Square root of 3
The square root of 3, denoted \sqrt{3}, is the positive real number r such that r^2 = 3.[1] It is an irrational number, meaning it cannot be expressed as a ratio of two integers.[2] Its decimal expansion is approximately 1.7320508075688772935274463415059.[3] \sqrt{3} is also known as Theodorus's constant, named after the ancient Greek mathematician Theodorus of Cyrene (c. 465–c. 398 BCE), who proved its irrationality as part of his work on the irrationality of square roots of non-square integers up to 17.[4] The proof of irrationality proceeds by contradiction: assume \sqrt{3} = p/q where p and q are coprime positive integers; then p^2 = 3q^2, implying p^2 is divisible by 3 (so p is divisible by 3), and similarly q is divisible by 3, contradicting the assumption that p/q is in lowest terms.[2] A geometric interpretation reinforces this, using areas of equilateral triangles to show that any rational approximation leads to a smaller positive solution, violating minimality.[2] In mathematics, \sqrt{3} has a simple continued fraction expansion [1; \overline{1, 2}], which is periodic and characteristic of quadratic irrationals.[5] This expansion provides good rational approximations, such as $7/4 = 1.75 and $26/15 \approx 1.73333.[6] Geometrically, \sqrt{3} is fundamental: the height h of an equilateral triangle with side length a is h = (\sqrt{3}/2) a, and the area is (\sqrt{3}/4) a^2.[7] These relations arise from dividing the triangle into two 30-60-90 right triangles, where the side ratios are $1 : \sqrt{3} : 2.[7] Beyond geometry, \sqrt{3} appears in trigonometry (e.g., \sin 60^\circ = \sqrt{3}/2) and various algebraic identities.[8]Fundamentals
Definition
The square root of 3, denoted mathematically as \sqrt{3}, is the positive real number whose square equals 3.[9] It represents the principal (non-negative) solution to the quadratic equation x^2 = 3, or equivalently x^2 - 3 = 0.[10] This value is approximately 1.732, though its exact form is the symbolic \sqrt{3}.[9] As a quadratic irrational number, \sqrt{3} cannot be expressed as a ratio of integers and satisfies a quadratic equation with integer coefficients, specifically the minimal polynomial x^2 - 3 = 0.[11]Numerical Value
The square root of 3, denoted √3, is a positive real number approximately equal to 1.73205080757.[8] This value lies between the consecutive integers 1 and 2, and its decimal expansion is infinite and non-terminating.[8] Simple bounds for √3 can be obtained by squaring decimal approximations and comparing to 3. For instance, since$1.732^2 = 2.999824 < 3
and
$1.733^2 = 3.003289 > 3,
it follows that $1.732 < \sqrt{3} < 1.733.[12] Early rational approximations provide tighter bounds through fractions whose squares bracket 3. One such pair is $19/11 \approx 1.72727 < \sqrt{3} < 26/15 \approx 1.73333, since (19/11)^2 = 361/121 \approx 2.98347 < 3 and (26/15)^2 = 676/225 \approx 3.00444 > 3. A closer approximation from the continued fraction convergents is $97/56 \approx 1.73214, which slightly exceeds √3 because (97/56)^2 = 9409/3136 \approx 3.000319 > 3.[13] Archimedes established even narrower bounds using polygonal methods related to circle approximations, yielding $265/153 \approx 1.732026 < \sqrt{3} < 1351/780 \approx 1.732177.[14]