Unique positive real number which when multiplied by itself gives 3
Square root of 3
The height of an
equilateral triangle with sides of length 2 equals the square root of 3.
Representations
Decimal
1.7320508075688772935...
Continued fraction
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.
As of December 2013[update], 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:
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 lower limit is an accurate approximation for to (six decimal places, relative error ) and the upper limit to (four decimal places, relative error ).
And, the height of a regular
hexagon with sides of length 1.
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[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.
The
vesica piscis has a major axis to minor axis ratio equal to . This can be shown by constructing two equilateral triangles within it.
Other uses and occurrence
Power engineering
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° apart, and two points on a circle 120 degrees apart are separated by times the radius (see
geometry examples above).
^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.
JSTOR2004806.