Golden Ratio Version 2

\[\begin {aligned} \frac{a}{b}&=\frac{a+b}{a}\\ a^2&=ba+b^2\\ a^2-ba&=b^2\\ a^2-2\left(\frac{b}{2}\right)a+\left(\frac{b}{2}\right)^2&=b^2+\left(\frac{b}{2}\right)^2\\ \left(a-\frac{b}{2}\right)^2&=b^2+\frac{b^2}{4}\\ a-\frac{b}{2}&=±\sqrt{\frac{5b^2}{4}}\\ a&=\frac{b}{2}±\frac{\sqrt5b}{2}\\ a&=b\left(\frac{1±\sqrt5}{2}\right) \end {aligned}\]

ϕ=defab=b(1±52)b=1±52\begin{aligned} \phi&\overset{\text{def}}{=}\frac{a}{b}\\ &=\frac{b\left(\frac{1±\sqrt5}{2}\right)}{b}\\ &=\boxed{\frac{1±\sqrt5}{2}} \end{aligned}

Note by Gandoff Tan
1 year, 8 months ago

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