Draft:Square root of pi

• Comment: We need reliable sources with in depth coverage. Stuartyeates (talk) 20:42, 21 January 2024 (UTC)

• Comment: Don't get me wrong, I'd love to approve the square root of pi as an article, but currently the sourcing is just not there. Half of the references exist to support the fact that "pi was discussed in pop culture X", and with a mathematical constant of this scale and magnitude, focusing on the technical aspects of this number within mathematics would be preferential.
  This article was deleted 18 years ago during an AfD; I think there's been plenty of time since then to give this another go in article-space, given that since then Pi has become a featured article and is the 3rd most viewed number article on Wikipedia only behind 0 and 1, receiving more pageviews than every other integer.
  Because of this, as well as Pi's Featured Article status, I don't think merging with Pi is the right call, as it would be overriding the precedent and existing content there. If there's any place to talk about the square root of pi, I feel it would be on its own standalone article.
  Still though, all things being equal, the sources that are provided seem to be routine coverage from math databases, and aren't currently enough to demonstrate the notability of this topic. Utopes _{(talk / cont)} 08:44, 9 September 2023 (UTC)

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Square root of pi

The function e^-2^x with the square root of pi being the area under the curve.
Rationality	Irrational
Representations
Decimal	1.7724538509055160272
Algebraic form	${\sqrt {\pi }}$

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 ${\textstyle {\sqrt {\pi }}}$, Γ(1/2) or sqrt(π). It is not the root of a non-zero polynomial and therefore is a transcendental number.

Approximations

• The Decimal approximation of ${\textstyle {\sqrt {\pi }}}$ to 65 decimal places is: 1.7724538509055160272981674833411451827975494561223871282138077898...
• 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, ${\textstyle {\sqrt {\pi }}}$ 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, ...]

Expressions

• Expressed as the area below the function ${\textstyle y=e^{-x^{2}}}$ as the integral:

$\int \limits _{-\infty }^{\infty }e^{-x^{2}}dx=\surd \pi$.

• Expressed in series as:

Right-angled triangle with sides of π/2 and a hypotenuse of ${\textstyle {\sqrt {\pi }}}$.

Multivariate normal distribution

$2\surd {\sum _{k=0}^{\infty }(-1)^{k}/1+2k}=\surd \pi$

• Can be expressed as:

$e^{-\log G(1/2)+\log G(3/2)}=\surd \pi$

• Geometrically expressed as the hypotenuse of a triangle with a base and height of ${\textstyle \pi /2}$.

Applications

• ${\textstyle {\sqrt {\pi }}}$ is the total area under the Gaussian integral, also known as the Euler-Poisson integral, ${\textstyle y=e^{-x^{2}}}$. 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)
• ${\textstyle {\sqrt {\pi }}}$ 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. This can be used to find relationships between data points that follow a joint normal distribution, particularly in statistics and computing.
• ${\textstyle {\sqrt {\pi }}}$ is the diameter of a sphere whose surface area is $\pi ^{2}$, or the radius of a circle with are of ${\textstyle \pi ^{2}}$.

Popular Culture

Reference in popular media

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

BELLA: So how bout 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.

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

RICK: Morty, t-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!

• The square root of pi, a song by Splooge God, Released in 2022.

See also

• Square root of 2
• Square root of 3
• Square root of 5
• List of numbers

References

Category:Transcendental numbers Category:Mathematical constants Category:Irrational numbers