# 5的算術平方根

 无理数 √2 - φ - √3 - √5 - δS - e - π 二进制 10.0011110001101111... 十进制 2.23606797749978969... 十六进制 2.3C6EF372FE94F82C... 连续分数 ${\displaystyle 2+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+{\cfrac {1}{4+\ddots }}}}}}}}}$

5的算術平方根是一个正的实数，為无理数[1]，一般称为“根号5”，记为 ${\displaystyle {\sqrt {5}}}$${\displaystyle {\sqrt {5}}}$乘以它本身的值为5

${\displaystyle {\sqrt {5}}}$黃金比值有關。5的算术平方根數值为：

2.23606 79774 99789 69640 91736 68731 27623 54406 18359 61152 57242 7089 ... （OEIS中的数列A002163

## 連分數表示法

${\displaystyle {\sqrt {5}}}$可以表示為連分數[2; 4, 4, 4, 4, 4...] （OEIS中的数列A040002）。最佳有理数逼近的數列如下：

${\displaystyle {\color {OliveGreen}{\frac {2}{1}}},{\frac {7}{3}},{\color {OliveGreen}{\frac {9}{4}}},{\frac {20}{9}},{\frac {29}{13}},{\color {OliveGreen}{\frac {38}{17}}},{\frac {123}{55}},{\color {OliveGreen}{\frac {161}{72}}},{\frac {360}{161}},{\frac {521}{233}},{\color {OliveGreen}{\frac {682}{305}}},{\frac {2207}{987}},{\color {OliveGreen}{\frac {2889}{1292}}},\dots }$

## 牛頓法

${\displaystyle {\frac {2}{1}}=2.0,\quad {\frac {9}{4}}=2.25,\quad {\frac {161}{72}}=2.23611\dots ,\quad {\frac {51841}{23184}}=2.2360679779\ldots }$

## 和黃金比例及費氏數列的關係

${\displaystyle {\sqrt {5}}=\varphi +\Phi =2\varphi -1=2\Phi +1}$
${\displaystyle \varphi ={\frac {{\sqrt {5}}+1}{2}}}$
${\displaystyle \Phi ={\frac {{\sqrt {5}}-1}{2}}.}$

${\displaystyle F\left(n\right)={{\varphi ^{n}-(1-\varphi )^{n}} \over {\sqrt {5}}}.}$

${\displaystyle {\sqrt {5}}}$ 除以${\displaystyle \varphi }$得到的商（或${\displaystyle {\sqrt {5}}}$和Φ的積）及其倒數的連分數有特別的模式，而且和費氏數列及盧卡斯數的比值有關[4]

${\displaystyle {\frac {\sqrt {5}}{\varphi }}=\Phi \cdot {\sqrt {5}}={\frac {5-{\sqrt {5}}}{2}}=1.3819660112501051518\dots =[1;2,1,1,1,1,1,1,1,\dots ]}$
${\displaystyle {\frac {\varphi }{\sqrt {5}}}={\frac {1}{\Phi \cdot {\sqrt {5}}}}={\frac {5+{\sqrt {5}}}{10}}=0.72360679774997896964\dots =[0;1,2,1,1,1,1,1,1,\dots ].}$

${\displaystyle {1,{\frac {3}{2}},{\frac {4}{3}},{\frac {7}{5}},{\frac {11}{8}},{\frac {18}{13}},{\frac {29}{21}},{\frac {47}{34}},{\frac {76}{55}},{\frac {123}{89}}},\dots \dots [1;2,1,1,1,1,1,1,1,\dots ]}$
${\displaystyle {1,{\frac {2}{3}},{\frac {3}{4}},{\frac {5}{7}},{\frac {8}{11}},{\frac {13}{18}},{\frac {21}{29}},{\frac {34}{47}},{\frac {55}{76}},{\frac {89}{123}}},\dots \dots [0;1,2,1,1,1,1,1,1,\dots ].}$

## 和丟番圖逼近的關係

${\displaystyle \left|x-{\frac {m}{n}}\right|<{\frac {1}{{\sqrt {5}}\,n^{2}}}}$

${\displaystyle \left|\alpha -{p_{i} \over q_{i}}\right|<{1 \over {\sqrt {5}}q_{i}^{2}},\qquad \left|\alpha -{p_{i+1} \over q_{i+1}}\right|<{1 \over {\sqrt {5}}q_{i+1}^{2}},\qquad \left|\alpha -{p_{i+2} \over q_{i+2}}\right|<{1 \over {\sqrt {5}}q_{i+2}^{2}}.}$

## 抽象代數中的意義

${\displaystyle \scriptstyle \mathbb {Z} \left[\,{\sqrt {-5}}\,\right]}$中的數均可表示為${\displaystyle \scriptstyle a\,+\,b{\sqrt {-5}}}$的形式，其中ab整數，而${\displaystyle \scriptstyle {\sqrt {-5}}}$虛數${\displaystyle \scriptstyle i{\sqrt {5}}}$。此環是一個整環，但不是唯一分解整環。例如在此環中，6的質因數分解方式就有二種：

${\displaystyle 6=2\cdot 3=\left(1-{\sqrt {-5}}\right)\left(1+{\sqrt {-5}}\right).\,}$

${\displaystyle {\sqrt {5}}=e^{\frac {2\pi i}{5}}-e^{\frac {4\pi i}{5}}-e^{\frac {6\pi i}{5}}+e^{\frac {8\pi i}{5}}.\,}$

## 拉马努金的恆等式

${\displaystyle {\cfrac {1}{1+{\cfrac {e^{-2\pi }}{1+{\cfrac {e^{-4\pi }}{1+{\cfrac {e^{-6\pi }}{1+\ddots }}}}}}}}=\left({\sqrt {\frac {5+{\sqrt {5}}}{2}}}-{\frac {{\sqrt {5}}+1}{2}}\right)e^{\frac {2\pi }{5}}=e^{\frac {2\pi }{5}}\left({\sqrt {\varphi {\sqrt {5}}}}-\varphi \right)}$

${\displaystyle {\cfrac {1}{1+{\cfrac {e^{-2\pi {\sqrt {5}}}}{1+{\cfrac {e^{-4\pi {\sqrt {5}}}}{1+{\cfrac {e^{-6\pi {\sqrt {5}}}}{1+\ddots }}}}}}}}=\left({{\sqrt {5}} \over 1+\left[5^{\frac {3}{4}}(\varphi -1)^{\frac {5}{2}}-1\right]^{\frac {1}{5}}}-\varphi \right)e^{\frac {2\pi }{\sqrt {5}}}}$

${\displaystyle 4\int _{0}^{\infty }{\frac {xe^{-x{\sqrt {5}}}}{\cosh x}}\,dx={\cfrac {1}{1+{\cfrac {1^{2}}{1+{\cfrac {1^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {2^{2}}{1+{\cfrac {3^{2}}{1+{\cfrac {3^{2}}{1+\ddots }}}}}}}}}}}}}}.}$

## 参考資料

1. ^ Dauben, Joseph W. (June 1983) Scientific American Georg Cantor and the origins of transfinite set theory. Volume 248; Page 122.
2. ^ R. Nemiroff and J. Bonnell: The first 1 million digits of the square root of 5
3. ^ Browne, Malcolm W. (July 30, 1985) New York Times Puzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1.
4. ^ Richard K. Guy: "The Strong Law of Small Numbers". American Mathematical Monthly, vol. 95, 1988, pp. 675–712
5. ^ Kimberly Elam, Geometry of Design: Studies in Proportion and Composition, New York: Princeton Architectural Press, 2001, ISBN 1568982496
6. ^ Jay Hambidge, The Elements of Dynamic Symmetry, Courier Dover Publications, 1967, ISBN 0486217760
7. ^ LeVeque, William Judson, Topics in number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1956, MR0080682
8. Khinchin, Aleksandr Yakovlevich, Continued Fractions, University of Chicago Press, Chicago and London, 1964
9. ^ Ramanathan, K. G., On the Rogers-Ramanujan continued fraction, Indian Academy of Sciences. Proceedings. Mathematical Sciences, 1984, 93 (2): 67–77, ISSN 0253-4142, doi:10.1007/BF02840651, MR813071
10. ^ Eric W. Weisstein, Ramanujan Continued Fractions at MathWorld