Latest revision as of 00:21, 24 May 2017

The golden ratio, $\varphi$, is the irrational algebraic number $$\varphi = \dfrac{1+\sqrt{5}}{2}.$$

Properties

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-The golden ratio $\varphi$ is the [[irrational]] [[algebraic number|algebraic]] number
+The golden ratio, $\varphi$, is the [[irrational]] [[algebraic number|algebraic]] number
 $$\varphi = \dfrac{1+\sqrt{5}}{2}.$$
 =Properties=
-<div class="toccolours mw-collapsible mw-collapsed">
+[[Limit of quotient of consecutive Fibonacci numbers]]<br />
-<strong>Theorem:</strong> The following formula holds:
+[[Relationship between sine, imaginary number, logarithm, and the golden ratio]]<br />
-$$2\sin(i \log(\varphi))=i,$$
+[[Relationship between cosine, imaginary number, logarithm, and the golden ratio]]<br />
-where $\sin$ denotes the [[sine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], and $\varphi$ denotes the [[golden ratio]].
-<div class="mw-collapsible-content">
-<strong>Proof:</strong>  █
-</div>
-</div>
-<div class="toccolours mw-collapsible mw-collapsed">
-<strong>Theorem:</strong> The following formula holds:
-$$2\cos(i \log(1+\varphi))=3,$$
-where $\cos$ denotes the [[cosine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], and $\varphi$ denotes the [[golden ratio]].
-<div class="mw-collapsible-content">
-<strong>Proof:</strong>  █
-</div>
-</div>
 =Videos=