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. Then √ can be expressed in lower terms as , which is a contradiction.  (The two fractional expressions are equal because equating them, cross-multiplying, and canceling like additive terms gives and hence , which is true by the premise. The second fractional expression for √ is in lower terms since, comparing denominators, since since since . And both the numerator and the denominator of the second fractional expression are positive since and .)
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 shape 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).
- Grant, M.; Perella, M. (July 1999). "Descending to the irrational". Mathematical Gazette 83 (497): 263–267. doi:10.2307/3619054.
- Julian D. A. Wiseman [Sin and Cos in Surds http://www.jdawiseman.com/papers/easymath/surds_sin_cos.html]
- S., D.; Jones, M. F. (1968). "22900D approximations to the square roots of the primes less than 100". Mathematics of Computation 22 (101): 234–235. JSTOR 2004806.
- 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.