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3の平方根

三角形の底辺を二等分すると、斜線が2、二等分した底辺が1、高さが3の比になる。
3は一辺の長さが 1正六角形の対辺の距離に等しい

3の平方根(さんのへいほうこん)は、平方して 3 になる実数である。正のものと負のものの2つがある。正の平方根は

と書き、「ルート3」と読む。その小数表示は

1.73205 08075 68877 29352 74463 41505 87236 69428 05253…

である(オンライン整数列大辞典の数列 A002194)。無理数であることが知られているので、この数字の並びは循環しない。

幾何学的には、一辺の長さが 2 の正三角形の高さに等しく、一辺の長さが 1正六角形の対辺の距離に等しい。また、一辺の長さが 1立方体の対角線の長さに等しい。三角関数を用いると、 とも表される。

小数部分の覚え方として、語呂合わせが知られており、代表的なものに「人並みに奢れや(ひとなみにおごれや)」がある。

性質

  • 代数的整数である。 の有理数体 上の既約多項式x2 − 3 である。
  • 連分数表示は

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