Rigorously Deriving the Golden Ratio

In this video we will find the value of the Golden Ratio rigorously. Everyone makes unfounded assumptions to derive its value, but here we will make no such unjustified assumptions. As far as I am aware, this is the first time this has been done.

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p.s. April Fools

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[EpicMathTime]

PS: April Fools

Let's recap:
#1) 0:48
It is perfectly fine to choose b = 1, and this is the standard way to calculate the golden ratio. As pointed out by Sam Hollins below, this is valid in the same exact way that it is valid to look at a 1 - 1 - sqrt(2) triangle to get the value of sin(45*). These are in _perfect_ analogy. We are seeking ratios, not the lengths themselves.

#2) 3:22 Clearly b^2 can be factored out and divided out. At this point we would have had the polynomial (a/b)^2 - a/b - 1. Notice that this is the exact same polynomial Andrew had under the assumption that b = 1 (except that it is a polynomial in a/b rather than a, but this is a superficial difference only).

#3) 5:57 this may very well be the first time anyone has gotten the value this way, but that's because it's a totally ridiculous way of doing things.

I'm aware that most of you caught that this was not real (as if the thumbnail and manually stopping the music during my gaffs wasn't evidence enough) but I felt the need to actually comment about, you know, my real take on the matter. 😂

Hope you guys got a kick out of it!

[ishaanvatus]

During the self quarantine, I was doing just fine.
Now I'm doing just _φine_

[serock]

I like to decide that a = (1+sqrt(5))/2 and go from there

[Mu_Lambda_Theta]

5:50 _Bold of you to assume the real numbers just exist_


*ZFC only. THAT is 100% rigorous.*

[gergodenes]

You really assumed that a and b are nonzero?
Damn bruh no unnecessary assumptions you say?

[haminatmiyaxwen]

I watch Epic Math Time for 2 reasons: 1. Math and 2. Tool playing in the background.

Simple.

[MathNerd]

Bold of you to assume b^2 isn't negative

This was brought to you by the complex numbers gang

[notabotta]

Andrew Dotson’s been real quiet since you posted this

[jeffreymerrick]

Alternatively, you could have factored out the b^2 before applying the quadratic formula. Let p = (a/b). Then you have


b^2 [ p^2 - p - 1 ] = 0


Then one solution is b = 0, but that can be ignored because that would make (a/b) undefined or indeterminate. So then you're left with a simple quadratic with constant coefficients.

[alkankondo]

Nice video editing! I used a slightly different approach. I took the equation a^2 - ab - b^2 = 0 and choose one of the variables to solve for in terms of the other. (In my case I choose to solve for a in terms of b.) This gave me an expression in one variable, and I just substituted that expression into the ratio a/b, which gave me the Golden Ratio.


Thanks for making a video on this topic. I, too, find it unsatisfying when proofs plug in a number prematurely. Usually, in such proofs (like this one), the unneeded variables will divide-out once enough algebraic manipulation is done. You just have to have faith, haha!

[thesecretguy]

Uhh rigour???

Unless you start with 0 and the successor function you're basically just an engineer...

[kenanwood]

I love the music! I'm noticing a lot of TOOL in your videos, which is awesome. Also, great video

[chemistro]

3:22 is where I realized what was going on. 😂

[dumbnerdstuff]

I like how you perfectly nail this "what if Captain Dissilusion switched to a math major and went to the gym" charakter. Amazing work, keep it up!






Also do you do all of your animations yourself? If yes how long did it take you and what did you use?

[applessuace]

The first math-themed superhero. What a cool concept you've developed. They should put you in the next Avengers.

[FreshBeatles]

I love the TOOL background music lol! Reflection!

[ahmeddjekhar]

me seeing the word rigorous i have to see it
me again wait I'm a physics student why am I excited
😐

[joaquinbadillogranillo]

In which point did you 'derive' the golden ratio? Here, I'll 'derive' φ
Let f(x, y, z, …) = φ
∂f/∂x = lim as h approaches 0 of (φ-φ)/h
∂f/∂x = lim as h approaches 0 of 0/h;
Given that 0/h is continuous and equal to 0 for all values except h=0, we can use the epsilon-delta definition of the limit and conclude that ∂f/∂x = 0
Since φ is a constant no matter with respect to which variable we derive φ the limit will always result in the 0/h form, due to the fact that f won't change with different values of x, y, z…
Thus by 'deriving' φ we obtain 0.

[suedoe]

So is the half tau logo on his shirt backwards in real life?

[angelmendez-rivera]

In Euclidean geometry, the golden ratio is equal to [1 + sqrt(5)]/2, but I wonder what it is equal to in other geometries, since curvature can consequently change the structure of the equation a/b = (a + b)/a into a different equation altogether.