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Square root of 3 : ウィキペディア英語版
Square root of 3
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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
:\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 \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 - \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 \frac for natural numbers and .
Therefore, multiplying by 1 will give an equal expression:
:\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:
:\frac
It follows that can be replaced with \sqrtn:
:\frac
Then, can also be replaced with \frac in the denominator:
:\frac-nq}
The square of can be replaced by 3. As \frac is multiplied by , their product equals :
:\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 \frac, which is a contradiction to the hypothesis that m/n was in lowest terms.
An alternate proof of this is, assuming \sqrt=\frac with \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)
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