"Let's call it phi. We don't know what phi is" **stares in suspicion**

Never have I clicked to the Numberphile2 video so quickly.

I just gotta say I NEED to see a fractal of these numbers between 1 and 10 with a different color representing each of the 5 possibilities.

The Sloane Ranger rides again. Showing us a tiny part of the horizon of math.

Neil: “This is a number. What is that number, would you like to guess?” Also Neil: “LeT’s JuSt CaLl It PhI.”

I would *very much* like some more videos on symbolic dynamics. This is absolutely fascinating.

It's gonna be the Golden Ratio, because it's another way of stating the 1+1/(1+1(.... continued fraction.

Maybe the real golden ratio was the different bases we made along the way

I wrote a quick code to do this and experimented with some numbers. Seems to me that if you start with any number of the form x+0.1, where x is a single-digit number, you converge to a constant whose powers, rounded to the nearest integer, generate a series of integers where a(n) = x*a(n-1) + a(n-2).

He has some interestingly labeled folders on that shelf above his desk.

Great. Now, suddenly, I'm slightly obsessed with symbolic dynamics... even though all that I know about it comes from this pair of videos!

this man is wearing a Jimi Hendrix shirt and the walls in his room are decorated like a circus. I like it.

This part was the most interesting, it should be on the main channel.

Fun fact: the infinite tower of 1.12 also converges to 2. It is the solution to x = 1 + 1/x + 2/x^2.

Hm I wonder if you can have complex bases, and if so, what would happen if you graphed the behavior for each base in an Argand diagram. First guess would be it might produce some sort of fractal

When he said 2 sub 2 = 2, isn't 2 sub 2 just nonsense? The sub 2 means it's in binary, and the symbol 2 doesn't exist in binary. Isn't that like asking for the value of Q in decimal?

Ah man, the most interesting bits are always in the extra video! Great seeing Neil again, always great!

I love it when a very simple game and corresponding sequence leads to a complete new area of mathematics.

This video has about 30% of the views of the preceding one. That's just due to inconvenience, not disinterest.

Wow! This is truly beautiful!

this was better than part 1! ! that explanation for 213.56 is a treasure

If you pronounce φ as"fee" then you gotta pronounce π as "pee".

I like this mystery better than the main video!

Was watching this to procrastinate. Then it simply explains what a limit cycle is. Exactly what I needed to help with my revision. Thanks numberphile.

The main video was interesting...this was way more engaging and fascinating. Thanks!

*Waiting for this sequel felt longer than entire year*

Ah, golden ratio. Pops up in random places just like pi does.

This is one of the best numberphiles I've seen...... and suddenly phi........ wtahhhh.... so we need another video on this!!

Wait a minute. How can you have 2 sub 2? Base 2 only has 0s and 1s!

I’m so glad it was the golden ratio because I was really starting to get confused

that's pretty cool

ϕ ϕ fo fum.

Numberphile is a drug to me man I can't get enough of this stuff lol

It would be neat to see another video more in depth on this.

Every time one of these sequence videos comes out, it blows my mind. Neil Sloane officially endorsed by the Funky Dungeon Dwellers.

Neil might be the best guest. What a gift he is to humanity!

Amazing!

Those notations give another way to express polynomial equations ... don't know whether it can be useful or just a notation view

The best parts are always on numberphile2

This vid is much much much better than the vid on Numberphile 1.

How could you not out this in the main video. This is the whole point of the prologue...

God, I love the golden ratio. I was suspicious as soon as he started calling it Phi but I still wasn't prepared.

The beauty.

Now thats golden

3:42 How is 2 base 2 even defined? In base 2, you don't have the digit "2", you only have the digits "0" and "1".

This may be the first time I've watched a Numberphile and the presenter seemed to be sad by the fact that we don't know something. It made me want to see if I can help!

I loved the video. I note there is an exception to the assertion that 2 (sub b) always equals "2". In base 1, 2 = "11", or "||" perhaps, depending on how you prefer your unary notation.

This reminds me a lot of the Mandelbrot set. I wonder if you similarly tried to create a visual for this function from 1->10 what it might look like. And of course, whether complex numbers would work in this function and if that has similar properties.

Again Neil proves himself the best numberphile guest for ASMR

A big part of this is that we interpret the base as a decimal number (so with something written with the subscript 12, we always interpret that as being double 6, and not say 12 base 3). What if that wasn't the case? 1.111... and 10 would be in a cycle whichever base we use (but the actual values of the two numbers would change), converging in the case of 2 being static in base 2. There's a lot more to explore here, simply by removing the requirement of reading the base as a base 10 number.

That's brilliant! And you can go farther by replacing 1.1 by 1.2 or 1.3 and so on... to get the nine first metallic ratios!

Numberphile 2 is usually just kinda extra optional stuff but I feel like this video should have been the conclusion to the first. the "numbers growing" bit and "not growing as large as the exponents" was very unsatisfying. Like, a non-result. This one demonstrating that some numbers converge to the golden ratio makes it interesting.

Funnily enough, I like it when phi pops up with absolutely no relation to "nature". It's literally everywhere for some reason, and it's usually in where we're doing arithmetic and modular stuff, but it'll just pop out of no where.

I wrote a program and noticed that the "converging point function" is not continuous on any finite decimals. For example, the chain of 1.999... converges to 4, but the chain of 2 is obviously 2.

The nature of humanity is just that every so often someone invents continued fractions again.

symbolic dynamics sounds badass

I would love to see some sort of mandelbrot esque graph to see which numbers resolve to what to see if there is some sort of fractal pattern

How interesting.

that smile is just big

lol, my totally random guess was: oh, how about the golden ratio, it always appears in random places... and it actually is...

Numberphile in 2020s: Brady be like "The viewers are ready. Advanced Math it is."

in general for a number between 1 and 10 that is A + B/10, then the equation should be A+B/x = x, and then x^2 = Ax+B, and then we get from the quadratic formula sqrt(A^2/4-B) - A/2, which should work so long as A^2 > 4B. So some stable numbers to investigate are... 1.1, 2.1, 2.2, 3.1, 3.2, 4.1, 4.2, 4.3, 4.4, 5.1 ... 9.9; for example 9.9 stabilizes around 9.9083 no matter how far you go, and holds the property 9+9/x = x

a_a immediately raises the question of how to interpret it. I'm rather tickled by the blithe way Neil leaps right past the question.

Neil Sloane is my favourite Numberphile personality. Well, and Cliff Stoll. Both treasures. Here's my question: this behaviour of numbers in symbolic dynamics sounds like they could be visualised. If you graphed all the numbers from 1 to 10, based on themselves to infinity, and the behaviour could settle or diverge -- that sounds an awful lot like fractals like the Mandelbrot Set in which every point on the complex Argand plain might be bounded or unbounded when an operator is applied to itself recursively, and which defines the shape of the fractal. Has anyone tried to see if there's a visual pattern to whether a particular number a base a becomes bounded or unbounded?

What happens if you use negative or complex bases? Is it still unpredictable or does it become regular again?

"Let's call it phi..." *narrows eyes*

5:05 i wonder if you can plot it similar to the mandelbrot set, how quickly it diverges?

Thanks for letting me know my phone still getting hacked

Can you do imaginary base numbers? 🤔 if so maybe you can plot on a 2d graph which numbers converge and which numbers diverge, like the mandelbrot set

What would happen if you worked with base in the dungeon number context? What about in a broader context? Would there be any pragmatic value to it or would it just make everything unnecessarily complicated and confusing without any helpful application?

Le magic pentagon number strikes again

I was messing around with my python code that was calculating it for any a, and I've found out that if a is 2.1 the result after 100 iterations was silver ratio, I tested it more and I've found this formula: a(n+0.1)=(n+sqrt(n+4)/2, where a(x) is the function mentioned in this video always* works. *Edit: It only works for n

What happens outside of the range (1,10)? If we don't know what's going on inside then does that mean that numbers outside of the range behave in some obvious way?

I started with (1.1 in base 1.1) which is 1 + (1/1.1) which is 1.9090909... and then (1.1 in base 1.9090909...) which is 1.90909... + (1/1.90909...) which is 2.4329 and then (1.1 in base 2.4329...) which is 2.4329... + (1/2.4329...) which is 2.8439... It eventually goes up to infinity, but it takes approximately n iterations to get up from approximately n to approximately n+1, so it increases at an ever decreasing rate.

This is a pretty pure example of mathematics being making up some rules and see what happens.

the "extra video" is phi^phi^phi^phi... better than the original video

Wouldnt the answer to Bradys question of what if we do it with 3.8 just be 3+8/phi=phi so 3x+8=x^2 which is just a quadratic that either does or doesnt (ima guess doesnt) equal the golden ratio so no

Can you extend to the complex non-reals?

Help, where that 5 come from at 2:56 ?

Does anyone have the OEIS code for the infinite subbing sequence?

What if the result of one of these cycles within a fixed range, but never settles into a loop? IE, Chaotic behavior? I did not see that possibility mentioned as one of the 5 outcomes (which I take to be zero, infinite, unchanged from input, converging, or cyclic). Am I missing something, or is it somehow known to not be possible? To me it seems not only possible, but almost unavoidable at the boundary of the other 5 outcomes.

I can't help but think this may tie in somehow to Riemann Zeta function.

I wonder what a graph of all of these limits might look like

Why did I get another notification for this?

Very interesting topic. The difficulty with this operation seems to be that "interpreting the decimal expansion of X in base b" is pretty clunky to express in general terms. Getting the nth digit of the decimal expansion (or any base of course) of a number is not a nice clean mathematical operation. It's something like `a[n] = floor(a*10^n)%10` with % standing for the modulo operation. So `a_b = sum(a[n] * b^n) = sum(floor(a*10^n)%10 * b^n)` which is no fun to work with. There probably is a nicer expression for a[n] than that, but in the end it just isn't very natural.

Let n = a.b, with a the integer and b the decimal part. Assume, n_(n_(n_...)) [where _ is the dungeon function]is equal to m=c.d where c is an integer and d is the decimal part. Assume a.b_c.d = c.d. Thus, a + b/c.d = c.d or axc.d +b = c.dxc.d = (c.d)^2. Hence we get (c.d)^2-axc.d-b=0 = m^2-axm-b. Since this is a quadratic equation, it has either 0, 1, or 2 positive solutions. If it's 0, it should diverge. If it's 1, it should converge. If it's 2, it should oscillate between two numbers. Examples: Let n=1.1. Then a=1 and b=1. This gives us the quadratic equation m^2-m-1=0 and m=the golden ratio. Let n=1.2. Then a=1 and b=2. This gives us the quadratic equation m^2-m-2=0 and m=2 (unique positive solution). Let n=2.3. Then a=2 and b=3. This gives us the quadratic equation m^2-2m-3=0 and m=3 (unique positive solution). Let n=3.4. Then a=3 and b=4. This gives us the quadratic equation m^2-3m-4=0 and m=4 (unique positive solution). Let n=a.(a+1). Then a=a and b=a+1. This gives us the quadratic equation m^2-am-a-1=0 and m=a+1 (unique positive solution). I honestly haven't seen examples of the other two since I haven't programmed this.

from what it seems, the numbers that dont diverge in this algorithm seem to be an overly complicated reinvention of finding the zeroes of a polynomial. the power of a polynomial being the same as the number of digits given. because 1,1 has 2 digits, it gave a quadratic equasion, for example.

3.35 sounds like the Mandelbrot set, isn’t it?

Is it even allowed to use a single digit for a? How should I interpret for example a 2 in binary? Or a 3 in tertiary?

So this is what mathematicians do when they're bored. Great cliffhanger, I just had to click the card...

My guess was wrong. I should have known a relation between something and "itself" would have given way to by. But I guessed at the phenomenon he mentioned here, "limit cycle." I had wondered about limts that have two major points. I'll have to look that up.

That wallpaper makes me want whataburger.

I wonder what happens if the base is tau or some such number.

okay but he's also wearing a Jimi Hendrix shirt, he's in the top Φ mathematicians in my book

i liked it

D'oh! I realized my limitation.

"phi equals root one plus phi equals root one plus phi equals root one plus phi equals root one plus - "*the proportion is divine~*" ♪ *YOU'LL FIND YOUR WAY TO PHI TO PHI TO PHI*

What happens as you approach pi as the start?

you can even get complex numbers!

No one is talking about that this video is UNLISTED.