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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Square root of 3 ： ウィキペディア英語版 Square root of 3 }}}} |} The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the principal square root of 3, to distinguish it from the negative number with the same property. It is denoted by :$\backslash sqrt.$ The first sixty digits of its decimal expansion are: :1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 5580... As of December 2013, its numerical value in decimal has been computed to at least ten billion digits.〔Lukasz Komsta: ''(Computations page )''〕 The rounded value of 1.732 is correct to within 0.01% of the actual value. A close fraction is $\backslash tfrac$ (1.732142857...). Archimedes reported (1351/780)2 > 3 > (265/153)2,〔.〕 accurate to 1/608400 (6-places) and 2/23409 (4-places), respectively. The square root of 3 is an irrational number. It is also known as Theodorus' constant, named after Theodorus of Cyrene. It can be expressed as the continued fraction () , expanded on the right. It can also be expressed by generalized continued fractions such as :$(-4,\; -4,\; -4,\; ...)\; =\; 2\; -\; \backslash cfrac\}\}$ which is (2,1, 2,1, 2,1, ... ) evaluated at every second term. ==Proof of irrationality== This irrationality proof for the square root of 3 uses Fermat's method of infinite descent: Suppose that is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as $\backslash frac$ for natural numbers and . Therefore, multiplying by 1 will give an equal expression: :$\backslash frac$ where is the largest integer smaller than . Note that both the numerator and the denominator have been multiplied by a number smaller than 1. Through this, and by multiplying out both the numerator and the denominator, we get: :$\backslash frac$ It follows that can be replaced with $\backslash sqrtn$: :$\backslash frac$ Then, can also be replaced with $\backslash frac$ in the denominator: :$\backslash frac-nq\}$ The square of can be replaced by 3. As $\backslash frac$ is multiplied by , their product equals : :$\backslash frac$ Then can be expressed in lower terms than $m/n$ (since the first step reduced the sizes of both the numerator and the denominator, and subsequent steps did not change them) as $\backslash frac$, which is a contradiction to the hypothesis that $m/n$ was in lowest terms. An alternate proof of this is, assuming $\backslash sqrt=\backslash frac$ with $\backslash frac$ being a fully reduced fraction: Multiplying by both terms, and then squaring both gives :$3n^2=m^2.$ Since the left side is divisible by 3, so is the right side, requiring that be divisible by 3. Then, can be expressed as $3k$: :$3n^2=(3k)^2$ :$3n^2=9k^2$ Therefore, dividing both terms by 3 gives: :$n^2=3k^2$ Since the right side is divisible by 3, so is the left side and hence so is . Thus, as both and are divisible by 3, they have a common factor and $m/n$ is not a fully reduced fraction, contradicting the original premise. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Square root of 3」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.