Representations | |
---|---|

Decimal | 1.7320508075688772935... |

Continued fraction | |

Binary | 1.10111011011001111010... |

Hexadecimal | 1.BB67AE8584CAA73B... |

The **square root of 3** is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as **√3** or **3 ^{1/2}**. It is more precisely called the

As of December 2013, its numerical value in decimal notation had been computed to at least ten billion digits.^{[1]} Its decimal expansion, written here to 65 decimal places, is given by OEIS: A002194:

- 1.732050807568877293527446341505872366942805253810380628055806

The fraction **97/56** (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 1/10,000 (approximately 9.2×10^{−5}, with a relative error of 5×10^{−5}). The rounded value of **1.732** is correct to within 0.01% of the actual value.

The fraction 716035/413403 (1.73205080756...) is accurate to 1×10^{−11}.

Archimedes reported a range for its value: (1351/780)^{2}_{} > 3 > (265/153)^{2}_{}.^{[2]}
The lower limit **1351/780** is an accurate approximation for √3 to 1/608400 (six decimal places, relative error 3×10^{−7}) and the upper limit **265/153** to 2/23409 (four decimal places, relative error 1×10^{−5}).

It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS).

So it's 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 1/2 and √3/2. From this the trigonometric function tangent of 60° equals √3, and the sine of 60° and the cosine of 30° both equal √3/2.

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including^{[3]} the sines of 3°, 12°, 15°, 21°, 24°, 33°, 39°, 48°, 51°, 57°, 66°, 69°, 75°, 78°, 84°, and 87°.

It is the distance between parallel sides of a regular hexagon with sides of length 1.

It is the length of the space diagonal of a unit cube.

The vesica piscis has a major axis to minor axis ratio equal to 1:√3, this can be shown by constructing two equilateral triangles within it.

In power engineering, the voltage between two phases in a three-phase system equals √3 times the line to neutral voltage. This is because any two phases are 120° apart, and two points on a circle 120 degrees apart are separated by √3 times the radius (see geometry examples above).

**^**Łukasz Komsta. "Computations | Łukasz Komsta".*komsta.net*. Retrieved September 24, 2016.**^**Knorr, Wilbur R. (1976), "Archimedes and the measurement of the circle: a new interpretation",*Archive for History of Exact Sciences*,**15**(2): 115–140, doi:10.1007/bf00348496, JSTOR 41133444, MR 0497462, S2CID 120954547.**^**Julian D. A. Wiseman Sin and Cos in Surds

- 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. doi:10.2307/2004806. JSTOR 2004806. - Uhler, H. S. (1951). "Approximations exceeding 1300 decimals for , , and distribution of digits in them".
*Proc. Natl. Acad. Sci. U.S.A*.**37**(7): 443–447. doi:10.1073/pnas.37.7.443. PMC 1063398. PMID 16578382. - Wells, D. (1997).
*The Penguin Dictionary of Curious and Interesting Numbers*(Revised ed.). London: Penguin Group. p. 23.

Wikimedia Commons has media related to Square root of 3.

- Theodorus' Constant at MathWorld
- [1] Kevin Brown
- [2] E. B. Davis