Difference between revisions of "Golden ratio"
From specialfunctionswiki
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| − | The golden ratio $\varphi$ is the [[irrational]] [[algebraic]] number | + | The golden ratio $\varphi$ is the [[irrational]] [[algebraic number|algebraic]] number |
$$\varphi = \dfrac{1+\sqrt{5}}{2}.$$ | $$\varphi = \dfrac{1+\sqrt{5}}{2}.$$ | ||
Revision as of 09:14, 5 June 2016
The golden ratio $\varphi$ is the irrational algebraic number $$\varphi = \dfrac{1+\sqrt{5}}{2}.$$
Properties
Theorem: The following formula holds: $$2\sin(i \log(\varphi))=i,$$ where $\sin$ denotes the sine function, $i$ denotes the imaginary number, $\log$ denotes the logarithm, and $\varphi$ denotes the golden ratio.
Proof: █
Theorem: 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.
Proof: █
Videos
The Golden Ratio & Fibonacci Numbers: Fact versus Fiction
