# 3的算術平方根

3的算術平方根是一个正的实数，它的平方等于3，记为：

命名 无理数${\displaystyle \color {blue}{\sqrt {2}}}$ - ${\displaystyle \color {blue}\varphi }$ - ${\displaystyle \color {blue}{\sqrt {3}}}$ - ${\displaystyle \color {blue}{\sqrt {5}}}$ - ${\displaystyle \color {blue}\delta _{S}}$ - ${\displaystyle \color {blue}e}$ - ${\displaystyle \color {blue}\pi }$ 邊長為2的正三角形高為3的平方根 1.7320508075688772935 3的主平方根 ${\displaystyle x^{2}-3=0}$ 1.7320508075688772935 1.101110110110011110101110… 1.732050807568877293527446… 1.BB67AE8584CAA73B25742D70…
${\displaystyle {\sqrt {3}}}$

1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 5580... （OEIS數列A002194

## 註釋

註:

1. ^ ${\displaystyle \!\ x=2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}}$ ， 由觀察可知${\displaystyle x=2+{\frac {1}{1+{\frac {1}{x}}}}}$ ，即${\displaystyle x^{2}-2x-2=0}$ ， 解方程，取正根，得${\displaystyle x=1+{\sqrt {3}}}$ ， 因此${\displaystyle {\sqrt {3}}=x-1=1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}}$

## 参考文献

• M. F. Jones, "22900D approximations to the square roots of the primes less than 100", Math. Comp 22 (1968): 234 - 235.
• H. S. Uhler, "Approximations exceeding 1300 decimals for ${\displaystyle {\sqrt {3}}}$ , ${\displaystyle {\frac {1}{\sqrt {3}}}}$ , ${\displaystyle \sin({\frac {\pi }{3}})}$  and distribution of digits in them" Proc. Nat. Acad. Sci. U. S. A. 37 (1951): 443 - 447.
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers Revised Edition. London: Penguin Group. (1997): 23