# Square root of 3

The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3. 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, its numerical value in decimal notation had been computed to at least ten billion digits. Its decimal expansion, written here to 65 decimal places, is given by :

1.732050807568877293527446341505872366942805253810380628055806

The fraction 97/56 (≈ 1.732142857...) is sometimes used as a good rational approximation with a reasonably small denominator.

 Binary 1.10111011011001111010… Decimal 1.7320508075688772935… Hexadecimal 1.BB67AE8584CAA73B… Continued fraction $1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ddots }}}}}}}}}}$ ## History

### Ancient Greco-Roman discoveries

Archimedes reported the following range for the value of 3:

(1351/780)2
> 3 > (265/153)2

## Expressions

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:

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

then when $n\to \infty$  :

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

It can also be expressed by generalized continued fractions such as

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

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

The following nested square expressions converge to 3:

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

## Decimal Value

### Computation Algorithms and Formulas

Further information: Methods of computing square roots

The most common algorithm for this, which is used as a basis in many computers and calculators, is a recursive method:

First, choose arbitrary values for a0 and a1. The choice of these values will affect the speed at which the estimates converge towards the correct value. Then iterate through the following recursive computation and algorithm:

Calculate nth estimate as (2an2 - 1) / (bn×2n)

where bn = an × bn-1
and b0 = 1

Next value of a = an+1 = 2an2 - 1

Then increase n by 1 and repeat.

...

The more iterations through the algorithm (that is, the more computations performed and the greater n), the better the approximation.

Starting with a0 = 1 and a1 = 2, the results of the algorithm are as follows:

1st estimate = (2 × 2^2 - 1) / (1 × 2^2) = 7/4 = 1.75000;

a2 = (2 × 2^2 - 1) = 7;

2nd estimate = (2 × 7^2 - 1) / (7 × 1 × 2^3) = 97/56 = 1.73214;

a3 = (2 × 7^2 - 1) = 97;

3rd estimate = (2 × 97^2 - 1) / (97 × 7 × 1 × 2^4) = 18817/10864 = 1.732050810;

(cf actual value of 1.732050807)

Each iteration roughly doubles the number of correct digits.

#### Rational approximations

The fraction 97/56 (1.732142857...) can be used as basic approximation. Despite having a denominator of only 56, it differs from the correct value by less than 1/10,000 (approximately 9.2×10−5). The rounded value of 1.732 is correct to within 0.01% of the actual value.

Archimedes reported a range for its value: (1351/780)2
> 3 > (265/153)2
; the lower limit accurate to 1/608400 (six decimal places) and the upper limit to 2/23409 (four decimal places).

Partial list of most useful and precise rational approximations: 7/4, 26/15, 97/56, 265/153, 362/209, 989/571, 1351/780, 2340/1351, 3691/2131, 5042/2911, 13775/7953, 18817/10864, 70226/40545, ...

## Proof of irrationality

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:

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

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:

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

It follows that m can be replaced with 3n:

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

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

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

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

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

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 3nmq/mnq, which is a contradiction to the hypothesis that m/n was in lowest terms.

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

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

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:

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

Therefore, dividing both terms by 3 gives:

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

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.

## Geometry and trigonometry

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, thus √3 = 2 × sin(60°) = tan(60°) = 3 × ctan(60°) = 2 × cos(30°) = 3 × tan(30°).

The square root of 3 also appears in algebraic expressions for various other trigonometric constants, including 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 i3 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.

There are many special right triangles containing √3 as one of its sides, for example:

1 : 2 : √3, 1 : √2 : √3, 1 : 3 : 2√3, 1 : 3√3 : 2√7, and etc...

For this and other reasons √3 is very useful and important in geometry and other fields of science.

### Square root of −3

Multiplication of 3 by the imaginary unit gives a square root of -3, an imaginary number. More exactly,

${\sqrt {-3}}=\pm {\sqrt {3}}i$

(see square root of negative numbers). It is an Eisenstein integer. Namely, it is expressed as the difference between two non-real cubic roots of 1 (which are Eisenstein integers).

## Other uses

### Power engineering

In power engineering, the voltage between two phases in a three-phase system equals 3 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 3 times the radius (see geometry examples above).