Waste less time on Facebook — follow Brilliant.

The Golden Ratio: Infinite Possibilities

Here is the first post introducing the golden ratio.

What else is there to the Golden Ratio? We will see that there are infinite possibilities with it. Here are a few.

From the first notable equation mentioned in the previous post, we can take a square root of both sides. We just get another relation involving \(\phi\). The left hand side is \(\phi\) itself, but the right hand side is some other expression containing \(\phi\). What happens if I substitute the \(\phi\) on the right side with something else. I can then repeat. Here is the math:

\[\phi\] \[=\sqrt{1+\phi}\] \[=\sqrt{1+\sqrt{1+\phi}}\] \[=\sqrt{1+\sqrt{1+\sqrt{1+\phi}}}\] \[=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}}\]

We can extend this operation so that it goes on forever. This is what's called an infinite radical. But it still always equals the golden ratio. This number is starting to seem more special than just some ratio.

Now, using the second notable equation, let's apply the same idea:

\[\phi\] \[=1+\frac{1}{\phi}\] \[=1+\frac{1}{1+\frac{1}{\phi}}\] \[=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}\]

This goes on forever too! And this one's called a continued fraction. Continued fractions are a well developed subject of mathematical study, where all the one's change to other numbers. Every number has it's own continued fraction. And, low and behold, the golden ratio has the simplest continued fraction of all.

Click here for the next post.

Note by Bob Krueger
3 years ago

No vote yet
1 vote


Sort by:

Top Newest

This is AWESOME !!!!!!!! Abhijeeth Babu · 3 years ago

Log in to reply

prove a^0=1 Devansh Shringi · 3 years ago

Log in to reply

@Devansh Shringi See, Devansh.........\(a^{b}*a^{-b}=a^{b-b}=a^{0}=a*\frac{1}{a}=1\) Satvik Golechha · 2 years, 10 months ago

Log in to reply

Can you prove these equation chains? Minimario Minimario · 3 years ago

Log in to reply

@Minimario Minimario Because I derived them using facts about the golden ratio, there is no need to prove them. They are already proven. Bob Krueger · 3 years ago

Log in to reply

Good.................. Ashik Sunny · 3 years ago

Log in to reply

Prove that 0! equals 1 Rishabh Gupta · 3 years ago

Log in to reply

@Rishabh Gupta How many ways are there to choose 0 objects from 0 objects?


(I got this from AoPS. It's not really a rigorous proof of 0!=1 but it explains it well.) Arkan Megraoui · 3 years ago

Log in to reply

@Rishabh Gupta nCn is equal to 1. so (n!)/(n-n)!.(n!) = 1 therefore (n-n)!=0! must be equal to 1. Rohan Mukhopadhyay · 3 years ago

Log in to reply

@Rishabh Gupta (n-1)! * n = n! For n = 1: (1-1)! * 1 = 1! 0! *1 = 1 0! = 1 César Maesima · 3 years ago

Log in to reply

@Rishabh Gupta ncn=1 ,implies n!/(n-n)! .n!=1.. so,0!=1 hence proved.. Saikarthik Bathula · 3 years ago

Log in to reply

Nice..challenging! Helen Canete · 3 years ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...