price hey, kittens are great!
Mind blown. I love all the feelings of awe Mandelbrot gives me.
This is a proof that Ygritte knows more things than Jon Snow !
Everybody knows that John Snow knows nothing.
tabularasa0606 At least he knows that one thing with the thong
" something messy that I don't want to calculate" wow ! I didn't know that was a thing ,option ...
Ginger based racism
Start at one. Keep going. It never ends. Brilliant.
I don't know if Numberphile has a video on this topic but I think you should make a video on why any number to the power of 0 equals 1
laaaaaaate replay
Well, nearly all (poor 0 :c)
x^(0) = x^(1-1) = (x^1)/(x^1) = x/x = 1
(plus apparently there are 20 or 11 comments in here *but* I can't see any for some reason so idk if it has been already written)
A cute red head and numbers, this is my fav't video of all time!
Now I'm Numberphilephile!
something i took from this is that, if a thing has some property that we define, it is interesting. And they complement the uninteresting cases.
I also learned that sometimes, we will find inconsistencies in these mathematical games we play, quite like glitches. Those glitches may point to deeper truths, is maybe one way you can put this concept
I am kind of off topic but 4 divided by 7 = 0.57142857142857142857142857142857 times 63 = 36 (63 mirrored) I thought this was the point of the show before I watched. Oh yeah this also works for 84, 42, 21, and probably others (like 4284 is 2448 and 5628 is 3216) notice the second digits or second group of digits are 1/2 of the first one in all cases. Well thought I'd share thank you!
Dr. Krieger is so totally the Dr. Mike Pound of Numberphile; charming, engaging and boss level Eli5 skills! :D
I'm disheartened that everyone seems to be commenting exclusively on how attractive she is. Can a good-looking woman discuss mathematics and spur a discussion on the actual topic like all of the other Numberphile hosts?
Is there anything wrong with appreciation of look?
No.
Special kind of trolls out there. And the marriage proposals... oi!
Sexual attraction is a reality of life. We should not be ashamed of it or feel we must apologize for, or not admit to feeling it.
get over it.
We need to see more of her.
While many of these mathematical patterns are sort of interesting I always wonder how much time and effort has been put into them and what value has come out of that work. I wish one of your questions on these was always "How can this be applied to the real world?" or "Now that we know that what else do we know?"
I think I cracked your question,
I think initially you asked why do you get the number very close to the 2^x (2,4,8,16,... you get it right?)
So here is my solution and explanation :-
You are just literally taking
f(f(f(...(f(x))...)))
And here the initial function is one away from the multiples of 2 that's the main reason you get that!
For example,
f(x) = 2x + 1
f(f(x)) = 2(2x+1) +1 = 4x + 3
f(f(f(x))) = 4(2x+1) + 3 = 8x +7
f(f(f(f(x)))) = 16+15
f(f(f(f(x)))) = 32x + 31
And so on
So, can you see the pattern here ?
Yeah! The coefficient of x is a 2^x sequence number and the addition factor is also the same number but one lesser so that's why you get that pattern.
And if you apply this logic you have learned, you can make infinite number of such series
Example,
let g(x) = 3x +2
You see what I did there ? One number away from multiple of 3!
Now, g(g(x)) = 3(3x+2) +2 = 9x + 8
g(g(g(x))) = 27x + 26
I think you see the Same pattern here too!
Yeah! It's not special it's just simple logic.....
That's gone under your radar!
Good day folks.
I tried -2 and it made me laugh.
me too too too too too... :)
Why exactly? I tried out the sequence but perhaps I calculated it wrong.
It's 0, -2,2,2,2,2,2,2,2...
...I think I saw a 2 D:
lol I tried it too, it's definitely funny.
what a triple threat! brilliant, beautiful, and charismatic!
do these dynamical properties of numbers extend to effects in physical systems? Great video, really like Holly. Please make more ....
“Idk I think that’s prime...” makes me feel a lot better.
would've been nice to have some sort of explanation as to why numbers in specific points on the Mandelbrot set are special. I understand the math is probably fairly difficult, but at least an overview.
I agree. This is the really surprising part to me. Would love a link.
My first true day of Summer Break, and I'm watching math videos on RUclips.
(x-63)/2=3
nice
"It's seems sort of fundamental and part of the reason why it's beacuse i'm hidding an example from you (giggles)"... Adorable
what happens if you use Pi or Phi?
"...alright well get ready..." my favorite numberphile quote
f(x)=x^2-2
0^2-2=-2
(-2)^2-2=2
(2)^2-2=2
And loops forever
Thanks
Are you sure?
Aaron Hollander it turns to 3 right after infinity
Thank you
i did that in my head in 5 seconds, am i smart enough yet?
Amazing voice for narration
....we need more women mathematicians. It was very refreshing listening to her explain this.
And shes from the town right next to mine; Illinois girls FTW!
It's disheartening how many people are commenting on just her looks, but it's heartening to see how many people are also bothered by this. That isn't always the case, and I think things are getting better. Keep pushing people to see the whole person and not just the gender stereotypes.
It's a lot easier to pay attention when 1) it's a pretty redhead and 2) you aren't being *required* to pay attention for some test.
Oh, and Dr. Krieger: If you want to square a two-digit number in your head, use:
(10a + b) ^ 2 = 100(a^2) + 20ab + b^2
Aurifeuillian factors are responsible for 63's "lack" of a new factor; evaluating
the 6th cyclotomic polynomial for an input of 2 just outputs 3; the output
of that function is divisible by 3 if the input lies below a multiple of 3,
so 3 is a trivial factor, but removing it just leaves 1, there are no factors left.
-7/4 = -1/1-1/2-1/4 = -1/4-2/4-4/4
That feels important somehow.
Three observations:
a) 63/(7/4) = 36. In writing, number 36 is just the opposite of number 63.
b) 63/(7/4) = 36 = 6^2.
c) 63-36 = 27 = 3^3.
I didn't know Ginny Weasley was a professor at MIT...
😂😂😂 lol
I think I just fell in love with this math teacher.