Square root of pi

Square root of pi
Rationality	Irrational
Representations
Decimal	1.7724538509055160272
Algebraic form	π

The square root of pi (Approximately 1.7724) is a positive real number, that when multiplied by itself, equals the number pi ( $π$ ). It is more precisely called the principal square root of pi to distinguish it from the negative number with the same property. It may be written in mathematics as $\sqrt{π}$ , Γ(1/2) or sqrt( $π$ ).^[1] It is not the root of a non-zero polynomial and therefore is a transcendental number.^[2]

Approximations

The Decimal approximation of $\sqrt{π}$ to 65 decimal places is: 1.7724538509055160272981674833411451827975494561223871282138077898...^[3]
The fraction 39/22 (≈1.7727272) can be used as a good approximation. It differs from the correct value by ~1/10000.
As a linear continued fraction, $\sqrt{π}$ is: [1; 1, 3, 2, 1, 1, 6, 1, 28, 13, 1, 1, 2, 18, 1, 1, 1, 83, 1, 4, 1, 2, 4, 1, 288, 1, 90, 1, ...]^[3]

Expressions

Expressed as the area below the function $y = e^{- x^{2}}$ as the integral:^[4]

$\int_{- \infty}^{\infty} e^{- x^{2}} d x = \sqrt π$ .

Expressed in series as:^[5]

$2 \sqrt \sum_{k = 0}^{\infty} (- 1)^{k} / 1 + 2 k = \sqrt π$

Can be expressed as:

$e^{- \log G (1 / 2) + \log G (3 / 2)} = \sqrt π$ ^[2]

Geometrically expressed as the hypotenuse of a triangle with a base and height of $π / 2$ .

Applications

$\sqrt{π}$ is the total area under the Gaussian integral, also known as the Euler-Poisson integral, $y = e^{- x^{2}}$ . Named after the German mathematician Carl Friedrich Gauss. This function has many applications. By altering some variables in the function the normalizing constant of the normal distribution can be calculated. The integral also appears frequently in quantum mechanics and statistical mechanics (see Gaussian integral)^[6]^[3]
$\sqrt{π}$ is the average distance between any two points in a multivariate normal distribution. Where the mean is 0 the variance is 1 and is in two dimensions.^[3] This can be used to find relationships between data points that follow a joint normal distribution, particularly in statistics and computing.
$\sqrt{π}$ is the diameter of a sphere whose surface area is $π^{2}$ , or the radius of a circle with are of $π^{2}$ .

Popular Culture

Reference in popular media

Referenced in dialogue between Bella and Edward in the 2008 film twilight (scene 44).

BELLA: So how about some answers?
EDWARD: Yes, no. To get to the other side, and 1 point 772453851.

BELLA: I didn't ask for the square root of pi.^[7]

Referenced in dialogue between Rick, Morty, Jerry and Beth in the 2013 TV series Rick and Morty (season 1, episode 1).

RICK: Morty, tell your parents the square root of pi.
MORTY: Oh, come on, Rick. You know I can't.
RICK: The square root of Pi, Morty. Go!

MORTY: 1.77245385... Whoa!

Referenced in dialogue with Larry Trudgeman in the 2001 TV series Lizzie McGuire (season 1, episode 25)

QUESTION: What is the square root of pi? LARRY: 1.98324 (answering incorrectly)

Referenced in dialogue between The Doctor and Sky in the 2008 TV series Doctor Who (series 4, episode 8)

THE DOCTOR: The square root of pi is 1.77245385090/5516027298167483341... wow! SKY: The square root of pi is 1.772453850905516027298167483341... wow!^[8]

The square root of pi, a song by Splooge God, Released in 2022.^[9]

References

↑ "index of important irrational constants". planetmath.org. Retrieved 2023-08-21.
↑ ^2.0 ^2.1 "Γ(Divide[1,2]) - Wolfram|Alpha". www.wolframalpha.com. Retrieved 2023-08-20.
↑ ^3.0 ^3.1 ^3.2 ^3.3 Sloane, N. J. A. (ed.). "Sequence A002161". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
↑ "The Square Root of Pi". digitaleditions.sheridan.com. Retrieved 2023-08-20.
↑ "sqrt( $π$ ) - Wolfram|Alpha". www.wolframalpha.com. Retrieved 2023-08-20.
↑ Cherry, G. W. (1985-09-01). "Integration in Finite Terms with Special Functions: the Error Function". Journal of Symbolic Computation. 1 (3): 283–302. doi:10.1016/S0747-7171(85)80037-7. ISSN 0747-7171.
↑ "Twilight Script at IMSDb". imsdb.com. Retrieved 2023-08-20.
↑ Doctor Who: The Writer's Tale: The Final Chapter (PDF). 14 January 2010. Archived from the original (PDF) on 2013-05-08. Retrieved 2023-08-20. Unknown parameter |url-status= ignored (help) Search this book on
↑ the square root of pi, 2022-05-01, retrieved 2023-08-20

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[1] "index of important irrational constants". planetmath.org. Retrieved 2023-08-21.

[:0-2] 2.0 ^2.1 "Γ(Divide[1,2]) - Wolfram|Alpha". www.wolframalpha.com. Retrieved 2023-08-20.

[OEIS_A002161-3] 3.0 ^3.1 ^3.2 ^3.3 Sloane, N. J. A. (ed.). "Sequence A002161". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

[4] "The Square Root of Pi". digitaleditions.sheridan.com. Retrieved 2023-08-20.

[5] "sqrt( $π$ ) - Wolfram|Alpha". www.wolframalpha.com. Retrieved 2023-08-20.

[6] Cherry, G. W. (1985-09-01). "Integration in Finite Terms with Special Functions: the Error Function". Journal of Symbolic Computation. 1 (3): 283–302. doi:10.1016/S0747-7171(85)80037-7. ISSN 0747-7171.

[7] "Twilight Script at IMSDb". imsdb.com. Retrieved 2023-08-20.

[8] Doctor Who: The Writer's Tale: The Final Chapter (PDF). 14 January 2010. Archived from the original (PDF) on 2013-05-08. Retrieved 2023-08-20. Unknown parameter |url-status= ignored (help) Search this book on

[9] the square root of pi, 2022-05-01, retrieved 2023-08-20

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Square root of pi

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