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
4 years, 3 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

This is AWESOME !!!!!!!!

Abhijeeth Babu - 4 years, 3 months ago

Log in to reply

prove a^0=1

Devansh Shringi - 4 years, 3 months ago

Log in to reply

See, Devansh.........\(a^{b}*a^{-b}=a^{b-b}=a^{0}=a*\frac{1}{a}=1\)

Satvik Golechha - 4 years, 1 month ago

Log in to reply

Can you prove these equation chains?

Minimario Minimario - 4 years, 3 months ago

Log in to reply

Because I derived them using facts about the golden ratio, there is no need to prove them. They are already proven.

Bob Krueger - 4 years, 3 months ago

Log in to reply

Good..................

Ashik Sunny - 4 years, 3 months ago

Log in to reply

Prove that 0! equals 1

Rishabh Gupta - 4 years, 3 months ago

Log in to reply

How many ways are there to choose 0 objects from 0 objects?

One: DO NOTHING!

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

Arkan Megraoui - 4 years, 3 months ago

Log in to reply

nCn is equal to 1. so (n!)/(n-n)!.(n!) = 1 therefore (n-n)!=0! must be equal to 1.

Rohan Mukhopadhyay - 4 years, 3 months ago

Log in to reply

(n-1)! * n = n! For n = 1: (1-1)! * 1 = 1! 0! *1 = 1 0! = 1

César Maesima - 4 years, 3 months ago

Log in to reply

ncn=1 ,implies n!/(n-n)! .n!=1.. so,0!=1 hence proved..

Saikarthik Bathula - 4 years, 3 months ago

Log in to reply

Nice..challenging!

Helen Canete - 4 years, 3 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...