Golden Ratio Version 1

\[\frac{a+b}{a}=\frac{a}{b}\overset{\text{def}}{=}\phi\]

a+ba=abab+b2=a2b2+2(a2)b+(a2)2=a2+(a2)2(b+a2)2=a2+a24b+a2=±5a24b=(1±5)a2\begin{aligned} \frac{a+b}{a}&=\frac{a}{b}\\ ab+b^2&=a^2\\ b^2+2\left(\frac{a}{2}\right)b+\left(\frac{a}{2}\right)^2&=a^2+\left(\frac{a}{2}\right)^2\\ \left(b+\frac{a}{2}\right)^2&=a^2+\frac{a^2}{4}\\ b+\frac{a}{2}&=±\sqrt{\frac{5a^2}{4}}\\ b&=\frac{\left(1±\sqrt5\right)a}{2} \end{aligned}

ϕ=ab=a(1±5)a2=2a1±5a=2(15)15=1±52\begin{aligned} \phi&=\frac{a}{b}\\ &=\frac{a}{\frac{\left(1±\sqrt5\right)a}{2}}\\ &=\frac{2a}{1±\sqrt5a}\\ &=\frac{2\left(1∓\sqrt5\right)}{1-5}\\ &=\boxed{\frac{1±\sqrt5}{2}} \end{aligned}

Note by Gandoff Tan
1 year, 10 months ago

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Why is the plus-minus sign flipped upside down? @Gandoff Tan.

A Former Brilliant Member - 1 year, 1 month ago

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