Square root of 3

The square root of 3 is the positive <a href="/facts/Real_number/R02gw5Pb">real number</a> that, when multiplied by itself, gives the number <a href="/facts/3_(number)/6QzW9RkE">3</a>. It is denoted mathematically as 
 
 
 
 
 
 3
 
 
 
 
 {\textstyle {\sqrt {3}}}
 
 or 
 
 
 
 
 3
 
 1
 
 /
 
 2
 
 
 
 
 {\displaystyle 3^{1/2}}
 
. It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property. The <a href="/facts/Square_root/AfrzfBdQ">square root</a> of 3 is an <a href="/facts/Irrational_number/Psv19TET">irrational number</a>. It is also known as Theodorus' constant, after <a href="/facts/Theodorus_of_Cyrene/cQKmQV0K">Theodorus of Cyrene</a>, who proved its irrationality.
In 2013, its numerical value in decimal notation was computed to ten billion digits. Its <a href="/facts/Decimal_expansion/HCLBt4UQ">decimal expansion</a>, written here to 65 decimal places, is given by <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A002194:

1.732050807568877293527446341505872366942805253810380628055806
The fraction 
 
 
 
 
 
 97
 56
 
 
 
 
 {\textstyle {\frac {97}{56}}}
 
 (1.732142857...) can be used as a good approximation. Despite having a <a href="/facts/Denominator/ghnQE9B3">denominator</a> of only 56, it differs from the correct value by less than 
 
 
 
 
 
 1
 
 10
 ,
 000
 
 
 
 
 
 {\textstyle {\frac {1}{10,000}}}
 
 (approximately 
 
 
 
 9.2
 ×
 
 10
 
 −
 5
 
 
 
 
 {\textstyle 9.2\times 10^{-5}}
 
, with a relative error of 
 
 
 
 5
 ×
 
 10
 
 −
 5
 
 
 
 
 {\textstyle 5\times 10^{-5}}
 
). The rounded value of 1.732 is correct to within 0.01% of the actual value.
The fraction 
 
 
 
 
 
 
 716
 ,
 035
 
 
 413
 ,
 403
 
 
 
 
 
 {\textstyle {\frac {716,035}{413,403}}}
 
 (1.73205080756...) is accurate to 
 
 
 
 1
 ×
 
 10
 
 −
 11
 
 
 
 
 {\textstyle 1\times 10^{-11}}
 
.
<a href="/facts/Archimedes/sIeMhSVF">Archimedes</a> reported a range for its value: 
 
 
 
 (
 
 
 1351
 780
 
 
 
 )
 
 2
 
 
 >
 3
 >
 (
 
 
 265
 153
 
 
 
 )
 
 2
 
 
 
 
 {\textstyle ({\frac {1351}{780}})^{2}>3>({\frac {265}{153}})^{2}}
 
. 
The lower limit 
 
 
 
 
 
 1351
 780
 
 
 
 
 {\textstyle {\frac {1351}{780}}}
 
 is an accurate approximation for 
 
 
 
 
 
 3
 
 
 
 
 {\displaystyle {\sqrt {3}}}
 
 to 
 
 
 
 
 
 1
 
 608
 ,
 400
 
 
 
 
 
 {\textstyle {\frac {1}{608,400}}}
 
 (six decimal places, relative error 
 
 
 
 3
 ×
 
 10
 
 −
 7
 
 
 
 
 {\textstyle 3\times 10^{-7}}
 
) and the upper limit 
 
 
 
 
 
 265
 153
 
 
 
 
 {\textstyle {\frac {265}{153}}}
 
 to 
 
 
 
 
 
 2
 
 23
 ,
 409
 
 
 
 
 
 {\textstyle {\frac {2}{23,409}}}
 
 (four decimal places, relative error 
 
 
 
 1
 ×
 
 10
 
 −
 5
 
 
 
 
 {\textstyle 1\times 10^{-5}}
 
).

Square root of 3 open-in-new

Square root of 3