Here is the previous post concerning the Golden Ratio. For a collection of all the posts concerning the Golden Ratio, click #GoldenRatio below.

Today we start delving into the algebra around the golden ratio. Remember that the golden ratio is derived from the equation

\[\phi^2=\phi +1\]

And is equal to \(\frac{1+\sqrt{5}}{2}\). Also, remember that we touched base a little on some interesting identities in one of the first posts. They were:

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

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

To move on, we need to first introduce Fibonacci numbers. But before we do this, I have a challenge for you. We already know that \(\frac{1+\sqrt{5}}{2}\) is equal to the golden ratio, but can you express \(\frac{1-\sqrt{5}}{2}\), \(\frac{-1+\sqrt{5}}{2}\), and \(\frac{-1-\sqrt{5}}{2}\) in terms of the golden ratio? Post your answers and work below.

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestThat wasn't hard.

\[\frac{1 - \sqrt{5}}{2} = 1 - \phi = -\frac{1}{\phi} = - \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{(\cdots)}}}\] \[\frac{-1 + \sqrt{5}}{2} = \phi - 1 = \frac{1}{\phi} = \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{(\cdots)}}}\] \[\frac{-1 - \sqrt{5}}{2} = - \phi = -1 - \cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{(\cdots)}}}\]

PS: This is what I was doing before going to sleep... I literally did the same exercises :D

Log in to reply

Can an admin edit this note so that the tags #CosinesGroup, #GoldenRatio, and #Algebra appear?

Log in to reply