Square root of 3

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:

${\displaystyle {\begin{bmatrix}1&2\\1&3\end{bmatrix}}^{n}={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}}$

then when ${\displaystyle n\to \infty }$ :

${\displaystyle {\sqrt {3}}=2\cdot {\frac {a_{22}}{a_{12}}}-1}$

It can also be expressed by generalized continued fractions such as

${\displaystyle [2;-4,-4,-4,...]=2-{\cfrac {1}{4-{\cfrac {1}{4-{\cfrac {1}{4-\ddots }}}}}}}$

which is [1; 1, 2, 1, 2, 1, 2, 1, …] evaluated at every second term.

The following nested square expressions converge to √3:

${\displaystyle \!\ {\sqrt {3}}=2-2\left({\frac {1}{2}}-\left({\frac {1}{2}}-\left({\frac {1}{2}}-\left({\frac {1}{2}}-\dots \right)^{2}\right)^{2}\right)^{2}\right)^{2}={\frac {7}{4}}-4\left({\frac {1}{16}}+\left({\frac {1}{16}}+\left({\frac {1}{16}}+\left({\frac {1}{16}}+\dots \right)^{2}\right)^{2}\right)^{2}\right)^{2}.}$

This irrationality proof for the √3 uses Fermat's method of infinite descent:

Suppose that √3 is rational, and express it in lowest possible terms (i.e., as a fully reduced fraction) as m/n for natural numbers m and n.

Therefore, multiplying by 1 will give an equal expression:

${\displaystyle {\frac {m({\sqrt {3}}-q)}{n({\sqrt {3}}-q)}}}$

where q is the largest integer smaller than √3. Note that both the numerator and the denominator have been multiplied by a number smaller than 1.

Through this, and by multiplying out both the numerator and the denominator, we get:

${\displaystyle {\frac {m{\sqrt {3}}-mq}{n{\sqrt {3}}-nq}}}$

It follows that m can be replaced with √3n:

${\displaystyle {\frac {n{\sqrt {3}}^{2}-mq}{n{\sqrt {3}}-nq}}}$

Then, √3 can also be replaced with m/n in the denominator:

${\displaystyle {\frac {n{\sqrt {3}}^{2}-mq}{n{\frac {m}{n}}-nq}}}$

The square of √3 can be replaced by 3. As m/n is multiplied by n, their product equals m:

${\displaystyle {\frac {3n-mq}{m-nq}}}$

Then √3 can be expressed in lower terms than m/n (since the first step reduced the sizes of both the numerator and the denominator, and subsequent steps did not change them) as 3n − mq/m − nq, which is a contradiction to the hypothesis that m/n was in lowest terms.[3]

An alternate proof of this is, assuming √3 = m/n with m/n being a fully reduced fraction:

Multiplying by n both terms, and then squaring both gives

${\displaystyle 3n^{2}=m^{2}.}$

Since the left side is divisible by 3, so is the right side, requiring that m be divisible by 3. Then, m can be expressed as 3k:

${\displaystyle 3n^{2}=(3k)^{2}=9k^{2}}$

Therefore, dividing both terms by 3 gives:

${\displaystyle n^{2}=3k^{2}}$

Since the right side is divisible by 3, so is the left side and hence so is n. Thus, as both n and m are divisible by 3, they have a common factor and m/n is not a fully reduced fraction, contradicting the original premise.

The height of an equilateral triangle with edge length 2 is √3. Also, the long leg of a 30-60-90 triangle with hypotenuse 2.

And, the height of a regular hexagon with sides of length 1.

The diagonal of the unit cube is √3.

This projection of the Bilinski dodecahedron is a rhombus with diagonal ratio √3.

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[4] 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. On the complex plane, this distance is expressed as i√3 mentioned below.

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.

• Square root of 2
• Square root of 5

• 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 ${\displaystyle {\sqrt {3}}}$, ${\displaystyle {\frac {1}{\sqrt {3}}}}$, ${\displaystyle \sin({\frac {\pi }{3}})}$ 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.

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