# 塑膠數

 $\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2} - \frac{1}{6}\sqrt{\frac{23}{3}}}$ 二進位 約為1.010100110010000010110111010011101100101 八進位 約為1.2462026723545104533260274211370405060463 十進位 約為1.324717957244746025960908854478097340734 十六進位 約為1.5320B74ECA44ADAC178897C41461334737F8172F

$\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2} - \frac{1}{6}\sqrt{\frac{23}{3}}}$

## 塑膠數的來源

$x^3-x-1 = 0\,$

$x = \frac{\lambda}{y}+y\,$

$y = \frac{1}{2}\sqrt{x^2-4\lambda}\,$

$-1-y-\frac{\lambda}{y}+\left(y+\frac{\lambda}{y}\right)^3 = 0\,$

$y^6+y^4\left(3\lambda-1\right)-y^3+y^2\left(3\lambda^2-\lambda\right)+\lambda^3 = 0\,$

$\lambda = \frac{1}{3}\,$，將其帶入上面方程，并設$z = y^3\,$，得到一個$z$二次方程

$z^2-z+\frac{1}{27} = 0\,$

$z = \frac{1}{18}\left(9+\sqrt{69}\right)\,$

$y^3 = \frac{1}{18}\left(9+\sqrt{69}\right)\,$

$y$有實數解

$y = \sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}\,$

$x = \sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2} - \frac{1}{6}\sqrt{\frac{23}{3}}}\,$