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
The first sixty digits of its decimal expansion are:
- 1.73205 08075 68877 29352 74463 41505 87236 69428 05253 81038 06280 5580... (sequence A002194 in OEIS)
The rounded value of 1.732 is correct to within 0.01% of the actual value. A close fraction is (1.732142857...).
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.
Proof of irrationality
Suppose that √ is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as for natural numbers m and n.
Therefore, multiplying by 1will give an equal expression:
where q 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:
It follows that m can be replaced with :
Then, √ can also be replaced with in the denominator:
The square of √ can be replaced by 3. As is multiplied by n, their product equals m:
Then √ can be expressed in lower terms than (since the first step reduced the sizes of both the numerator and the denominator, and subsequent steps did not change them) as , which is a contradiction to the hypothesis that was in lowest terms. 
An alternate proof of this is, assuming with being a fully reduced fraction:
Multiplying by n both terms, and then squaring both gives
Since the left side is divisible by 3, so is the right side, requiring that m is divisible by 3. Then, m can be expressed as :
Therefore, dividing both terms by 3 gives:
Since the right side is divisible by 3, so is the left side and hence so is n. Thus, as both n and m are divisible by 3, they have a common factor and is not a fully reduced fraction, contradicting the original premise.
Geometry and trigonometry
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 1/2 and √/2. From this the trigonometric function tangent of 60 degrees equals √, and the sine of 60° and the cosine of 30° both equal half of √.
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°.
The vesica piscis has a major axis: minor axis ratio equal to the square root of three, this can be shown by constructing two equilateral triangles within it.
Square root of −3
In power engineering, the voltage between two phases in a three-phase system equals √ times the line to neutral voltage. This is because any two phases are 120 degrees apart, and two points on a circle 120 degrees apart are separated by √ times the radius (see geometry examples above).
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- Julian D. A. Wiseman [Sin and Cos in Surds http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html]
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- Uhler, H. S. (1951). "Approximations exceeding 1300 decimals for , , and distribution of digits in them". Proc. Nat. Acad. Sci. U. S. A. 37: 443–447. PMC 1063398.
- Wells, D. (1997). The Penguin Dictionary of Curious and Interesting Numbers (Revised ed.). London: Penguin Group. p. 23.