The Golden Ratio (why it is so irrational) - Numberphile
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π = 3 + a bit. Going to use this in all of my code from now on.
π = 3; π += a bit;
private double aBit = Math.random(); private double giveOrTake = Math.random(); if (giveOrTake > aBit) { aBit += giveOrTake; } if (giveOrTake < aBit) { aBit -= giveOrTake; } private final static double PI = 3 + aBit;
pi = pi + bit
> 'in my code' > has MissingNo as a profile pic I love you :^)
so π will be either 6 or 7 depending on the value of that bit :P
This was one of the best Numberphile videos ever.
Ersen Couldn't agree more
I never liked the golden ratio because the way I learned it was: 1. greek dude came up with a series 2. divide 2 following numbers in it 3. WOW! flowers grow this way 4. the end This was a very unsatisfying explanation, because the whole 'WHY?!?' was missing. Thanks for giving me some love for the golden ratio.
I agree. This one is up there as a top candidate for the best one ever. (And I have of course seen every single one, as we all have. Surely.)
I was going to post the same thing. But I knew someone else already must have. So I found your comment, liked it, and...
asd i mean he looks like a judge who dropped his wig in the mud
The idea that numbers can be "more" or "less" irrational kind of blew my mind.
Mark O did you go to school marko
Imchattingabsolutefuckingshit username checks out
@pyropulse You seem upset. You wanna talk about that?
pyropulse pretty rude for no reason
pyropulse I mean I found it interesting that this could be a way to visualize how closely can an irrational number be approximated by smaller numbers, and I think using the language he used to explain - “more or less irrational” is an easy way to express my thoughts in this context
This is the absolute best explanation of the Golden Ratio I have ever seen. Thank you!
For me it was the sound dinosaurs that did it.
The golden ratio is 1,618
Establishment is training Al to learn from revised/censored reality of mediiaa and internet (mostly peaceful 👍), manufactured by NewNormal agenda. Starting the systemic use of Al with special interest focus is pure corruption at the root,. thanks Sillyc0nVally
Nature is so precise. And yet many people still call themselves "atheists".
@@RaineriHakkarainenApproximately. It’s really 1,618…. Since it’s irrational (as indicated by its infinite continued fraction and its precise formula: (1 + sqrt(5))/2 (which is just sqrt(5) with some rational tweaks); as 5 is not a square number, its square root has to be irrational), and a bunch of other things, I’m sure), it has an infinite, non-periodic decimal expansion. 1,618 is a rational number that can be expressed as a precise fraction: 809/500.
"I'll be there in a bit" = "I'll be there in a pi minus 3"
Nice
no, the rhs is equivalent to "I'll be there in a a bit"
400th like
You made me smile and giggle just alittle bit lol. :)
So you'll be there in a 1/(pi-3)-7?
That was mind-blowing, watching that animation run. You could see the whole-number fractions passing as the animation proceeded. It's almost like watching some part of the universe that you can't normally see, but which was somehow exposed by this video. A bit unsettling, yet completely fascinating. I can't quite describe it.
Great. Glad you (sort of) liked it.
.
Is the code for the animation, or anything like it, available anywhere…maybe on GitHub?
AGREED! (and yes, that required caps.. lol)
yea but to me math is like super complicated but at the end of the day it seems to me its just like a never ending mandelbrot set. it seems we have came up with infinite amounts of knowledge to describe something we should have already known all along lol idk man im having one of those "bruh i just figured out how the universe works" moments.
"A BIT is not a mathematical recognized terminology" - CS major: sweats profusely
@Arbnora Vezaj Elsi
Lol
CS major? Counter strike major?
@@ryanolsen294 Obviosly not. It's Coconut Science major.
@@Padeir0 at my school it's called ECS (Engineering in Coconut Science of course)
"Flowers can cancel fractions" - Ben Sparks, 2018
That "bad flower" with no rotation is just a legume.
It uses the least irrational number: 1
S.C. Wood why isn’t 0 the least irrational number?
@@worldisfilledb how is nothing less irrational than something?
@@Good_Hot_Chocolate Why should there be something
@Dirty Sack it does noth exist
Instructions Unclear. Accidentally produced an infinite spin.
Lesson 5 Johnny
eventually, he stopped thinking
@@meowmeowmeowmeow1 Magenta Magenta
@@meowmeowmeowmeow1 ... the yen to ken zen means, nothing's gonna change your world ...
How. It's impossible
This is perhaps in the absolute TOP3 episodes of numberphile... everything is so great, I've watched it at least 4 times now over the past couple of years since I got into the channel. The content is fascinating, I love this dude, the animations and the music is soooo freaking perfect - even the little snaps when it pauses for a sec' ... just a wonderful peice of art created here
Everyone: I finally understand the golden ratio Me: SPINN
SPEEEENNN
Nyo-ho!
@@skyebluee S P E E E E E E E E E E E E N
SPaIN
SPEEEEEEEEEN
Believe in the rotation, Johnny.
The Masked Man The spin is the power of infinity!!
GYROOOOOOO
arigato, gyro
Is... Is this?! Could it be? Is this a Jojo's reference?!
I opened this video only to see if there was a JoJo reference in it, thank you
Hadn't heard of the golden ratio being the "most irrational" number before, that's pretty cool.
Spectrally Mathologer did a video on this once. I think it was even titled "the most irrational number"
An unpublished interview with Steve Mould had him mentioning the Golden Ratio as the most irrational number.
It doesn't make lot of sense , if (1+sqr(5))/2 is the most irrational number , multiplying this number by 2 & substracting 1 shouldn't drasticly change it's properties , does it mean sqr(5) is extremely irrational ?
Sqrt(5) has the continued fraction 2+1/(4+1/(4+1/(4+1/(4+... It's a similar beast in that it has a continuing fraction that repeats the same number over and over again. You can prove it in a similar way to the way he proved the all-1's continued fraction equals phi.
sqrt(5) = 2+1/(4+1/(4+1/(4+...), so according to this video, it would be more rational than sqrt(2) and sqrt(7). It has to do with proximity to a perfect square. sqrt(2) = 1+1/(2..., and sqrt(5) = 2+1/(4..., and sqrt(10) = 3+1/(6... If you look at the numbers, you get sqrt(1+x^2) = 1+1/(2x..., which means that when you take the square root of a number, the closer that number is to a perfect square, the more rational it will be (according to this video).
"Gondor has no king, Gondor needs no king." *Boromir* "Rule of Gondor is mine !" *Denethor* "So passes Denethor, son of Ecthelion..." *Gandalf* 1:59 "So if I jumped, say, to a tenth of a turn, would you care to predict what you would see ?" *Denethor*
*IF YOUR HEART WAVERS, DO NOT SHOOT*
Is that a JoJo's reference
Zero Two it’s making me head spin
@@iforgottbh4488 it’s making my nails spin
Then a new gate will open to you
The corpse was "JEs0s" for some reason
All of differential calculus is based on "and a bit", It is perfectly ok to use, it just sounds better with Δ, δ, ε
Let epsilon < 0
Turn the sign around.
Fovlsbane no
TURN IT!
RainbowMash Noooo! Don't turn it!
Glad to see that the steward of Gondor is alive and well!
Nailed it Bro!
Flee for your lives!
And killing it at maths
That's exactly what i thought when i saw the thumbnail. lol
I said that too!
Johnny you've gotta watch this Numberphile video to learn the power of the spin, Trust me Johnny
"johnny johnny" "yes gyro"
6:10 also, if you count the number of seeds on one of the 3 spirals starting at the center, then the 7th seed will always line up with one of the 22 spokes - and 22/7 is approximately pi, amazing!
I have heard nearly everything in here before, but I've never seen such a succinct, logical explanation for all of it. This was freaking amazing.
r/iamverysmart
I'd never actually seen the derivation of (1 +- sqrt5 ) / 2 before. This was very helpful!
Well he didn't said he understood everything @@simonshugar1651
Same. I knew about most of the properties of the golden ratio that were shown here, but I was never quite able to put together _why_ it was the case. When he went from the continued fraction representation straight to "x = 1 + 1/x" it just blew my mind.
Johnny, you must spin your nail based in the shape of the golden rectangle！
I can't do it Gyro!
@@worldisdoomed9994 Say that 3 more times then I'll give it to you
Pizza mozzarella
cant ruin 314 likes :/
No
Nature: *Exists* Mathematicians: _That's Irrational_
Irrational in math means something else compared to irrational in reality. Rational typically means "in accordance with logic." In that sense it's latin root. Irrational in that sense means not according to logic. However, that is not the etymological root of mathematical rational and irrational. The english started using ratio, which has the same root at rational, to refer to a relationship(by division) between two numbers. Rational in that sense means able to be described in a ratio and irrational simply means unable to be described in a ratio, not that the number is illogical - since numbers kind of can't be illogical because of how they're defined.
@@jhomastefferson3693 thanks for explaining
@@jhomastefferson3693 but you forgot to sat ratios of integers or other rational numbers. All are a ratio
This was BEAUTIFUL! You made me fall in love with mathematics. I come and see this video every once in a while to keep being motivated to learn.
What a wholesome comment. I’m going back to school soon and I’m going to remember this to motivate myself.
We need an hour long animation of the flower at the end.
Libor Kundrát yes.
Libor Kundrát same
slower, or looping?
MamboBean Slower. Imagine it spinning slowly with a hypnotyzing music as it crosses the milestones. (the larger fractions, the golden ratio, etc.) Looping wouldn't really have much meaning.
Make it 10 hours.
PI=3+a bit I knew it, PI is something between 3 and 4.
Nobel prize incoming...
*Fields Medal
Between 3 and 4 Between 3 and 3 2/5 (or 12/5) Between 3 and 3 1/7 (or 22/7) and so on
Isnt pi = 3.2? Yeah..
embrace yourself and pi=22/7 is coming!
I have no idea how I came across this video nor have I the slightest clue of anything that was just explained, but, I watched all 15:12 in complete wonderment.
Wonderment lies in the question more than the answer, doesn’t it?
I love how when at 10:48 he mentions Matt Parker, there is a tiny flash of Parker Square in the bottom right corner :-)
Wow!!! I already thought I knew a lot about the golden ratio, but I've never thought of one irrational number as being "more irrational than another". The way they calculated phi from that infinite fraction is something I've never seen before and it was absolutely awesome!
It's a different metric but I thought of transcendental numbers as the most irrational but things like "e" and pi are close to 3 so will make curly spokes if you try to use 1/e or 1/pi to space seeds.
Some hard thinking has gone into this, I would never have thought of this!
They actually taught us exactly this in uni at a number theory course
Quite different to be taught something compared to be the one that figured it out. Uff, one part of me wants to study again since i never got much education. Now at older age knowledge interest me more :D
Mine was actually pretty easy it was simply a matter of all actions being compressed into a series of yes no and i do not know from there you simply compress the possible repetitive calculations down to a reasonable form like holographic in a particular way then no matter what question you have as long as the answer is yes or no you have a direct path from question and answer in the fractal patterns of that holographic that eventually themselves repeat and the world becomes yours. Took me less than a day to figure out and usually just takes a few seconds on paper.
Andrew Kelley what exactly have you find out? and how again the world becomes yours?
just do the equation I have in my post
I have spent years of academia studying the golden ratio and yet this is the best and clearest explanation I have ever seen on its irrationality! Incredible!
This was so interesting and I really couldn't understand exactly why the golden ratio was so important and this really blew my mind, thank you!
Best Numberphile video in a while
Easily.
Absolutely brilliant.
Agreed
Denethor is that you?
What would You think are the other best videos on this channel? Or better, what are, in your opinion, the best videos from Numberphile, Computerphile and other science-related channels? I would even go as far as asking what are the best videos You've ever seen on KZhead?
Best Numberphile video yet!
What would You think are the other best videos on this channel? Or better, what are, in your opinion, the best videos from Numberphile, Computerphile and other science-related channels? I would even go as far as asking what are the best videos You've ever seen on KZhead?
I love how he just mentioned "you can count the spokes and if you do you get fibonacci numbers"
Compelled to come here because of Steve Mould’s 1 million subscriber video. Great content.
That flower animation at the end really creeped me out for some reason.
Mister Apple You have floweranimationmathsthingphobia?
I want it as my screensaver.
Mister Apple the way it changes the circles' size is really disturbing
Sentinels from matrix
It's more likely the music.
Brilliant video! Captivating from start to finite end. We are off to go measure flowers now.
If there's an end, 100% of the times it will be finite. There is no such thing as "infinite" with an end. I'm still confused about your thought process.
@@oscarpritzker6278 did you have high expectations for a kids channel?
@@oscarpritzker6278 i mean... you can technically “complete” an infinite series
@@doublecircus No we can't. There's a reason why it's called infinite, but I agree that there's always an end, we just can't calculate it, so it's correct to say that infinite just means immeasurable and not endless.
@@oscarpritzker6278 I was referring to something like Zeno’s paradox, and probably could find a few other examples
Oh boy! I'm ready to watch a nice video and learn about the golden ratio! I sure do hope there aren't any references to this Japanese cartoon in the comment section!
😂😂😂😂😂😂😂😂
Username checks out
@@sinbad4696 omg chica I am a huge fan
@@diavoloisamasochist4986 aren't you miyuki's dad omg i'm a huge fan
@@sinbad4696 YES! I AM!
Flowers canceling fractions is the coolest sentence I've heard today.
So flowers are smarter than me, thanks.
..."me (period )Thanks capital " T " .
your "stupidity" is a learned behavior. GL
Selective processes are the way to go!
flowers had 250 million years of trial and error to get phi. you basically understood 250 million years of work in about 15mins.
No, flowers are just like that after billions of years of Evolutionary trial and error. The real question is why is the universe so specifically, logically ordered such that this is the universal best ration of flower petals, among other things...
This guy is a fantastic teacher. He clearly understands what he's talking about. For me, the subject is quite interesting in the first place, but even still he's so clear and concise in his explanations. Great video!
Thanks and glad you enjoyed it.
The most clearly explained video on Numberphile. Was following along quite nicely all the way through!
You guys rock! My favorite numberphile video yet. Ben you’re awesome, I found you through the Mandelbrot set video which I loved!
This is, by far, the best explanation about how math helps to explain natural occurences. I am a high school geometry teacher with a degree in secondary mathematics education. I always feel that when I start to talk about Fibonacci numbers, the Golden Ratio, etc, I tend to lose people. Most high schools students, and students beyond high school, really sort of start to glaze over when talking about sequences. I absolutely love this explanation and animation. I feel like anyone could understand it because it's so beautifully done. Also, to be honest, I never thought about the fact that some irrational numbers are more irrational than others. This video was so cool! Thank you!
5d75dv6e
Considering that flowers have had about 250 million years of evolutionary trial and error to progressively find more and more efficient seed packing methods, is it any surprise that eventually they would get to the most perfect method mathematically possible?
Not even then, since flowers don't have any sort of "choice" about where they grow seeds anyway. That's all determined by the behavior of the proteins down at the molecular level.
Exactly. When I said flowers "found" the best solution, I didn't mean consciously. I meant in the sense that a repeating computer algorithm might "find" the best solution to something if it cycles through the problem enough times.
Then every plant should have this structure. But they dont.
@@SuperQuwertz not every plant has the same goal... Other plants with different goals find other uncannily mathematical sequences. Like how leaves grow on a tree for example.
@@brokenwave6125 the goal should be to survive. therefore after millions of years everything should be more or less equal. there is no need to be "beautiful". bees dont care about the perfect geometry of a flower. Or do you think the lotus is repelling dirt and is using perfect geometry in order to survive better?
This is the most *brilliant* and *fascinating* video on Phi that I’ve ever seen in my life!! Well done!!
This guy is by far the best explainer you have on numberphile
Nah gyro Zeppeli is better
That is your opinion and you should state it as such. Other people may have other opinions because liking someone is not easily quantifiable.
Lol the Parker square.
Love such easter egg.
Parker square will never die, love you guys :D
It'll never be let goooo.
Parker Square flash for a frame or two, then suggesting the video for it immediately afterward. You cheeky sorts. 😂
it has become like one of those small fish that lives on the surface of a much bigger fish. chances are the small fish will show it self whilst you are admiring the big fish
The best Acid trip I ever had on youtube.
I believe you.
did anyone ever make a video of the animation with that music yet?
Mandelbrot Deep Zoom would like to have a word with you...
i watched this tripping and it was entertaining af
Do mandelbrot zoom in
_Lesson 4: Pay your respects_
I've had a backstage obsession with the golden ratios (and other numbers/mathematical anomalies), and this blew my mind. Thank you sir
All the videos with Ben Sparks have been fantastic on numberphile, looking forward to more!!
Here’s his playlist: bit.ly/Sparks_Playlist
The true power of lesson 5...
which lesson?
@@alanlowen2766 you probably wont get it
@@alanlowen2766 Lesson 5 Johnny, it was the most roundabout path
GAH NOOO NO JOJOS IN MY MATH AAAAAAAAA
Watched all this and really enjoyed it...now I'm going to watch again and code my own version. I love when mathematical concepts show some element of symmetry or beauty when you never expect it.
what a time to be alive!! thank you for this video, it actually helps my eyes understand what I been seeing... I've been seeing the spirals but my eyes used to readjust focus (which hurts), but this model lets me know my eyes weren't broken
Next time I get in a fight with a romantic partner, I’m going to shout “(1+/- sqrt(5))/2 “ to let them know JUST HOW IRRATIONAL they’re being.
*Bahaha* 👏🏼😂 me too!
Might want to save that for a time when they are being the most irrational.
Please don't spoil the comment section by making such bad jokes
But what if you never have another romantic partner ever again?
I was wondering if anyone in the comments was going to connect this to people and how they act in relationships. Do I really need to elaborate?
"I'm not saying flowers are thinking about this", ibelieve you
One of the coolest videos on Phi I have seen. Hats off!
Ty for zooming through all the possible curves at the end, very illustrative
"Hey are you the gold ratio, because you're behaving extremely irrational right now"
Probably not your go-to pickup line, not gonna lie
Daamn
Message from the Save the Adverb Foundation: *irrationally
embustero71 where can I join the Save the Adverb Foundation. I am from the Adjective Protection Agency.
hey are you a fire alarm, because you are loud and annoying
Ben (or "Mr Sparks" as he was to me) was my teacher for the first year of A Level maths. Sadly he wasn't there for my second year. Needless to say I did a lot better the first year than the second. He was one of the best teachers I ever had, and that's a pretty high bar!
Hi James!
Ben is my favourite contributor to this channel. All his videos are fantastic
Arigato, Gyro.
I had already heard that the golden ratio was found everywhere in nature, but I never could understand why. This video made me see why! I think that the explanation is thorough, understandable and very well presented. Great video!
That is one simple, clear and truly amazing video you put together here Brady !
I have a pic of a sunflower that demonstrates the golden ratio extraordinarily well... Just looks really amazing. Started noticing the other flowers in that batch had some of that also.
Excellent video!! What makes this video so great is the way quadratic equation is solved, so slick. This is why I watch this channel!
Continued fractions are always fun. They make me wonder if a musical interval of 1/phi should be the harshest possible ratio, not the tritone (which is 1/sqrt(2)). But if you try to make 1/phi, what you hear is a sharp minor sixth, closely approximated by 8 semitones and 33 cents. The next question is on the 36 tone scale, where would this golden interval resolve?
Great video. I really dislike the rectangle explanation of the golden ratio, it makes it seem so arbitrarily. Saying "it's the least rational number" is a much better way of highlighting its importance.
I think that must be a legacy from the Ancient Greek mathematicians. For them, numbers were for quantifying lengths (and areas and volumes), so the shape or aspect ratio of a rectangle comes out naturally.
The Golden ratio is our lesson for this semester. Thanks for the clear explanation ☺️
THANK YOU for that snap frame, it made it a lot easier to stop on the frame to study it a bit
flower seem to be better at math than me.
They use an evolutionary algorithm. They do not know that they are solving a maths problem, but nature put in the constraints and they just blasted out that optimum. Maybe there were Root Two seeder sunflowers kicking around for a while before their Golden Ratio cousins took over.
+Thumper Maybe you're right. But if it's the Root Two seeders followed by the Golden Ratios, who's next? Maybe the Eulers? (Check out the evolution at 1/e, or around 0.36788, where the seeding looks maybe even more random than at 0.618...)
Its biology whose smarter then u
idling dove nice thinking on evolution, poor thinking mathematically. Did you even watch the video the Golden ratio was the ultimate randomness factor because it is the 1/(1+1/(1+...)))... so 1/e would be less 'irrational'. On the other hand their could have been 1/e sunflowers already that also got beaten out by the Golden Ratio ones. Also maybe some sunflowers are closer to the golden ratio than others and they are still evolving towards that perfect design (only to go extinct due to completely unrelated climate or ecological changes).
I believe there are still some plants that use a slightly less irrational number in the angular spacing of their branches. Maybe evolution is satisfied or is still busy optimising
This exchange was absolutely captivating; consequently, I was completely entranced by the lecturer's presentation of the subject matter. I could listen to this man speak about mathematics all day. These statements are coming from someone who has historically always had a feeling of dread when approaching math. This man's enthusiasm overrode the dread and made me want to learn and participate.
This is amazing!! I always wanted to know a bit about the sunflower patterns. Can you post the GeoGebra file of the simulation... It would be great. Thank you.
"The words a bit are not mathematically recognized terminology" Computer scientists: :/
0:23 that joke, though
So this was the point of lesson 5... Arigato, Gyro...
Darkness! Get back to Kazuma!
It took 6 comments to get to a JoJo reference. It’s a new record
"Arigatou, Gyro..." "I think that's all I can say" ~Johnny Joestar
unexpected jojo
We tend to think in decimals, that's why it seems so weird to us. But just by reinterpreting the value (here by presenting it by fractions within fractions) it suddenly makes much more sense. It's just there, it's natural and intuitive. We just don't use the perfectly intuitive number system. I wonder if there is one that would allow us to advance the mathematics faster, but I guess, we already have the knowledge to experiment on other systems and it has already been done for some time now,
I've studied about the golden ratio many times and nothing in this video is new to me, but this is an amazing summary and really blew my mind. I love it!
Cheers
Now if only I could actually figure out the arcane nightmare that is Quadratic Equations as explained by a rather garbage textbook, I could at least say I had that much in math.
@@hariman7727 its easy, watch a video about it
My most excellent digital guitar effects pedal has a setting on it called the "Golden Ratio" (related to the number and style of delayed repeats to an input tone) and I always wondered why it sounded so musical and pleasing no matter what I played into it. I got a lot out of this video and even broke out my trusty hp 12c calculator to follow along. Thanks. Am a new subscriber.
This is a wonderful explanation of irrational numbers, and it gives me an entirely different perspective on "the golden ratio". Thank you!
just WHOW. best video ive seen today. wont be able to get it out from mind and keep thinking about how gorgeously this man explains.
This channel never ceases to amaze me
completing the square - man, that's throwing back the years to my school life
I first came across this property of phi in golden angle-based MRI approaches as part of my doctoral studies. The basic idea is that when you're scanning, the thing you're scanning is evolving with time, but you can only scan one point in k-space at a time. (k-space is a spatial frequency space, but you could think of it as real/image space without losing the take-home here.) If you want to get the "most uncorrelated" data and therefore use your scan time most wisely, or if you want to be able to bin your scans and create a timeseries that "shares" data in a window as it evolves, you should scan in golden angle spirals.
13:39 right around there you can see it unwrapping as it becomes 1/2.
I want an app that will let me do that spirally thing.
I think they probably used Processing (processing.org)
The app is called... just about any programming language. As an added bonus, they also lets you do every else that is computable.
I‘m pretty sure you‘ll find something like that on Wolfram Demonstrations...
I wonder why the hippies didn't use this video to represent hippiness?
let an indian do it for 15$
The bumps on my popcorn ceiling were moving similarly to 3:30 when I was on shrooms 🤔
That was nicely done, someone has finally made math interesting to me.
Thanks much prof Ben . I loved this video and i cant stop seeing it again n again
the pattern of the last flower animation was 1, 5, 4, 3, 5, 2, 2, 5, 3, 4, 5, 1 amount of spokes
this is incredibly interesting, and explained very well.
I can't explain what I've felt watching this video... simply amazing, thanks a lot
Thank you, finally this is explained easily! They always mention this in math classes and nature shows, and I've always seen this explanation of cutting golden rectangles into pieces forming a spiral that looks nothing like a flower, and then some sort of a half-assed explanation of, "see, you can form a spiral with the golden rectangle, so spirals in nature contain golden ratios and fibonacci sequences," always leaving me thinking the golden spiral looks nothing like the spirals in sunflowers, and that any rectangle can be cut into a spiral, thus a totally useless explantion. Thank you for fixing this. Finally!
I don't understand half of it but it sounds and looks cool! Nice video
I need to learn more math. This actually is super interesting!. I've been lacking omitvation to learn triangles in highschool but these videos really motivate me to get to know the intrications of our numeric system
Excellent and very clear explanation of the particular irrationality of Phi, very illuminating!
Great Job! Well explained!!! Thanks for the excellent and cool explanations.
Thanks! I known this. And yet... Now I have the feeling I deeply understand it, which is quite different. This is the power of the Numberphile videos. So, thanks again