Dank Hill Really recommended you try Khan website. It's fantastic and FREE to use. Also try You Tube's HEGARTYMATHS CHANNEL. It is based on 🇬🇧exams, but the ideas are explained clearly, and the videos are quick to view. Regards Samantha Wallace ps Good Luck!
Wow... Got given a programming problem to solve looking at fibonacci's sequence, read the equation and could not see where the hell it came from. Spotting something about "The golden ratio", I decided to go searching for that, which in turn led me here... How the hell can such a simple idea of having a line with the ratio of "a to b" being the same as "a+b to a" produce something so fricking awesome!? Mind officially blown. Thanks muchly ;)
I know it's been 5 years but is this programming problem from the book SICP? I'm currently reading it now and found the golden ratio being mentioned in a Fibonacci sequence problem.
This is great! I'm frustrated though that math is being dumbed down in public school. Like the math in this video isn't that complex, but when I show it to my teacher in ninth grade HONORS its treated like some kind of disease and I'm being told that it's too "hard" to do in class. Like seriously we should be allowed to learn at our own pace and not be put down to learn simple formulas that we've learned in the sixth grade!
+Jacob Rajacich it´s the same shit here in germany. But I always think that it´s just annoying to me, because 98% of my math class doesn´t want to learn more or anything AT ALL about math.
The association of the main numbers in the field of mathematics with each other, reflects numerical sequences that correspond to the dimensions of the Earth, the Moon, and the Sun in the unit of measurement in meters, which is: 1' (second) / 299792458 m/s (speed of light in a vacuum). Ramanujan number: 1,729 Earth's equatorial radius: 6,378 km. Golden ratio: 1.61803... • (1,729 x 6,378 x (10^-3)) ^1.61803 x (10^-3) = 3,474.18 Moon's diameter: 3,474 km. Ramanujan number: 1,729 Speed of light: 299,792,458 m/s Earth's Equatorial Diameter: 12,756 km. Earth's Equatorial Radius: 6,378 km. • (1,729 x 299,792,458) / 12,756 / 6,378) = 6,371 Earth's average radius: 6,371 km. The Cubit The cubit = Pi - phi^2 = 0.5236 Lunar distance: 384,400 km. (0.5236 x (10^6) - 384,400) x 10 = 1,392,000 Sun´s diameter: 1,392,000 km. Higgs Boson: 125.35 (GeV) Golden ratio: 1.61803... (125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97 Circumference of the Moon: 10,916 km. Golden ratio: 1.618 Golden Angle: 137.5 Earth's equatorial radius: 6,378 Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2. (((1.618 ^137.5) / 6,378) / 6.67) x (10^-20) = 12,756.62 Earth’s equatorial diameter: 12,756 km. The Euler Number is approximately: 2.71828... Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Golden ratio: 1.618ɸ (2.71828 ^ 6.67) x 1.618 x 10 = 12,756.23 Earth’s equatorial diameter: 12,756 km. Planck’s constant: 6.63 × 10-34 m2 kg. Circumference of the Moon: 10,916. Golden ratio: 1.618 ɸ (((6.63 ^ (10,916 x 10^-4 )) x 1.618 x (10^3) = 12,756.82 Earth’s equatorial diameter: 12,756 km. Planck's temperature: 1.41679 x 10^32 Kelvin. Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2. Speed of Sound: 340.29 m/s (1.41679 ^ 6.67) x 340.29 - 1 = 3,474.81 Moon's diameter:: 3,474 km. Cosmic microwave background radiation 2.725 kelvins ,160.4 GHz, Pi: 3.14 Earth's polar radius: 6,357 km. ((2.725 x 160.4) / 3.14 x (10^4) - (6,357 x 10^-3) = 1,392,000 The diameter of the Sun: 1,392,000 km. Numbers 3, 6 & 9 - Nikola Tesla One Parsec = 206265 AU = 3.26 light-years = 3.086 × 10^13 km. The Numbers: 3, 6 and 9 ((3^6) x 9) - (3.086 x (10^3)) -1 = 3,474 The Moon's diameter: 3,474 km. Now we will use the diameter of the Moon. Moon's diameter: 3,474 km. (3.474 + 369 + 1) x (10^2) = 384,400 The term L.D (Lunar Distance) refers to the average distance between the Earth and the Moon, which is 384,400 km. Moon's diameter: 3,474 km. ((3+6+9) x 3 x 6 x 9) - 9 - 3 + 3,474 = 6,378 Earth's equatorial radius: 6,378 km. By Gustavo Muniz
Very straight forward. Just look at the basic definition: φ = a/b. Thus if a = 10, φ = 10/b = 1.618, or b = 10/φ And that is even more fun if you remember that 1/φ = φ - 1, so you can do that in your head. φ - 1 = 0.618 times 10 = 6.18 units (cm for this case) So side A = 10, side B = 6.18... GREAT video!! PS, obviously I'm only using the first 3 digits of φ for calculations.
The Golden ratio is also found the our DNA helix, the bronchial tree structure of the lung, the heart valves, etc..... This is not just a ratio... It is nature's ratio. Very intriguing indeed.
Sal I love how excited you get about math and concepts like this. I feel like these things can be mind-blowing and it's so cool to have someone to share this with, even if it's just me at home watching your videos. So proud of everything Khan Academy has become
you could go this way as well: a^2 -ba -b^2 = 0 and solve for a, choosing the positive answer: a= b(1+sqrt(5))/2. Then use this in a/b and get the golden ratio as (1+sqrt(5))/2
Because that is what the golden ratio is. 2 numbers in this ratio added together, divided by the bigger one, exactly equals the bigger one. Spend some time with it to understand it. The golden number, 1.6180339.... is the only number that you can square by adding 1, and find the reciprocal by subtracting 1. a over a does =1. b over a =.6180339, those added = golden number. To make the ratio to any desired accuracy, you can add any 2 numbers, add that result to th previous, and the more times>
Having never studied them myself, I have to take Discovery Channel's word for it that the Golden Ratio was used extensively in ancient architecture; the Parthenon being one of them. Also as I understand it, there are a great number of archways with this ratio measuring from base to sides and up to just where it terminates with the arch its self. It’s also in the windows of some churches. (as I understand it)
>you do that, the more exactly 2 consecutive numbers will = phi. Ex: 1+2=3;2+3=5; 3+5=8; 5+8=13--and so on. The more u do that the more accurately the ratio of the last 2 numbers will = phi. Try it. If you do it around 20 times it will be to the accuracy of an 8 digit calculator. The fascination things about th ratio are endless.
I personally found this video very confusing. It didn't really explain what phi was in the beginning of the video and went right into some equation that was afterwards explained as the golden ratio. I was beyond confused. I also would like to see that the work is more organized, it looks like its written it all over the place.
shapes make it easier, single pointedness 1, eye or flame 2, triangle 3, square 4, pentagon 5, hexagon 6 ect all these shapes hold the triangle in them, and is easier to see when 3d :) its metatrons art and language of numbers :)
I read 1/2 of the book about the golden ratio by Mario Livio, I gave up on it because I wasn't really grasping all that was said, but this video really cleared things up, I'll probably head back into it now, thanks so much for your time and effort, the way you use colors and simple explanations is very helpful and clear.
Cool timing for this video. I am preparing to build a door to my house, and was considering incorporating the golden ratio into the design for fun. You've explained some things here that I didn't know about it. Thanks!
Pretty simple, since we know that phi^2 = phi + 1. then we multiply both sides by phi and we get: phi^3 = phi^2 + phi but we already know that phi^2 = phi + 1, so: phi^3 = phi + 1 + phi phi^3 = 2phi + 1
φ² = φ + 1 φ = (√φ)² φ² = (√φ)² + 1² this is the Golden triangle (Kepler's triangle), written in Pythagorean form, so it is a right angle triangle, which hypotenuse is φ. By the way from here and the unit square we can see the connection between φ and the REAL VALUE of π. Thus π = 4 : √φ = 3.144...
The number 4 is always used in the quadratic formula :) If you're still confused, just google the quadratic formula and plug the values in for a, b, and c.
Have you also noticed that u can raise phi to any positive odd integer power, find the reciprocal, and that equals the decimal part of the number. Example: 1/phi^1=1.618033....minus 1. (naturally!) But notice that 1/phi^7 exactly equals the decimal portion of phi^7. Hallelujah! Also one can find a (little more complicated) similar relationship with phi raised to even positive integer values, but i leave that to you.
Sal - I seem to recall the golden ratio has something to do with the sub-linear implementation of Fibonacci in programming. I never took the time to learn it. I'd love it if you can make a video about how to calculate Fibonacci numbers is sub-linear time based upon this. Awesome Video! Thanks!
the number of solutions is the mass results of a ratio, which means the effort using numbers is more relative and has its educating spiral, if this inteligence itself is a manifestation, its the number of the golden ration, which makes the fibonacci numbers interesting, the missing fibonacci numbers are the skipped sections to speed up patterns :) this phi number is 3 if its cycling, trinity :P triangle is in any shape ;)
@Jeorney Dude, it's not that mystery actually. Since, 2cos(Pi/5) = 2cos(36deg) = Phi 666 = 36deg x 18.5 or you increase your circumference distance a multiple of 36 degree 6*6*6 = 36deg x 6 (same analogy) As a result, the sine and cosine of evil number will be associate with Phi. However, it's very cool and flashy when you got the number from the calculator ;-)
If this interests you at all, look up Metatron's Cube and the Flower of Life. It's almost scary how the Universe's patterns have been known for a while now.
@theterabyte Maybe you mean the fact that the quotient of two adjacent fibonacci number converges to the golden ratio? (quotient of 89/55 (10th/9th fib.nbr.) already is 1.618). You can use this fact in a way like this: f(n) = floor(phi^n / sqrt(5) + 1/2). btw. you can start with any two integers and apply the fibonacci algorithm and the quotient will still converge to the golden ratio! It really is somehow magical ;)
As metnioned, it is studied thorughout traditional art, and occurs throughout all visual pieces places of nature. The Vitruvian‑Man is recognized by most everyone, but is an early example of the basic physical form regarding proportion. Other studies go deeper into the semetry of entire facial features, and beauty as recognized universaly innate.
*BEST introductory treatment I have found, yet! (and I have been THOROUGHLY researching it.) The major interrelationships are presented in an excellent order and fashion. Also, the largely mathematical treatment is refreshing, in light of the "sacred connections", "wild speculation", and misinformation (albeit, unintentional) found all around. Thank you Khan Academy for another GREAT JOB! P.S. For the record, the nautilus shell is indeed an "equiangular spiral", but alas NOT on phi. Close, but no cigar :(
Nice lecture. The number of nuture itself ! I always feel strange when i hear greek words or letters pronounced in this weirdo english way... Fie and pie :| Its just φ=fi and π=pi
Thanks Sal! This is a really great video. But is it possible that you could not rely on colors so much? I think I speak on behalf of your colorblind audience when I say that it's quite difficult to notice when a line starts and ends. Once again, awesome video :)
Yes! Can someon please explain how he does this here? He goes all out and squares things and turns that 1 into a positive number and stuff. How did he do it? Please help!
I am only 3 minutes into this lesson but I am stopping the video here as I want to savor it. I think this video combined with Donald Duck in Mathemagical Land could easily account for a good creative semester!
I will always laugh how teachers always had a terrible answer for how math was ever going to be used in real life. Always "how u gonna calculate interest or be architect or something" not "look at a pinecone, if you count the points, it's the fibonacci sequence. Math is nature, its everywhere"
I realize how interesting math can be now that I am not under pressure to learn it.
Me too
Do u remember this?
@@lgndary5715 bro they are responding to a 9 year old comment
I hate math at school. It's stressful and I dread doing the work. But when I'm learning about it on my own (like watching this video) I love it!
school is not for learning.
you learn math at school to pass a standerized test so the school doesn't lose funding.
good videos sal
Dank Hill Really recommended you try Khan website. It's fantastic and FREE to use. Also try You Tube's HEGARTYMATHS CHANNEL. It is based on 🇬🇧exams, but the ideas are explained clearly, and the videos are quick to view. Regards Samantha Wallace ps Good Luck!
True
Wow... Got given a programming problem to solve looking at fibonacci's sequence, read the equation and could not see where the hell it came from. Spotting something about "The golden ratio", I decided to go searching for that, which in turn led me here...
How the hell can such a simple idea of having a line with the ratio of "a to b" being the same as "a+b to a" produce something so fricking awesome!? Mind officially blown. Thanks muchly ;)
I know it's been 5 years but is this programming problem from the book SICP? I'm currently reading it now and found the golden ratio being mentioned in a Fibonacci sequence problem.
@@formulaint I think it is, I got it too
This is great! I'm frustrated though that math is being dumbed down in public school. Like the math in this video isn't that complex, but when I show it to my teacher in ninth grade HONORS its treated like some kind of disease and I'm being told that it's too "hard" to do in class. Like seriously we should be allowed to learn at our own pace and not be put down to learn simple formulas that we've learned in the sixth grade!
+Jacob Rajacich it´s the same shit here in germany. But I always think that it´s just annoying to me, because 98% of my math class doesn´t want to learn more or anything AT ALL about math.
Lol stop boi, you just want attention and for people to think ur smart. It doesn't take any intelligence for a 9th grader to understand this conceptź
+Swagg Kidd given your name and how you try to type things,I would say you are the one who wants attention
I'm pointing something out
+Amazing Ike ikr
The association of the main numbers in the field of mathematics with each other, reflects numerical sequences that correspond to the dimensions of the Earth, the Moon, and the Sun in the unit of measurement in meters, which is: 1' (second) / 299792458 m/s (speed of light in a vacuum).
Ramanujan number: 1,729
Earth's equatorial radius: 6,378 km.
Golden ratio: 1.61803...
• (1,729 x 6,378 x (10^-3)) ^1.61803 x (10^-3) = 3,474.18
Moon's diameter: 3,474 km.
Ramanujan number: 1,729
Speed of light: 299,792,458 m/s
Earth's Equatorial Diameter: 12,756 km. Earth's Equatorial Radius: 6,378 km.
• (1,729 x 299,792,458) / 12,756 / 6,378) = 6,371
Earth's average radius: 6,371 km.
The Cubit
The cubit = Pi - phi^2 = 0.5236
Lunar distance: 384,400 km.
(0.5236 x (10^6) - 384,400) x 10 = 1,392,000
Sun´s diameter: 1,392,000 km.
Higgs Boson: 125.35 (GeV)
Golden ratio: 1.61803...
(125.35 x (10^-1) - 1.61803) x (10^3) = 10,916.97
Circumference of the Moon: 10,916 km.
Golden ratio: 1.618
Golden Angle: 137.5
Earth's equatorial radius: 6,378
Universal Gravitation G = 6.67 x 10^-11 N.m^2/kg^2.
(((1.618 ^137.5) / 6,378) / 6.67) x (10^-20) = 12,756.62
Earth’s equatorial diameter: 12,756 km.
The Euler Number is approximately: 2.71828...
Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2.
Golden ratio: 1.618ɸ
(2.71828 ^ 6.67) x 1.618 x 10 = 12,756.23
Earth’s equatorial diameter: 12,756 km.
Planck’s constant: 6.63 × 10-34 m2 kg.
Circumference of the Moon: 10,916.
Golden ratio: 1.618 ɸ
(((6.63 ^ (10,916 x 10^-4 )) x 1.618 x (10^3) = 12,756.82
Earth’s equatorial diameter: 12,756 km.
Planck's temperature: 1.41679 x 10^32 Kelvin.
Newton’s law of gravitation: G = 6.67 x 10^-11 N.m^2/kg^2.
Speed of Sound: 340.29 m/s
(1.41679 ^ 6.67) x 340.29 - 1 = 3,474.81
Moon's diameter:: 3,474 km.
Cosmic microwave background radiation
2.725 kelvins ,160.4 GHz,
Pi: 3.14
Earth's polar radius: 6,357 km.
((2.725 x 160.4) / 3.14 x (10^4) - (6,357 x 10^-3) = 1,392,000
The diameter of the Sun: 1,392,000 km.
Numbers 3, 6 & 9 - Nikola Tesla
One Parsec = 206265 AU = 3.26 light-years = 3.086 × 10^13 km.
The Numbers: 3, 6 and 9
((3^6) x 9) - (3.086 x (10^3)) -1 = 3,474
The Moon's diameter: 3,474 km.
Now we will use the diameter of the Moon.
Moon's diameter: 3,474 km.
(3.474 + 369 + 1) x (10^2) = 384,400
The term L.D (Lunar Distance) refers to the average distance between the Earth and the Moon, which is 384,400 km.
Moon's diameter: 3,474 km.
((3+6+9) x 3 x 6 x 9) - 9 - 3 + 3,474 = 6,378
Earth's equatorial radius: 6,378 km.
By Gustavo Muniz
Very straight forward. Just look at the basic definition: φ = a/b.
Thus if a = 10, φ = 10/b = 1.618, or b = 10/φ
And that is even more fun if you remember that 1/φ = φ - 1, so you can do that in your head.
φ - 1 = 0.618 times 10 = 6.18 units (cm for this case)
So side A = 10, side B = 6.18...
GREAT video!!
PS, obviously I'm only using the first 3 digits of φ for calculations.
The Golden ratio is also found the our DNA helix, the bronchial tree structure of the lung, the heart valves, etc..... This is not just a ratio... It is nature's ratio. Very intriguing indeed.
Sal I love how excited you get about math and concepts like this. I feel like these things can be mind-blowing and it's so cool to have someone to share this with, even if it's just me at home watching your videos. So proud of everything Khan Academy has become
you could go this way as well: a^2 -ba -b^2 = 0 and solve for a, choosing the positive answer: a= b(1+sqrt(5))/2. Then use this in a/b and get the golden ratio as (1+sqrt(5))/2
Started watching this as a goof to myself, and just couldn't stop watching. Now THAT is teaching.
Because that is what the golden ratio is. 2 numbers in this ratio added together, divided by the bigger one, exactly equals the bigger one. Spend some time with it to understand it. The golden number, 1.6180339.... is the only number that you can square by adding 1, and find the reciprocal by subtracting 1. a over a does =1. b over a =.6180339, those added = golden number. To make the ratio to any desired accuracy, you can add any 2 numbers, add that result to th previous, and the more times>
One of the best Khan Academy videos ever!!! I have watched dozens of times!! Every time I think of φ, I return to this video just for fun!!
I followed everything it was so awesome. I was doing a research essay on this topic and I got an A+ on it.
Such a beautiful equation I ever see in my life. I love Mathematics now!!!!!!!!!
😢😢😢
Very interesting ................. a great wonder i just liked it a lot Thanx for dis wonder creation.!!!
I consider myself as a pretty good amateur at investigating this number already. but you've blown my mind away
Having never studied them myself, I have to take Discovery Channel's word for it that the Golden Ratio was used extensively in ancient architecture; the Parthenon being one of them. Also as I understand it, there are a great number of archways with this ratio measuring from base to sides and up to just where it terminates with the arch its self. It’s also in the windows of some churches. (as I understand it)
>you do that, the more exactly 2 consecutive numbers will = phi. Ex: 1+2=3;2+3=5; 3+5=8; 5+8=13--and so on. The more u do that the more accurately the ratio of the last 2 numbers will = phi. Try it. If you do it around 20 times it will be to the accuracy of an 8 digit calculator. The fascination things about th ratio are endless.
I have always loved Math. I do stem talks at my kids schools and I tell them math is the key to the universe which it is.
10:10, "Let me scroll down a little bit," scrolls up.
+Monzur Rahman Some mouses are actually inverted.
Maybe he is in Australia
I think it's safe to guess Sal uses a Mac.
Monzur Rahman lmaooooooo
@@xxxXLopesXxxx in order to understand this, he needs to buy a mac first.
this vid should be viewed in every math class
I personally found this video very confusing. It didn't really explain what phi was in the beginning of the video and went right into some equation that was afterwards explained as the golden ratio. I was beyond confused. I also would like to see that the work is more organized, it looks like its written it all over the place.
shapes make it easier, single pointedness 1, eye or flame 2, triangle 3, square 4, pentagon 5, hexagon 6 ect
all these shapes hold the triangle in them, and is easier to see when 3d :) its metatrons art and language of numbers :)
yes. it is the result of Fibonacci numbers being added to the previous numeral in the sequence. then dividing it
I read 1/2 of the book about the golden ratio by Mario Livio, I gave up on it because I wasn't really grasping all that was said, but this video really cleared things up, I'll probably head back into it now, thanks so much for your time and effort, the way you use colors and simple explanations is very helpful and clear.
Cool timing for this video. I am preparing to build a door to my house, and was considering incorporating the golden ratio into the design for fun. You've explained some things here that I didn't know about it. Thanks!
This video is the best fundamental mathematical explanation of the golden ratio online. THANKS a lot!!!!!
The best part is, is that there's still so much more to it.
holy crap thats amazing!
Pretty simple, since we know that phi^2 = phi + 1.
then we multiply both sides by phi and we get:
phi^3 = phi^2 + phi
but we already know that phi^2 = phi + 1, so:
phi^3 = phi + 1 + phi
phi^3 = 2phi + 1
A cool way to find phi with any starting number other than 1.
Let: x be a number not equal to 1.
Phi = ((1/x + 1)
this is the best video i have honestly seen. It is educational and took my thinking to a new level i never though achievable. THANK YOU
This guy is SOOOOOOO SMART!!!!!
Reading JoJo helps with this. :D
Well jojo reminded me that this exists and I wanted to regain the knowledge how to calculate it.
Thanks, this Video was really helpful.
φ² = φ + 1
φ = (√φ)²
φ² = (√φ)² + 1² this is the Golden triangle (Kepler's triangle), written in Pythagorean form, so it is a right angle triangle, which hypotenuse is φ.
By the way from here and the unit square we can see the connection between φ and the REAL VALUE of π. Thus π = 4 : √φ = 3.144...
just discovered this channel, three thousand eight hundred and four more video's to go
The number 4 is always used in the quadratic formula :) If you're still confused, just google the quadratic formula and plug the values in for a, b, and c.
Have you also noticed that u can raise phi to any positive odd integer power, find the reciprocal, and that equals the decimal part of the number. Example: 1/phi^1=1.618033....minus 1. (naturally!) But notice that 1/phi^7 exactly equals the decimal portion of phi^7. Hallelujah!
Also one can find a (little more complicated) similar relationship with phi raised to even positive integer values, but i leave that to you.
Sal - I seem to recall the golden ratio has something to do with the sub-linear implementation of Fibonacci in programming. I never took the time to learn it. I'd love it if you can make a video about how to calculate Fibonacci numbers is sub-linear time based upon this.
Awesome Video! Thanks!
the number of solutions is the mass results of a ratio, which means the effort using numbers is more relative and has its educating spiral, if this inteligence itself is a manifestation, its the number of the golden ration, which makes the fibonacci numbers interesting, the missing fibonacci numbers are the skipped sections to speed up patterns :) this phi number is 3 if its cycling, trinity :P triangle is in any shape ;)
TI-85 is the best TI. Great lesson!
we're actually using this video in our trig class
an excellent source for studying maths
This is gr8 it makes maths a fun way of learning and I was so bad at maths before but now I have improved
THANK YOU SO MUCH I FINALLY HAVE INFINITE ROTATION
@Jeorney Dude, it's not that mystery actually. Since, 2cos(Pi/5) = 2cos(36deg) = Phi
666 = 36deg x 18.5 or you increase your circumference distance a multiple of 36 degree
6*6*6 = 36deg x 6 (same analogy)
As a result, the sine and cosine of evil number will be associate with Phi. However, it's very cool and flashy when you got the number from the calculator ;-)
I feel super smart for understanding this whole thing. Lovely!!
This may be the coolest video on the internet. Thanks, Sal!
Proud of Shridhar Acharya who developed the quadratic formula
If this interests you at all, look up Metatron's Cube and the Flower of Life. It's almost scary how the Universe's patterns have been known for a while now.
this ratio is really exciting! that spiral thing seems paradoxical to me.I hope that Sal will upload more videos about it.
This is very interesting and educative. Golden ratios are asthetically perfect ratios.
Wow, I watched that whole entire video... But anywhoo, Thanks for the info! You explained it better than any other site I visited
This video is pure sunflower 🌻
***looking agrily upward... shaking fist*** KHAAAAAN! I love that you did this. Rock on, man.
@theterabyte
Maybe you mean the fact that the quotient of two adjacent fibonacci number converges to the golden ratio? (quotient of 89/55 (10th/9th fib.nbr.) already is 1.618). You can use this fact in a way like this: f(n) = floor(phi^n / sqrt(5) + 1/2).
btw. you can start with any two integers and apply the fibonacci algorithm and the quotient will still converge to the golden ratio! It really is somehow magical ;)
A cosmic gem of comedy
Fantabulous! I'm speechless.
awsome explanation and an intiguing concept!!!
Complexity is intriguing, you'll learn that one day.
This isn't only amazing to mathematicians, it's amazing towards everyone!
What thing did U use to create this?????? ITS SO AMAISING!!!
This was really helpful. Thankss
this is pretty amazing. and your teaching skills are pretty awesome too. thanks for sharing
MIND BLOWN
As metnioned, it is studied thorughout traditional art, and occurs throughout all visual pieces places of nature. The Vitruvian‑Man is recognized by most everyone, but is an early example of the basic physical form regarding proportion. Other studies go deeper into the semetry of entire facial features, and beauty as recognized universaly innate.
The first time Math made me smile 😀
*BEST introductory treatment I have found, yet! (and I have been THOROUGHLY researching it.) The major interrelationships are presented in an excellent order and fashion. Also, the largely mathematical treatment is refreshing, in light of the "sacred connections", "wild speculation", and misinformation (albeit, unintentional) found all around.
Thank you Khan Academy for another GREAT JOB!
P.S. For the record, the nautilus shell is indeed an "equiangular spiral", but alas NOT on phi. Close, but no cigar :(
Nice lecture. The number of nuture itself ! I always feel strange when i hear greek words or letters pronounced in this weirdo english way... Fie and pie :| Its just φ=fi and π=pi
You're suggesting that we know of a propper way to pronounce them?
actually it sounds like π = pee and φ = fee. Its just that natives will always notice the differences :p
This is awesome... I saw the same thing in wiki and I didnt get it until I watched this.
This video is so elegant!
He actually has multiplied both sides by ⱷ:
ⱷ times ⱷ = ⱷ^2
1 times ⱷ = ⱷ
1/ⱷ times ⱷ = 1
hence ⱷ^2=ⱷ + 1
perfect!!! Thank u!!
That just blew my mind
I will have to watch this again sir.. too much information like this makes my head hurt and makes me cry sometimes
THIS GUY IS BOSS
Awesome. I hadn't considered either the continued fraction or continued root form of phi.
Math is so amazing and it's all over the universe , but I don't understand it at all . I wish I had a mathematically inclined mind .
Thanks 😀
supreme mathematics at work!
Recursive and phi is so cursive looking. Once again my mind is blown
Thanks Sal! This is a really great video. But is it possible that you could not rely on colors so much? I think I speak on behalf of your colorblind audience when I say that it's quite difficult to notice when a line starts and ends. Once again, awesome video :)
Mind= AMAZED
What mic do you use? It's amazing.
Yes! Can someon please explain how he does this here? He goes all out and squares things and turns that 1 into a positive number and stuff. How did he do it? Please help!
Thanks
I am only 3 minutes into this lesson but I am stopping the video here as I want to savor it. I think this video combined with Donald Duck in Mathemagical Land could easily account for a good creative semester!
Now that is amazing, I'm in love with the Golden Ratio
Beautiful ❤️ mathematics
Nice sir iam enjoyed well😊😊😊
Genius, damn you have so much knowledge.
my head hurts but this is soo fun to watch..
I love math
I love you Mr. Purple Mousepad. I love you.
This video needs way more views, this is literally the mathematics of beauty.
This makes me wish I could math.
I will always laugh how teachers always had a terrible answer for how math was ever going to be used in real life. Always "how u gonna calculate interest or be architect or something" not "look at a pinecone, if you count the points, it's the fibonacci sequence. Math is nature, its everywhere"
Thank you so much!
The golden ratio has infinite possibilities ( If the value can change each time when changing the values for a and b)