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Concept
Square root of 3
Summary
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as or . It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality.
its numerical value in decimal notation had been computed to at least ten billion digits. Its decimal expansion, written here to 65 decimal places, is given by :
1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 55806
The fraction (1.732142857...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than (approximately , with a relative error of ). The rounded value of 1.732 is correct to within 0.01% of the actual value.
The fraction (1.73205080756...) is accurate to .
Archimedes reported a range for its value: .
The lower limit is an accurate approximation for to (six decimal places, relative error ) and the upper limit to (four decimal places, relative error ).
It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, 1, ...] .
So it is true to say:
then when :
It can also be expressed by generalized continued fractions such as
which is [1; 1, 2, 1, 2, 1, 2, 1, ...] evaluated at every second term.
The square root of 3 can be found as the leg length of an equilateral triangle that encompasses a circle with a diameter of 1.
If an equilateral triangle with sides of length 1 is cut into two equal halves, by bisecting an internal angle across to make a right angle with one side, the right angle triangle's hypotenuse is length one, and the sides are of length and . From this, , , and .
The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.
Official source
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