# Square root of 3

Binary | 1.10111011011001111010… |

Decimal | 1.7320508075688772935… |

Hexadecimal | 1.BB67AE8584CAA73B… |

Continued fraction |

The **square root of 3** is the positive real number that, when multiplied by itself, gives the number 3. It is more precisely called the **principal square root of 3**, to distinguish it from the negative number with the same property. It is denoted by **√3**.

The square root of 3 is an irrational number. It is also known as **Theodorus' constant**, named after Theodorus of Cyrene, who proved its irrationality.

The first sixty digits of its decimal expansion are:

As of December 2013, its numerical value in decimal has been computed to at least ten billion digits.^{[1]}

The fraction 97/56 (142857…) for the 1.732square root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than 1/10,000 (approximately ×10^{−5}). The rounded value of 1.732 is correct to within 0.01% of the actual value.
9.2

Archimedes reported (1351/780)^{2}_{} > 3 > (265/153)^{2}_{},^{[2]} accurate to 1/608400 (six decimal places) and 2/23409 (four decimal places), respectively.

It can be expressed as the continued fraction [1; 1, 2, 1, 2, 1, 2, 1, …] (sequence A040001 in the OEIS), expanded on the right.

It can also be expressed by generalized continued fractions such as

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

The following nested square expressions converge to √3:

## Contents

## Proof of irrationality[edit]

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:

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:

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

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

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

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 3*n* − *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

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 3*k*:

Therefore, dividing both terms by 3 gives:

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[edit]

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 −3[edit]

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

(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[edit]

### Power engineering[edit]

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).

## See also[edit]

## Notes[edit]

**^**Lukasz Komsta. "Computations | Łukasz Komsta".*komsta.net*. Retrieved September 24, 2016.**^**Knorr, Wilbur R. (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.**^**Grant, M.; Perella, M. (July 1999). "Descending to the irrational".*Mathematical Gazette*.**83**(497): 263–267. doi:10.2307/3619054.**^**Julian D. A. Wiseman Sin and Cos in Surds

## References[edit]

- 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 √3, 1/√3, sin(π/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. More than one of`|pmc=`

and`|PMC=`

specified (help) - Wells, D. (1997).
*The Penguin Dictionary of Curious and Interesting Numbers*(Revised ed.). London: Penguin Group. p. 23.

## External links[edit]

- Theodorus' Constant at MathWorld
- [1] Kevin Brown
- [2] E. B. Davis