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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.[citation needed]
In 2013, its numerical value in decimal notation was computed to 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.[citation needed]
The fraction (1.73205080756...) is accurate to .[citation needed]
Archimedes reported a range for its value: .[2]
The lower limit is an accurate approximation for to (six decimal places, relative error ) and the upper limit to (four decimal places, relative error ).
^Komsta, Łukasz (December 2013). "Computations | Łukasz Komsta". komsta.net. WordPress. Archived from the original on 2023-10-02. Retrieved September 24, 2016.
^Knorr, Wilbur R. (June 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. Retrieved November 15, 2022 – via SpringerLink.
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