Fibonacci Mystery - Numberphile

[@TairyHesticles]

As someone that has always struggled greatly with mathematics, I find myself entranced with these videos. I never thought I'd see the day when math became interesting. Great work, guys.

[@davedude6415]

Same here. Could have used these back in school. Way better than any math teacher I ever had.

[@RobMarchione]

This is the kind of thing that shows how fundamentally flawed our education system is.

[@renger6002]

To be fair, while I agree that talking about cool, more abstract mathematics is great motivation for learning math, you'd still need to do all the stuff that many people struggled with because otherwise there's no useful way of understanding these things. Ie. it's cool to know at a very shallow level about the mysterious aspects of math, but you can't really understand it or use it unless you have the basics down, and we should be setting up a foundation where student will one day be able to get to a place where they can do proofs, and apply these ideas to problems if they so wish. Ie. teach kids algebra :P

[@RobMarchione]

I think there's a lot more to a combined set of learning where you begin teaching young children old myths as stories, then talk about early civilizations roughly, then move into the Egyptians, then the Greeks, all the while introducing mathematical concepts as they come into the story. I feel like in this way you could have kids not only doing but understanding the function and necessity of algebra by about the age of 10. there's a whole world out there and a great moral failing of ours is how truncated and compartmentalized we make that world for children at a young age.

[@nekogod]

Have always loved maths, I could watch these videos all day long

[@numberphile]

didn't realise we had been controversial!!!

[@rachnamallick9360]

Why isn't there a reply to this comment? I LOVE BRADYDDYYDYDYDYDYDDYYDYDYDYDYYDYDYYDYDYDYDDYYDYAOWEOWOSOSPPXKJCJCJCJCJCJCJCJCJJCJCICIF

[@pranavlimaye]

@@discreet_boson Hi

[@MindstabThrull]

I know this video is over 7.5 years old as I write this comment but: Is there any reason we can't start the Fibonacci sequence at 0,1 instead of at 1,1? It still progresses the same way, and then you don't end up with the oddness of the first two terms both being 1 to get started - each number in the sequence is larger than the previous term, and 0,1 follows that as well.

[@HunGredy]

@@MindstabThrull I guess for the same reason any other sequences don't start with 0. It is just not needed.

[@forestofrandom2498]

@@MindstabThrull 0 is the 0th Fibonacci number, so the sequence can start there. In fact, it is interesting to generalise the sequence and start looking at the terms before the 0th term. Look up Binet’s formula for the nth Fibonacci number, it is pretty amazing.

[@micheall1127]

The joy this man gets from Math is just infectious.

[@whatdoiputhere224]

james is singingbanana

[@joeschmoe2697]

Almost a decade later and I still love coming back to these videos. I’m done with college and working full time but seeing these videos make me want to learn again

[@carterpryor3880]

Funny to think this sequence started as a medieval joke about rabbit reproduction.

[@TheTrainSideGraffiti-x5n]

Biology. another example of his sequence being present in nature. it also quantified the theory making it relevant in not only recognizing pattern in terms of logical reasoning but also in terms of science and math.

[@Leyrann]

It wasn't a joke though. It was an attempt to predict rabbit population growth.

[@ckmym]

@Leyrann only if the rabbits were immortal

[@r0bw00d]

@jocaguz18 "We cant measure exactly a meter..." Sure we can. It's called using a ruler.

[@alfienade6738]

@@r0bw00d a euler ruler

[@ogracer]

i wish there were enthusiastic math teachers when i was young. all we seemed to have was grumpy, unkept men who smelt like tobacco.

[@Macsk8ing]

ogracer thats because math is a complete hoax

[@teyxen]

EElectric_M That's a bold statement. How so?

[@Macsk8ing]

teyxen it's hard to explain, it was basically a bunch of nonsense invented in the past to try to interpret real things and assign a value. Math in and of itself does not mean anything.

[@bm-br3go]

EElectric_M That's because math is a language. I wouldn't call it a hoax, but I agree that without proper interpretation, it is meaningless. This property is the same for all languages, the only reason why they mean anything is because we make them mean something. Math just happens to be a language that we discovered rather than created ourselves. As Niel Degrasse Tyson said, "math is the language of the universe"

[@TheCompleteGuitarist]

it's the nicotine that keeps their cerebral functionality so high. No nicotine, no math(s)

[@GoggyboiYT]

Tool also wrote a song that incorporates the fibonacci sequence in both the vocals and the rythm. The song is called Lateralus.

[@TheCheezFace]

I'm so glad someone commented this.

[@XzFreaKzX]

TheCheezFace Go away

[@TheCheezFace]

XzFreaKzX Gab that to my mug m8 n not online and see wot comes about

[@pagangreen6350]

goge1807 i thought schism maybe was 2

[@ryeschultz84]

As a non-mathematician I have to say this channel is teaching me a lot. For working on a spreadsheet I learned Modulus math to solve a problem. I am stoked that I connected that math to this before they showed how the remainders switched back after the divisor when adding them like fib. You guys are making a difference out here. I am certainly a bit stronger math wise than before.

[@Meier-dx7wq]

10 years after release, this vid found me. As I watch educational stuff in math, physics and tech to calm me down after a panic attack or when my anxiety goes through the roof. And today you did a great job by doing so. Thank you. You matter, even in ways you might not think.

[@HerrBaton]

One of cool things about Pisano Periods is there are types of them. F.ex n=5,6,7,9,14 can be 'cut' in two halves, and the respective digits add up to the divider. There are amazing orders in this sequence yet to be discovered I'm sure.

[@peterluth]

MATHEMATICS IS POETRY

[@kitty-rf9zu]

I love maths sooooooo much, it makes me feel so happy

[@Michael010298]

peterluth Poetry is Mathematics*

[@polpottopg]

peterluth was the caps lock really necessary?

[@peterluth]

WHY NOT?

[@hanspeterrrholtvigerust6503]

CAPS LOCK IS CRUISE CONTROL FOR COOL

[@fmshazam]

Theres a great song by Tool using the fibonacci sequence to arrange the vowels called Lateralus

[@HulkRemade]

That's how I learned of the Fibonnaci sequence! Also, my 8th grade math teacher had a poster of it :P

[@DrKaii]

@@HulkRemade Fibonaccissimo

[@CraigMansfield]

I like that he knows about the history of mathematicians.

[@AmberAmber]

+Numberphile - I love your videos so much

[@ffggddss]

@ Numberphile: As a fellow mathematical person, I really dig what you're doing with these vids -- they really bring out the FUN of math, and that should go a long way toward infecting others who haven't seen that aspect, hitherto! A few observations to consider including, if a new edition of this is ever done: • The "natural" start to the Fibonacci sequence is F₀=0, then F₁=1. • Also, when doing one of these cyclic deals, as soon as you get "1" followed by "0," you know you've come to the end of the current cycle and the start of the next one, because the "0" is the zero-point of the new cycle, and F₋₁ = 1 ensures you're going to get the F's again, rather than some non-unit multiple of them. Also, as a fellow musical person, I'd just like to insert that while 7 is a musically interesting modulus, because it's the repeating length of a (Western) musical scale; 12 is the next musically interesting modulus, because it's the repeating length of a chromatic scale. Perhaps your musical friend can play around with that, if he hasn't done already. I know you know, but for benefit of others, the period of the F's mod 12 is 24, versus the 16-cycle for mod 7; so the Fibonacci'd chromatic scale could make an interesting musical pattern, as well.

[@rillloudmother]

idk if anyone has said this, but the reason the musician divided by 7 is because that's how many notes are in a key or tonality in western music.

[@tunarules88]

It's also interesting to note that when dividing by: 10; the period is 60, or 6 times the divisible (10) 100; the period is 300, or 3 times the divisible (100) 1000; the period is 1500, or 1.5 times the divisible (1000) The period length halves every time we jump up the base ten system. I assume when dividing by 10 000 the period length will be 7500, or 0.75 of the divisible (10 000) Just an interesting pattern I just noticed right now. Also if numberphile ever read this I love you guys, I'm in school for philosophy but I find your videos so very interesting :)

[@TheRelicARTS]

Rich Colmer I wrote a short programm to test that and you are kind of right. The first digits of the leftover seem to line up correctly. But 7501,7502,7504,7505,7507,7508 and so on have 5000 more leftover. So the starting from 7500 the leftovers look like this: 0, 5001, 5001, 2, 5003, 5005, 8, 5013, 5021, 34, 5055, 5089, 144 ... Trying to figure out why that is

[@TheRelicARTS]

Rich Colmer When dividing by 10 000 the period length is 15 000

[@tevadevere895]

wow I just notice how sun burned he is

[@VK-pk8uz]

You've never been to the UK, have you? They get sunburned when they turn on the television.

[@tevadevere895]

lol

[@chhavigupta2802]

Whatever...he's cute.

[@pepehimovic3135]

@@chhavigupta2802 he needs sunscreen he's red like a tomato

[@ViratKohli-jj3wj]

Because sun never sets in british empire

[@iabervon]

Additionally, it must repeat from the beginning, because you can work out the sequence backwards in a unique way: if there's a section that goes a, b, c, and you know b and c but not a, you can work out a = (c - b + n) mod n. This means that the sequence can't loop back on itself in the middle (or there would be two different numbers before the b and c which you loop back to. Therefore, it must return to the beginning.

[@OvyGr]

I expected some mentioning of the "golden angle"

[@Ulkomaalainen]

I don't know whether it has been requested yet, but I'd really love to see a video on the connection between the Fibonacci sequence and the fraction 1/89.

[@helloitsme7553]

And if you divide by infinity , and the remainder pattern will be the fibonnaci sequence

[@aidangarvey7049]

Woah, dude!

[@mrwardy4475]

Jeftakels jeez

[@supercool1312]

Jeftakels by 0

[@screwhalunderhill885]

No you can't just divide by infinity that doesn't make sense. When handling infinity you let a pattern grow big. The remainder will be the Fibonacci sequence if you divide by the biggest prime number not any big number. Let's say my pattern is written as 2n as n gets big then 2 will always divide some numbers even if n does not and you don't get the Fibonacci sequence. Infinity as such does not exist.

[@ingeniayexplora]

Actually, if you talk about tendency to infinite, each division will tend to 0, so that can't be able beacuse, as the video's dude said, each pattern can only have one zero, two zeros or four zeros...

[@ewan.cartwright]

This and many other Videos are proving what's wrong with School! Education can be fun despite what the teachers say!

[@juliasmith5146]

7:00 Period of 60, then period of 300, then period of 1,500...would dividing by 10,000 have a period of 7500? 8:00 It's basically the last digits of the fibonacci sequence in base 7, right?

[@henk6172]

Making it base-7 and taking the last digit is the same as taking the remainder of dividing by 7

[@trequor]

Julia Smith but the first three examples are also just multiplying by 5

[@me_hanics]

Julia Smith No there is no general formula to be honest, or mathematicians haven't found it yet and didn't even have a guess so surely there isn't any "nice" formula for it.

[@femilor]

I just tested this. Interestingly enough, Number 7,501 is the first X0,000 (after the obvious first one) but it doesnt repeat there quite yet. The next number in the sequence is X5,001. The correct period is 15,000. So Number 15,001 is X0,001, Number 15,002 is X0,001 and so on.

[@femilor]

and in general it seems that when dividing by 10^n the second zero is at ( 7,5*10^(n-1) )+1 and that the pisano period is 1,5*10^n for n > 2. Note that i have no proof for this holding true, i just checked for n up to 11 or dividing by 100,000,000,000. I'd love to hear the reason for this, especially as James said that there is no general formula for the length of the period. So somehow the powers of 10 must be special.

[@TacoSt8]

James always have his pupils dilated

[@paulcervenka]

+Omar St Math gets him a little...stimulated :3

[@SpaceGuru5]

+Paul Cervenka Seems like a transcendental experience.

[@NoriMori1992]

+SpaceGuru5 Maybe he sπs some π?

[@bobbobson2318]

+NoriMori dat pun doh

[@NoriMori1992]

+Klapaucius Fitzpatrick Which makes it all the more mysterious that his pupils are so large.

[@michaels4340]

I've noticed the video said that there's a Pisano period of 5^0*60 for 10^1, Pisano period of 5^1*60 for 10^2, and a Pisano period of 5^2*60 for 10^3... is that exact? Is that a random coincidence or a true pattern?

[@yourrealdad9442]

Tool - Lateralus

[@horatiotrismegistus616]

+Pikaboss You're overanalyzing.

[@yourrealdad9442]

Horatio Trismegistus or am I overthinking?

[@horatiotrismegistus616]

Pikaboss Not sure, but I'm pretty sure your duality is showing.

[@yourrealdad9442]

Horatio Trismegistus nah, it's just that my body is separated from my mind, and it's withering intuition is leaving opportunities behind...

[@horatiotrismegistus616]

Pikaboss The only thing I can tell you is to just spiral out, see what happens.

[@JesperoTV]

The comments: 20% "I never thought I would like math, but then I found this channel" 2% "Wow, James is so sunburnt!" 78% "THERE IS ALSO A SONG BY TOOL THAT USES THE FIBONACCI SEQUENCE HAVE U HEARD ABOUT IT?"

[@anhbayar11]

Really uninteresting comments.

[@DrKaii]

@@anhbayar11 i know I once had a dream where it was all so uninteresting and it was my house but it wasn't my house

[@DavidHodge-z9v]

The fibonacci sequence done in an Ulam spiral gives some really cool new patterns :)

[@element1192]

I noticed that when you were dividing by 10, then 100, then 1000, the length of the period increased by 5x every time, there’s definitely some formula you could use to determine the length of the period based on this

[@RyanMacMcC]

The Hungarian composer Bela Bartok used the fibonacci series in many of his major works from 1907 onwards, esp. "Sonata for Two Pianos and Percussion," "Music for Strings Percussion and Celeste," etc. You should find a music theorist to do a video about Bartok's musical structural use of the series!

[@jordyb4862]

He said at the end that a formula for the period was unknown, but just in my head I predicted that the 1000'th would have a period of 1500 because 10 had a period of 60 and 100 (10 * 10) had a period of 300 (60/2 * 10) and therefore 1000 (100 * 10) had a period of (300/2 * 10) = 1500. I'm guessing that 10,000, therefore, has a period of 7500.

[@naimulhaq9626]

Dr. Grime, this is for you.I have posted similar question before, without anybody answering it. I will be extremely glad if you kindly do.The question is:What is so special about the number system of base 10 (modulus 10)? Nature follows this base when it adapts Fibonacci sequence or the golden ratio both of which are defined for base 10. Again, if you think of prime numbers in base 10, then you can easily see the first few prime numbers, but if you think base13(say) you will get a head ache,and will need paper and pencil.Why? I will be glad if your next video is on this theme.

[@coopergates9680]

Have you seen their dozenal / duodecimal video yet (base twelve)? You can write any real number in any base you wish.

[@dinocogaming7912]

I think that there is a fibonscci sequence in other bases, e.g. base 2 : 1, 1, 10, 11, 101. Humans use base 10 because it is what we grow up with and because we have 10 fingers. This are just my thoughts

[@telotawa]

there isn't and i have no idea what you're talking about with those examples, fibonacci sequence and golden ratio have nothing to do with base 10, the primes are easy for you to see in base 10 because you live and work with base 10 and are more used to it

[@iabervon]

For any numbers a, b, and n: (a + b) mod n = (a mod n) + (b mod n). This means that you can use the remainders in the rule for the sequence instead of taking the remainders after calculating the full values. Since all of the remainders mod n are less than n, and only two numbers are used in calculating the next number, there are only n*n combinations available. If you look at n*n numbers, you must have a cycle.

[@josuadosreis2269]

James is my favourite, he’s always so excited!!

[@donnythedingo]

In school Iearned Fibonacci's sequence as 0,1,1,2.... and it confused the f*** out of me. I just kept wondering where the first 1 came from?

[@egormatuk3786]

Your school taught you wrong. If the Fibonacci sequence will start with a 0 it will look like this : 0, 0, 0, 0, 0, 0, 0 ... and so on because you can't make a 1 out of thin air XD

[@JohnWeyrauch]

define x_0 = 0, x_1=1 and then you have the same sequence

[@russellcheng2378]

+John Weyrauch 0_0 = x_x = @_@ = T_T ok sorry, just messin with ya

[@J7Handle]

+donnythedingo If you go backwards in the fibonacci sequence (subtracting the numbers) You'll find that beyond zero to the left the fibonacci sequence repeats again, but this time with alternating positive and negative.

[@JohnWeyrauch]

Another fun fact. Ratios of Fibonacci numbers appear in the stock markets and can be used to obtain information about when to enter or exit a trade. i.e. x_(k)/x_(k+1), x_k/x_(k+2)....

[@dilhanmanawadu9649]

if you divide by 100, the pattern repeats at 300 if you divide it by 1000, the pattern repeats by 1500 s it a mere coincidence that we are getting round numbers?

[@sergey1519]

Actually if you divide it by 10 you get 60 if you divide it by 100 you get 300 then 1 500 then 15 000 150 000 1 500 000 15 000 000 and so on

[@DrKaii]

@@sergey1519 I need a few more terms, like 20

[@bbrk24]

3:30 or 12 - the number of "semitones" in an octave.

[@matthewbertrand4139]

Well, based on the length of the periods you listed for the powers of ten, there does seem to be a general formula for those particular values. The length for a period when dividing by 10 ^ n is 60(5 ^ [n - 1]). So the period for dividing by 10^1 is 60, 10^2 is 300, 10^3 is 1500 (and this matches the three period lengths you said for them), 10^4 is 7500, and so on.

[@lesnyk255]

I just discovered this channel, so haven't explored it much yet. I have a Fibonacci question about the Golden Mean. While playing around with Excel one day, I discovered that it doesn't matter what values you seed your "Fibonacci" sequence with - the ratio between two consecutive elements of any such series seems to approach the same Golden Mean. You can start with pi and -e, and still approach 1.618... What gives? I think I'm going to like it here.

[@hoekz]

Hey Numberphile! Your video actually inspired me to investigate these patterns further, so I wrote a program that would find the remainder for a set of numbers between 2 and x (I went as high as 100,000) and it would find the pattern and it's length. I found something really cool in the graph. It's a little hard to explain, but I could make a video about it if you guys wanted to see. Granted, I'm sure it's probably already known about amongst you guys.

[@sebastianorellana3912]

I realize it's years later, but you can probably submit it to the OEIS.

[@talkalexis]

Yeah🎉

[@random9gager]

But it seems there is a formula for the length of a sequence in the 10s 10 has a sequence of 60 numbers 100 has a sequence of 300 (60 x 5) 1000 has a sequence of 1500 (300 x 5) Does it keep going like that?

[@Colgatelicious]

16 numbers? that's insane because in music, time is split up into beats. and to human ears, 16 beats is the ideal length of a hook in a song.

[@UndecimeBeatitudo]

I've also noticed a weird thing about Fibonacci sequence. The quantity of numbers going in a row that have the number of digits that is a multiple of 4 equals 4. For example: 1597, 2584, 4181, 6765 There are four digits in each consecutive number. And the number of these numbers is 4. Next, let's take numbers with the number of digits is a multiple of 8, which, in turn, is also a multiple of 4 14930352, 24157817, 39088169, 63245986 Also 4 of them. And the pattern repeats for each group where the number of digits is a multiple of 4. For the rest it's 5 numbers in a row 2 digits:13 21 34 55 89 3 digits: 144 233 377 610 987 6 digits: 121393 196418 317811 514229 832040 And so on. I don't know why did I find it out and what are the practical applications of this, but whatever.

[@peter.dolkens]

I discovered these back in primary school in a special class they had for students that enjoyed maths where they introduced us to new concepts, and let us experiment with them. Great to find out that it was actually a thing after all these years!

[@KatySwiere]

So would the remainders of the fibonacci sequence when diving by seven be the same as the fibonacci sequence in base 7?

[@robin-vt1qj]

yap

[@DrKaii]

@Joji Joestar what are the last digits? just before infinity

[@nekogod]

So for the last 1 digit the pattern is 60 numbers long, last 2 digits is 300 and last 3 digits is 1500, does that mean last 4 digits will be 7500, and last 5 digits will be 37500?

[@beppe9638]

+nekogod only for 10 multiple, there is no formula for any "n" i guess

[@femilor]

even for multiples of 10 this doesn't work. Last 4 digits is 15000. I responded to a comment by Julia Smith in more detail, because there does seem to be a pattern

[@nekogod]

Interesting I wonder what the period is for last 5 digits

[@femilor]

nekogod The period for last 5 digits is 150,000. In general it seems that when dividing by 10^n the second zero is at ( 7,5*10^(n-1) )+1 and that the pisano period is 1,5*10^n for n > 2. Note that i have no proof for this holding true, i just checked for n up to 11 or dividing by 100,000,000,000. I'd love to hear the reason for this, especially as James said that there is no general formula for the length of the period. So somehow the powers of 10 must be special.

[@DrKaii]

@@femilor that's about how many fingers and toes there are in the world at time of commenting

[@dementedp]

I whish you were my teacher for mathematical theory at my university :P

[@dustinmartin9]

why does the camera man always film so close to their faces haha

[@143mailliw]

he probably just zooms into their faces

[@douglastaylor2652]

Because he forgot to use the Fibonacci and it got all goofy then..

[@3ZEBRA]

This has become one of my favorite KZbin channels

[@WorldWaterWars14]

Correct me if I'm wrong, but did James divide 5 by 5 and not get 1?

[@Bismvth]

When? Most of the video he's talking about the remainder, not the division answer. So 5/5 is equal to exactly 1, therefore rem is 0, whereas 5/6 is like 5/(5+1), remainder is 1

[@TheTenthBlueJay]

He is talking about the remainders

[@highpath4776]

Its the remainder = 0

[@DrKaii]

@@Bismvth these other comments took the remainder of your comment

[@RainAngel111]

math is trippy.

[@djxjxixsmjxjskjzxn1853]

"math" is maths

[@user-vk9kx5nh3j]

Why are your pupils so dilated...

[@LostElsen]

Dan Cruz It seems that he`s very excited

[@wolken_bruch]

numbers are a great drug :D

[@regg2943]

Dan Cruz Some pupil have naturally large pupils.

[@korean007coin]

regg2943 Some pupil have naturally large peoples

[@IYTLIPYAW]

korean007coin Some people have naturally large peoples.

[@CeeKay64]

1:09 math class be like: Teacher: 1 + 1 = 2 5:18 *a few minutes later* Teacher: Fn / Fm if and only if n/m Classmates: ???

[@ericmelton22]

When dividing the numbers and taking the remainder, ever factor of 10 increase (1,10,100,1000) also increases the sequence length by a factor of 5. Ex. Divide by 10, length of sequence 60. Divide by 100, length of sequence 300. Divide by 1000, length of sequence 1500.

[@sdhananjay1993]

Is it like subjecting Fibonacci numbers to modulo 7 ??

[@FourthDerivative]

The Fibonacci tune kinda sounds like something from an N64 Rareware game.

[@lucasm4299]

FourthDerivative It sounds like the level Frantic Factory in Donkey Kong 64

[@DrKaii]

@@lucasm4299 Frantic factory? i bet you are trying to remember when you commented this. click here to be taken back in time

[@DarkMoonDroid]

It's nice to have a non-controversial numberphile again. Thank you Brady and James.

[@ericmelton22]

Not having too much free time at the moment I only checked divisions of 4 & 40. Divide by 4, length 6. Divide by 40, length 60. (For every factor of 10 increase, length also increases by factor of 10). Early conclusion shows different increase rates for different divisors. I wonder if you double to 8 if the period would also double? I will come back to this when I have time. Keep me posted if you discover anything. Thanks.

[@shannonmarbut3648]

Back in school, I remember graphing some of these patterns for some reason. The only one that sticks out in my mind is dividing by 4 since the graph looked like a heartbeat on an ekg.

[@SmileyMPV]

cant you start the sequence with 0,1 instead of 1,1 the rest goes the same, right?

[@anthonyhall9791]

I am 51 years old and this is the first time in my life I have been genuinely enthused by maths. I have found my channel! ( My aim is to understand what on earth Godel was on about but its too complicated for me at this time)

[@DavidParket-g1h]

Have you picked up a biography or maybe a book that summarizes or explains him? I guess you could have gone deeper into the realm of mathematical logic.

[@christophertrott8663]

Your point about Fn|Fm if n|m has an exception F2 is the same as F1 5 cannot be divided by 2 but 5 but can be divided by 1. Could someone explain this to me

[@MrDannyDetail]

Whatever the nth Fibonacci number is, that number will also divide perfectly the '2n'th Fibonacci number, the '3n'th and so for every multiple of n. So the first Fibonacci number is 1, which therefore means every 'oneth' Fibonacci number (i.e. all of them) must be divisible by 1. The second Fibonacci number is also 1, so every other Fibonacci number must be divisible by one. The third Fibonacci number is 2, so every third Fibonacci number (e.g. the 6th, 9th, 12th and so on) will be divisible by 3. The fourth Fibonacci number is 3, so every fourth Fibonacci number is divisble by 3. Then the fifth and sixth Fibonacci numbers are covered in the video. It was little unforunate that the first example they used to illustrate this property was the 5th Fibonacci number, which happens to actually be 5, so it kind of clouded the fact that the two fives were coming from two different places (the 5th number, and the fact that it actually was 5) and that for most of the other example there would be two different numbers involved.

[@devondevon4366]

2:08 , the fibonacci mod 7 gives 16 periods and mod 5 , mod 2 and so on all give different periods. Certainly one day , they will find formulae to determine the length of the period based on the mod.

[@TheIZIKBG]

Fibonacci sequence is also related to the golden ratio 1.61800 , if you divide a big number with the priveous one .

[@epicpolyphony]

Wait, so we DON'T have a formula for the length of the periods based on the dividing number? Interesting :)

[@jasonsavory8688]

epicpolyphony There seemed to be a pattern in the examples, divide by 10 = period of 60, divide by 100 = period of 300, divide by 1000 = period of 1500. /100 = 5* /10, /1000 = 50*/100. I wonder if you /10000 is the period 500* /1000 ?

[@JordanMetroidManiac]

Challenge accepted.

[@andrewshirley8045]

Jason Savory It would be interesting to know if a similar pattern exists for the other numbers. Unfortunately I prefer to watch videos after someone else has figured it out.

[@bengtbengt3850]

If you go to Wikipedia and search for pisano period there will be a big list of the lenghts of the periods and after you have seen those numbers im pretty sure you will understand that er havent come up with a general formula yet..

[@ThrottleKitty]

Tool used the Fibonacci in their 2001 song Lateralus =]

[@missanime1234]

wait.... 3:50

[@TheDeathlord21]

+michelle bostic they are looking for the remain of the division; so 5/5=1 R0 -> 0

[@missanime1234]

TheDeathlord21 we know but the way he said it was funny. because he shouldve said "the remainder of 5 divided by 5.." or whatever.

[@abuhassanabo]

+isabel love he corrected himself instantly

[@lillyparks4450]

So the fibonacci numbers are:1 1 2 3 5 8 13 21 34.... If you add 1+1+2+3+5+8+13, the result is 33 (one less the number after 21 ,in this case) in another case if you add 1+1+2, the answer is 4 ( one less the number after 3 in this case).... Why is this???

[@DaleGerdemann]

Lilly Parks Have a look at Identity 1 in the book "Proofs that Really Count" by Benjamin and Quinn. Since it's at the beginning of the book, you can read the short, easy proof in the book excerpt at your favorite online book seller. But if you catch Fibonacci fever, this is one book you have to buy.

[@gokai8610]

You are the only person whom I have ever known to actually make math interesting,

[@Kneedragon1962]

"No idea what significance this is or where I would use it in life" Yes, rather a common feature of mathematics, that one. Very occasionally, somebody will discover a piece of maths because they needed it. Much more commonly, someone will discover a piece, like this thing, and then some time later, a bright spark will notice something in the real world which corresponds to the odd number fact, then we have a tool to deal with it. What is odd and striking is how quickly these uses for strange bits of number stuff can turn up, and just how very important and useful they can be.

[@GodsDesire]

Every 5th number is divided by 5 because the 5th number of the fibonacci sequence is 5 Every 6th number is divided by 8 because the 6th number of the fibonacci sequence is 8 Every 7th number is divided by 13 because the 7th number of the fibonacci sequence is 13 Every 8th number is divided by 21 because the 8th number of the fibonacci sequence is 21

[@MEdalFan26]

Whoo ar yuo talken to

[@sharcblazer99]

I can't explain it, but I just love this! Symmetry!... Sorta!

[@thegreeneyedbubu]

Sharc Blazer it is perfectly symmetrical. Find the digital root of the numbers, it loops.

[@DrKaii]

@@thegreeneyedbubu do you still teach math to KZbin commenters?

[@MrHenryG123]

Not sure if this corresponds to anything but when solving the rubiks cube on the last 4 corner positioning, you can either have 1, 2 or 4 corners in at once, same with the last layer edge positioning.

[@RixDobbs]

Hello Brady; People always describe the root of the sequence as 1,1 but it really starts at 0,1. So this shows that you can start at a positive offset and the sequence still achieves the golden ratio. You can start at ANY offset - positive or negative, integer or fraction, stepping in the positive direction or negative direction. You can even seed it with a positive AND a negative (dissimilar) number and it will seek the golden ratio. Remarkable. Why? and why does nature favor Fibbonacci numbers?

[@duck6872]

I don't want to brag but I discovered some of these patterns and the sequence on my own when I was 10 years old

[@nickdziura7373]

Duck most of the patterns are easy to figure out, but the implications of the patterns are where the hard part is, like Pascal's triangle and binomial theorem.

[@duck6872]

True

[@Darkstar159]

Divide them by 1 :p

[@RJiiFin]

My first thought.

[@coopergates9680]

Lolz remainders are 0 and period has length..... 1 I think? Dividing by two gives 1,1,0,1,1,0,1,1,0,1,1,0,1,1,0....

[@Kaepsele337]

Every whole number is divisible by 1, so the remainder is always zero. You'd get: 0 0 0 0 0 0 0 0 ... with Period 1

[@TheCoin100]

dividing by 2 gives the binary blueprint for world domination

[@james5604]

Cooper Gates Ever consider what would happen if you turned this into machine language?

[@kanchenjunga9461]

Math blows my mind.

[@TheMerchantDS]

Interesting that using 7 (number of notes in a major scale) results in a period of 16, a convenient number of notes in music (exactly 4 measures in 4/4 time).

[@十二夜夢]

5:35 the rule seems like it doesn't work when n=2 because F_2=1 divides any number while 2 cannot divides odd numbers

[@themanwiththepan]

Now divide by G64.

[@GabrielConstantinides]

😂

[@GabrielConstantinides]

Then divide by 0

[@HulkRemade]

I mean we can just say everything is approximately 0 until like...infinity - 1...

[@GabrielConstantinides]

But infinity is greater than G64 to the power of G64, done G64 times.

[@HulkRemade]

***** G64 is just too big man, and infinity isn't a number.

[@MRKArden-zz4sw]

Lateralus by Tool uses this too.

[@kenmargo8262]

Why are people so fixated on the 'uniqueness' of Fibonacci's series? The real magic lies in the general series x,y,x+y... ,irrespective of the values of x and y Here is a non Fibonacci series , also obtained by adding the last and previous (x,y,x+y...) but ,starting with 2,4.. 6,10 16,26,etc.Except for the first one which is the 4th in the series,every 5th number thereafter ends ends in zero.. Start with 1,4 except for the third number, every 5th thereafter ends in 0 or 5..Start with 1,3 every third number ends in 4,8,6,2. And then the most amazing property-that any number divided by its previous gives a closer and closer approximation to the golden ratio.(1,61803..) as the series progresses. irrespective of what two numbers you choose to generate the series, positive and negative whole numbers and fractions .

[@chrisg3030]

***** Hi Ken and Zen. That's an interesting debate. I share Ken's frustration with the narrow focus on the classic Fibonacci as opposed to variations, particularly when it comes to seeing the pattern in biological structures like pine cones, bee family trees, cell division. My interest at the moment is generalising the addition algorithm to x,y,z,x+z ("Narayana's cows") and further, like x,y,z,a,x+a (OEIS AOO3269)and so on. These incorporate delays and lags which might well characterise less obvious natural reproduction and growth patterns. And it's great to see that video on Pisano periods in the Fibonacci case, just because such cycles are fascinatingly different in the above mentioned Narayana and other cases.

[@RalphDratman]

+Zacharie Etienne (Zen) We all know nothing about everything in the end, Zacharie Etienne. We are born knowing nothing about everything, and of course we want to learn. Later we may come to understand that all the things we have learned do not make much difference in our lives.

[@kenmargo8262]

+Ken Margo . Its the Golden ratio that is found so often in Nature. The fibonacci numbers occur occasionally only in pine cones and some petals. But any others, like spirals (eg spiral galaxies), ratios of the human body etc are relations of the Golden Ratio obtained from the GENERAL series of which Fibonacci sequence is only one of an infinite number. Its the Golden Ratio which is 'awesome' for example .The Golden Ratio is the solution of the equation x^2-x-1=0, its solution is (rt5+1)/2-the elements of which ,rt5,1 &2 are the sides of the second most basic natural number right angled triangle! If Fibonacci had started with say 1,6 ,or any other two numbers he would have generated another sequence which also created the Golden Ratio, and that series would have then become for posterity the 'famous Fibonacci sequence. .

[@DrKaii]

@@kenmargo8262 nobody replied

[@mitchstilborn]

@dihan, the pattern is clearer if you start at 10. Divide by 10, repeat is 60. Divide by 100, repeat is 300. Divide by 1000, repeat is 1500. Repeat length increased by 5x each time.

[@VladTepesh409]

6:35 Interesting. Babylonian Cuneiform numerals used were base-60.

[@OneMoreMileAnHour]

Tool made a song based on the Fibonacci sequence; it's called schism. The syllables he sings are in the Fibonacci sequence

[@drakethorn1339]

And Lateralus :)

[@JoseRobledo]

It's the whole Laeteralus album.

[@KyosBlog]

As pointed out, Lateralus is the Fibonacci song. Schism has a different kind of math to it with the measure also switching back and forth constantly, oftentimes alternating 5/8 and 7/8 as well as 6/8 and 7/8 measures. You can easily check for the absence of Fibonacci numbers by just the first two lines: "I know (2) the pieces fit (4) cause I watched them fall away (7) Mildewed (2) and smoldering (4) Fundamental differing (7)"

[@miloblackmetalhate]

I embrace my, desire to Feel the rhythm, To feel connected, enough to step aside and weep like a widow, to feel inspired- to fathom the power, to witness the beauty, to bathe in the fountain, To Swing on a spiral, To swing on the SPIRAL \m/ P.S. : Spiral out, Keep going \m/

[@Bunny99s]

***** Yes, Lateralus is also one or my favourite Tool songs ^^.

[@thranduiloropherion7756]

he reminds me of freddie in charlie and the chocolate factory.

[@nadineahmedelnaggar2946]

Tool fans hello

[@fireinthehead4280]

the song Lateralus by Tool, has the fibonacci code in it. what a beautiful song.

[@sonicpsycho13]

The song "Lateralus" by Tool uses a Fibonacci sequence for its lyrics. The entire album can also be rearranged using a spiral pattern to generate a new "album", affectionately known as "The Holy Gift".

[@ncamen68]

I am 9yrs old and I can do math like in a instant (nearly)

[@sawansinghmahara]

k

[@breenge]

Jay Ess has just solved internet arguements.

[@JeanPierreBro]

💪Good job, stay in school, don't do drugs.

[@supersniper698]

I can do math in an instant and I am still in the womb

[@bollywongaloid]

I can do math in an instant and I'm 27 with a masters degree in mathematics

[@ultrapoci]

And what happens if we divide every Fibonacci's number by 1? There's no pattern that repeats :)

[@avinashiyer3089]

You realize remainder for every number is 0 so actually it's a repeating sequence of length 1

[@DrKaii]

this comment, 4 years older than the parent, has double the likes of the parent. I beat the odds

[@Arjetube]

It shows the endless cicle of creation and death

[@koenth2359]

Might have mentioned the following simple observation: although there may be no formula known for the repetition period when working with remainders of division by a number t, it is easy to see that there is an upper bound to it. There are t possible remainders, so t^2 possible combinations for two consecutive remainders. The next occurrence of these exact same consecutive values defines the period. So that can never be longer than t^2.

[@walidrahman2]

This channel makes me fall in love with math

[@Johntub3]

woooooooow

[@dhananjaysawant4646]

6:27 doesn’t this theorem break when the index is 2

[@MadisonKirkman]

My friend pointed out to me that the Fibonacci numbers can give you the golden ratio by picking a number and dividing it by the number before it. He also said that the larger the number, the closer you get to the actual golden ratio. Does this have any significance, or was the how the golden ration was created?

[@martiniverson750]

not sure if this was implied when you said it, but with /10, it repeats after 60, then /100=300 and then /1000=1500, so it multiplies by 5 each time... maybe that has something to do with it.

[@ccerrato147]

Best explanation of Pisano sequence I've seen to date!