What is Pi: Definition and 513 Discussions

The number π () is a mathematical constant. It is defined as the ratio of a circle's circumference to its diameter, and it also has various equivalent definitions. It appears in many formulas in all areas of mathematics and physics. The earliest known use of the Greek letter π to represent the ratio of a circle's circumference to its diameter was by Welsh mathematician William Jones in 1706. It is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century, and is spelled out as "pi". It is also referred to as Archimedes' constant.Being an irrational number, π cannot be expressed as a common fraction, although fractions such as 22/7 are commonly used to approximate it. Equivalently, its decimal representation never ends and never settles into a permanently repeating pattern. Its decimal (or other base) digits appear to be randomly distributed, and are conjectured to satisfy a specific kind of statistical randomness.
It is known that π is a transcendental number: it is not the root of any polynomial with rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge.
Ancient civilizations, including the Egyptians and Babylonians, required fairly accurate approximations of π for practical computations. Around 250 BC, the Greek mathematician Archimedes created an algorithm to approximate π with arbitrary accuracy. In the 5th century AD, Chinese mathematics approximated π to seven digits, while Indian mathematics made a five-digit approximation, both using geometrical techniques. The first exact formula for π, based on infinite series, was discovered a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics.The invention of calculus soon led to the calculation of hundreds of digits of π, enough for all practical scientific computations. Nevertheless, in the 20th and 21st centuries, mathematicians and computer scientists have pursued new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits. The primary motivation for these computations is as a test case to develop efficient algorithms to calculate numeric series, as well as the quest to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and sciences having little to do with geometry of circles, such as number theory and statistics, as well as in almost all areas of physics. The ubiquity of π makes it one of the most widely known mathematical constants—both inside and outside the scientific community. Several books devoted to π have been published, and record-setting calculations of the digits of π often result in news headlines.

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  1. S

    B Sequences of digits in Pi

    As the value of Pi is taken to more decimal places, does any sequence of digits become equally likely? I'm thinking of sequences like ...123456789.... ...333333... and so on. best regards ... Stef
  2. diogenesNY

    Why Do We Celebrate Pi Day on March 14?

    Happy Pi Day everyone :) https://en.wikipedia.org/wiki/Pi_Day https://www.dogpile.com/serp?q=pi+day&sc=ZBfGciaK2GX310
  3. Delta2

    I The infinite sequence of the digits of pi

    Is there a way to prove if the 0s (or 1s 2s or 3s or ... or 9s) that are present in the infinite sequence of digits of pi in the decimal system are finite or infinite? If they are infinite they are countable infinite or uncountable infinite? My intuition tells me that they are countable infinite...
  4. P

    Admissions Who should I pick for my third letter of recommendation?

    I have two PIs who will write the first and second letter. And I'm torn between asking my direct supervisor from one of those labs (who was a grad student when I was working there and we became quite close) and a professor that I had for an upper div class I took during my last year. My...
  5. Euge

    POTW Does the Taylor series for arctan converge at x = 1?

    Show that $$\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$$
  6. S

    [ Raspberry Pi or Android ] --> WiFi Router --> USB printer

    A couple of years ago, I connected a USB printer to the USB socket on my broadband router. After a lot of trial and error (of which I remember no details) I was able to print from my windows laptop. Here are the port settings that currently work from the laptop: 1) Based on the above...
  7. V

    Cannot understand what ##(mod ~ \pi)## means in the given formula

    My understanding is that it probably means either ##+ \pi## or ## - \pi##, so that ##\pi## is either added to or subtracted from the first term, just like ##|x| = \pm x##.
  8. T

    Are Pi-mesons and W-bosons Identical? Exploring Particle Decay

    They decay to identical particles but are they identical?
  9. S

    Mathematica "Evaluation > Debugger" missing (Mathematica on Raspberry Pi 4)

    I don't have the menu item Evaluation > Debugger . Could it be a limitation of the Raspberry Pi version of Mathematica? Or are there other potential reasons for it? Can anyone with a RasPi using Mathematica check if they have the debug menu?
  10. S

    Raspberry Pi 4 / Debian : Static IP on one SSID and Dynamic on another

    My Raspberry Pi 4 has a GUI widget that is supposed to let us specify WLAN settings on a per-SSID basis, but it doesn't work. (All SSIDs end up with the same setting). Some articles on the net say that one can modify the dhcpd.conf file with an entry headed by the SSID name and then put in...
  11. O

    How do PI controllers achieve steady state?

    Summary: A question about how PI controllers reach steady state Hello PhysicsForums, I need a little help understanding how a PI controller works when operating in steady state. Here is the equation in State Space form: Where In standard form, the equation is: Where The question I’m...
  12. S

    Changing mouse cursor on Raspberry Pi OS (Debian Linux)

    I used GIMP to edit copies of certain cursor files called left_ptr and hand2 from the "cursors" folders within the theme directories /usr/share/icons/[theme name]/cursor. This was for the themes "Adwaita" and "DMZ-white". Then I renamed the original cursor files and put the new edited versions...
  13. K

    How to Estimate Pi Without Trig Functions

    Have neither seen an estimation nor derivation of pi that does not use trig functions. This is problematic as trig functions require radian inputs, via the relation pi radians = 180 deg. But if looking for pi, then how to get the input for the trig functions without pi? Sure there are...
  14. Trysse

    Surface area of a sphere ##= \pi * (a^2+b^2+c^2+d^2)##

    I am not very good at proofs. The only thing I have come up with is the following regularity. However, I am not sure how this can be related to the above problem. Given a sphere ##S_a## with a center ##C## and a diameter of ##a##. I can now construct a line segment ##b## with the endpoints...
  15. A

    Hilbert Transform, Causality, PI Controller

    I was told that PI controller is a causal filter, and has frequency response represented by H(w) = Ki/(iw)+ Kp. I was also told that causal filter should satisfy this relationship H(w) = G(w) -i G_hat(w) where G_hat(w) is the Hilbert transform of G(w). Does this mean that we cannot freely...
  16. bland

    I Is Riemann's Zeta at 2 Related to Pi through Prime Numbers?

    I just saw that one of the ways of calculating Pi uses the set of prime numbers. This must sound crazy even to people who understand it, is it possible that this can be explained in terms that I, a mere mortal can understand or it is out of reach for non mathematicians?
  17. C

    Missing factor of 2 pi in dielectric loss?

    Hi. I am sure that this is something to do with it being Friday afternoon...but can anyone explain what I am missing here?
  18. fresh_42

    Explore Alan Turing's Computable Numbers & Generate Pi with Python

    I found this article about Alan Turing and his concept of Turing machines on the AMS website. Since we often get questions about countability and computability I thought it is worth sharing. https://blogs.ams.org/featurecolumn/2021/12/01/alan-turing-computable-numbers/ It also contains a Python...
  19. curious__

    Find the locus of the points arg((z+1)/(z+2)) = pi

    I tried the following proof and got -2 < x < -1 and y = 0 but my prof said that there should be something else I am missing. I have no idea what that is. Thank you.
  20. W

    A Using Statistics to Test for Normality of Pi

    Is there a " reasonable" way to test for the normality of ##\pi## , i .e., that every digit occurs with the same frequency? Someone suggested randomly sampling strings of size 20 and outputting the frequency. Then I guess we could average the frequencies among samples , use a chi-squared test...
  21. Astronuc

    B Find Optimal Int Ratios to Approximate Pi

    On July 22, V50 wrote https://www.physicsforums.com/threads/happy-pi-day.985513/post-6518554, and it got me thinking about the ratio of two integers approximating pi. I presume someone has done it, and that there is an algorithm for an optimal calculation. Then I found -...
  22. nduka-san

    Electrical Raspberry Pi Zero W receiver / transmitter

    For a project that I'm making it has a raspberry pi 0w as its cpu for running autonomously, But I want to make use the raspberry pi 0w as the receiver as well. The range needed is about 3000 feet minimum. Currently I have been looking at some rc and drone receivers that would work as well, but I...
  23. F

    I Calculating the digits of pi by colliding boxes

    I came across this and it is rather fun! Assume there is a floor and a wall. There is a 1 lkg box on the left and a box to its right as shown in the diagram. Assume there is no friction and that no energy is lost during any collision. Set the right box to 1 kg and cause it to move to the...
  24. S

    Raspberry Pi bluetooth audio problem after upgrade

    Raspberry Pi 3 with Raspberry OS... Recently did apt-get update and upgrade. It did an upgrade of the desktop after asking for permission to replace the file piwiz.desktop. It then ran a wizard to redo various options and preferences, including WiFi and password (which I had set up initially on...
  25. A

    I Base Pi Integers: Isomorphism with Rationals?

    A proper number is expressed in \pi in a similar way as a decimal integer is expressed in base 2. For example, 4375_{\pi} = 4{\pi^3}+3{\pi^2}+7{\pi}+5. The only exception I make is that the 10 digits are included when expressing a number with \pi. To clarify, the first positive such numbers are...
  26. R

    B Can physics deal with the existence of Pi?

    Hi. I'm not sure if physics/cosmology can deal with my question. I suspect not, but I'll ask it anyway. The answer could be "No" and that would be "end of". Is there any situation, where Pi = 3.142...does not exist as a fact? Thanks. Rich
  27. G

    How can pie help you remember pi?

    I always have a hard time remembering pi past 3.14 but this pie makes it easy because I see pictures in my brain, I can look see 3.14159 any time I want.
  28. murshid_islam

    I A question about a proof of the irrationality of pi

    Here is the proof I was reading: https://mathschallenge.net/full/irrationality_of_pi I have a question about this very last inequality at the end: How did they get that "less than 1" bit? .
  29. sbrothy

    Apropos Raspberry PI DIY Mass Spec using Raman Scattering

    https://hackaday.io/project/1279-ramanpi-raman-spectrometer It's a pretty nifty gadget this Raspberry thingy. A computer the size of a pack of cigs. :) And a mass spectrometer... I'm not sure what such a device costs but I'm sure it pretty frigging expensive!
  30. S

    Raspberry Pi : Security of online account logins

    Should one avoid logging into important accounts like Google / Gmail / e-commerce from a Raspberry Pi -- or is it just as safe as logging in from a regular PC? Or, is it actually safer, considering that it is Linux?
  31. F

    Admissions What should a PI look for in a physical sciences PhD candidate when hiring?

    What should a PI look for in a physical sciences PhD candidate when hiring? Among diligence, knowledge about the field and cleverness, willingness to follow their boss, complementary skillsets if they are from another discipline, what's your favorite? What else?
  32. E

    Prove the irrationality of pi by contradiction

    Normally you can do reduction formula type questions with I.B.P., but here that just results in something like $$I_n = \left[-f(x) \cos{x}\right]_0^{\pi} + \frac{q^n n}{n!} \int_0^{\pi} \cos{(x)} x^{n-1} (\pi - x)^{n-1} (\pi - 2x) dx$$I can't seem to get anywhere from here though. Substitutions...
  33. Tone L

    Running a Raspberry Pi with a Solar Panel

    The objective: Determine the daily power draw of a Raspberry Pi (Watts per Day) and determine the average daily power generation of a solar panel. Will energy generated be greater than energy drawn? Lets begin, I have a 9 Watt, 18 Volt solar panel. This solar panel is charging a sealed lead...
  34. Greg Bernhardt

    Computer Raspberry Pi 4 Website Up-Time Monitor

    For a quarantine project I bought a Raspberry Pi 4 and a breadboard accessory kit. It won't arrive for another couple weeks, but I as super excited to dive in. My first project will be to create a simple LED up time monitor for PF :) Anyone have some "must read" tutorials? I'll be using python.
  35. BillTre

    Celebrate Pi Day with Funny Pi Jokes!

    Happy Pi Day: Pi jokes! A mathematician, a physicist, and an engineer are all given identical rubber balls and told to find the volume. They are given anything they want to measure it, and have all the time they need. The mathematician pulls out a measuring tape and records the circumference...
  36. Merlin3189

    Can Pi be Calculated Through Colliding Blocks and Quantum Search?

    How Pi Connects Colliding Blocks to a Quantum Search Algorithm Just a link a friend sent me. I like the simulation videos! Whether it would help calculate pi, I doubt! Just needed a short, catchy title.
  37. Eclair_de_XII

    B Is this method of estimating Pi too similar to Archimedes' method?

    The recurrence relation was given as: ##p_k=2^{k+1}\cdot h_k## where ##h_0^2=2## ##h_{k+1}^2=(\frac{1}{2}h_k)^2+(1-\frac{1}{2}h_k \cdot \cot(2^{-k}\cdot \alpha))^2## and ##\alpha=\arctan(1)##. This is not exactly an original or noteworthy derivation, is it? I feel that it's been done...
  38. A

    B Is there a simple way to graph this transcendental expression with pi?

    Hello everyone, I recently derived an arithmetic expression in the following form: aπ² + bπ² ± cπ / dπ² = 1, Φ³ (4.23606797...) I have two question about this: i. Does this equation/form resemble anything known in physics/math? (it looks like a psuedo-quadratic to me, but unsure) ii. Is...
  39. jedishrfu

    B Grant Sanderson on PI, Galperin's Puzzle and QC Grover's Search

    https://www.quantamagazine.org/how-pi-connects-colliding-blocks-to-a-quantum-search-algorithm-20200121/ and related papers: Play Pool with PI (2003 paper by Galperin) https://www.maths.tcd.ie/~lebed/Galperin. Playing pool with pi.pdf and Playing Pool with |ψ⟩: from Bouncing Billiards to...
  40. r12214001

    Bond energy of a Pi bond and a sigma bond

    It is well known that sigma bond is stronger than Pi bond. However in bond energy table that C=O is 745 KJ and C-O is 358 KJ. we can calculate the Pi bond energy is 745-358=387! The Pi bond(387KJ) is stronger than sigma bond(358KJ) ? How could this happen?
  41. E

    Mathematician Daniel Tammet counts pi to 22,500 digits

    Summary: wow! Mathematician and autism-spectrum savant Daniel Tammet counts pi to 22,500 digits.
  42. th078

    Radio frequency shielding for Raspberry Pi Zero

    So I'm trying to put together a Pi Zero mini computer to use for security-oriented applications, for example encrypting/decrypting messages, working with encryption keys, etc. It's cheap, very small and thus easy to carry around. However, I'm not entirely comfortable with the presence of the...
  43. K

    My PI questions - I hope they are valid?

    1) If Pi has my birthday digit sequence starting at digit at ~ 200k position (I did check, strange experience), can It also contain digital encoded (Ascii) Shakespeare's Hamlet? or Any other human creation ever done now and in the future? 2) Pi square ~ 9.86960440109 , we can calculate...
  44. stidiek

    B Exploring Velocity/Acceleration with Pi

    Imagine a compass that can move in two ways: 1- Opening it to make a Radius 2- Draw a circle. If we can imagine it going at the same velocity (or accelaration), can we create a phormula of velocity and aceleration base on Pi... and see what it implies?
  45. X

    B How can I accurately calculate pi using polygons and the concept of infinity?

    Since the ratio of perimeter and diagonal in a polygon( with a side number can be divided whole by 2) is a×sin(180° /a),and a is the side number of the polygon. And if we want the number of sides are always a double number we can say that there are 2a sides, and the equation can be...
  46. Matt Benesi

    B What are cosine and sine functions called in relation to Pi?

    1)* What are sine and cosine functions called in relation to Pi? 2) What is the exponential function called in relation to cosine and sine functions? 3) What are the other smooth, continual nested (or iterative) root functions (that are similar to sine and cosine) called in relation to...
  47. PI emerging from collisions

    PI emerging from collisions

    One of the weirdest ways to calculate pi. Personally think it is weirder than Monte Carlo.
  48. JacobPhys

    Experimental Derivations of Pi in Physics

    Greetings I'm currently designing a lab-script intended for comprehension and use at an undergraduate level. I was extremely frustrated during my undergrad to be dealing with a plethora of uninspired and dull experiments so I decided to take a slightly unconventional (or extremely conventional...