July 21, 2012 | 21

Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter

I hope you’re ready for your big Pi Approximation Day party tomorrow. You might have observed Pi Day on March 14. It gets its name from 3.14, the first three digits of the ratio of a circle’s circumference to its diameter. Always on the lookout for excuses to eat pie, some geeky math types also celebrate the number on July 22. The fraction ^{22}⁄_{7} has a value of 3.142857, so it has the same first three digits as pi.

Both 3.14 and ^{22}⁄_{7} are approximations of pi, so the two days deserve the same title. In fact, ^{22}⁄_{7} is closer to pi than 3.14 is. So if you’re an aspiring pedant, you can choose to celebrate July 22 as Pi Day and March 14 as Not Quite as Close to Pi Day. (Either way, you’ll enjoy more pie.) But what does it mean to be an approximation of pi—and why does it matter?

Pi is irrational. That is, the decimal expansion never ends and never repeats, so any number of decimal places we write out is an approximation. (Of course, we can write the number exactly using just one symbol: π.)

Each decimal digit we know makes any computation involving pi more precise. But how many of them do we actually need for sufficient accuracy? Of course it depends on the application. When we round pi to the integer 3, we are about 4.51 percent off from the correct value. If we use it to estimate the circumference of an object with a diameter of 100 feet, we will be off by a little over 14 feet.* When we add the tenths place, and use the approximation 3.1 for pi, our error is only about 1.3 percent. The approximation 3.14 is about ½ percent off from the true value, and the fairly well known 3.14159 is within 0.000084 percent.

If you were building a fence around a giant circular swimming pool with a radius of 100 meters and used that approximation to estimate the amount of fencing you would need, you would be half a millimeter short. Half a millimeter is tiny compared with the total fence length, 628.3185 meters. Being within half a millimeter is surely sufficient, and the tools you are using to make the fence probably introduce more uncertainty into your structure than your approximation of pi.

What about something with higher precision standards over much larger distances? I asked a NASA scientist how many digits of pi the agency uses for its calculations. Susan Gomez, manager of the International Space Station Guidance Navigation and Control (GNC) subsystem for NASA, said that calculations involving pi use 15 digits for GNC code and 16 for the Space Integrated Global Positioning System/Inertial Navigation System (SIGI). SIGI is the program that controls and stabilizes spacecraft during missions.

Pi appears most often in formulas involving circles or periodic motion, and it infiltrates some fundamental physical constants. These constants appear all over physics: masses of elementary particles, the number of molecules in a volume of a gas, the forces holding matter together, and so on. (Pi itself is not considered a fundamental physical constant.) The fine-structure constant, or “coupling constant,” which measures the strength of the electromagnetic force that governs how electrons and muons interact with photons, involves pi, and the permeability of free space, which describes how a magnetic field forms in a vacuum, is 4π×10^{-7}. It is important to know highly accurate values of the fundamental constants to make good predictions of phenomena involving physics, and the experimental determination of the constants can even help improve our understanding of the physical laws that govern the universe.

Believe it or not, there is a committee that makes recommendations about the values of these fundamental constants. The Committee on Data for Science and Technology, or CODATA, an interdisciplinary group from the International Council for Science, periodically publishes a set of accepted values of the fundamental physical constants. The most recent version, 2010CODATA, was published in June 2011.

Peter Mohr, a physicist who works for the Fundamental Constants Data Center at the National Institute for Standards and Technology, which is involved in calculating and disseminating the accepted CODATA values, says that the institute uses 32 significant digits of pi in their computations. (For programming geeks, this is called “quadruple precision.”)

So NASA scientists keep the space station operational with only 15 or 16 significant digits of pi, and the fundamental constants of the universe only require 32. Yet in 2006 Akira Haraguchi of Japan recited 100,000 digits of pi from memory in 16 ½ hours, stopping for five minutes every hour to replenish his strength with onigiri rice balls. And the world record for number of digits of pi computed is 10 trillion, at least as of October 2011. Pi computation can be used to test computer precision, but I think this is a symptom of pi-mania rather than a legitimate need for pi. Other numbers could be used just as meaningfully, but we choose to use pi.

It seems that we know, and strive to discover, many, many more digits of pi than we need for any practical application on Earth, or even in the part of space we can hope to get to right now. I guess the endlessness of the decimal representation just fascinates people. Haraguchi, the pi reciter, told The Japan Times that his memorization of pi is part of his quest for eternal truth. For some, it is probably a challenge: How far can I go? We want to push our limits, and memorizing pages of numbers seems pointless until we give it the halo of pi.

Coming back down to Earth, Pi Day and Pi Approximation Day are great reasons to have some dessert and contemplate the most transfixing transcendental. Perhaps on July 22, we should make some approximate pie: crisps, cobblers, and Brown Bettys for all!

**Correction (7/23/12): This sentence was edited after posting to correct an error.*

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My favorite approximation of Pi is 355/113, which works out to 3.141592920353982…

Actual Pi is 3.14159265…, so 355/113 is accurate to about 0.0000086%, which is an order of magnitude more accurate than 3.14159.

If you can remember 113355, and then rearrange the numbers, you’ve got it.

Link to thisWhat and informative and well written article!

just a note…irrational numbers don’t necessarily equate with infinite decimal points. Our current level of understanding indicates they are infinite but that is just ‘educated guestimate’..we can’t experiment with infinite decimal points.

Link to thisIt’s not an “educated gusesstimate” that irrational numbers have an infinite number of random digits. If the digits end at some point, you can just take represent the number as some large integer divided by a power of ten, but that is a rational number. And it’s not as if people have gone and looked at so many digits and said that, because there are so many, they must be infinite. Logical methods can be used to prove they are infinite in number by simply proving the number not rational. By their nature, rational numbers have infinitelly repeating decimal portions, so anything which is not rational must have infinite digits but not repeating. Much of mathematics may start out observational, but, in the ideal, it becomes all a logical system, applying rules that hold even out at infinity to establish the absolute nature of an entity. In that way, you can “expeiment with infinite points” in mathematics. Since, for example, an infinite random, non-repeating, sequence represents an irrational number, multiplying it by any integer will never produce a repeating sequence, all the way out to infinity, since that would make the number rational.

Link to thisSheldon, is that you?

Link to thisIs 32 digits of accuracy for pi enough? Why? If the universe is really quantized then doesn’t that mean it is not continuous and at some precision of the universal constants, say 32, that is all that is necessary, and any more digits are noise and just an artifact?

Link to thisTo julianpenrod and geojellyroll -

another way of looking at it is that pi and other irrational numbers can be expressed as an infinite series, with the terms getting smaller-and-smaller but never zero.

Musn’t this produce a number infinitely long? I express this tentatively, because mathematical logic is rigorous (see julianpenrod), and I’m no master of it.

Link to thisBy the way, isn’t quadruple precision 32 bit, rather than 32 (decimal) digits.

Link to thisFrom a failed geek.

63 decimal places of pi suffice to calculate the circumference of the observable universe to an accuracy of one planck length. More is a waste of computing cycles.

Link to thisThus why past a certain decimals point it is about mathematics and not necessarily a reflection of reality…irrational numbers do not equate with infininte decimal points. Pi is about measuring is real phenomena based don matter and energy.

Link to thisQuadruple-precision has 128 bits, which ends up being around 32 decimal places. Single-precision has 32 bits. It’s a bit confusing because the number 32 shows up in both.

Link to thisTheBiboSez, maybe we should celebrate at 3:55 on April 13 (April 12 in leap years) and 1:13 on December 21 (December 20 in leap years). Those are the 113th and 355th days of the year.

Link to thisWhy all the hoopla about pi? How about e, or the square root of 2, or the square root of -1?

Link to thisExcellent article! However I believe you made a mistake in the following statement – “When we round pi to the integer 3, we are about 4.51 percent off from the correct value. If we use it to estimate the circumference of an object with a diameter of 100 feet, we will be off by 4 ½ feet.”

Using integer 3 to approximate Pi to calculate the circumference of a 100′ diameter circle will result in an error of 14.16 feet. Your mistake was in applying the % difference of 4.51% to 100 feet instead of (Pi x 100).

Link to thisIN PARAGRAPH #7 ABOVE IN THIS HIGHLY LEARNED AND SOPHISTIKATED (SIC) MAGAZINE— THE PLURAL OF FORMULA IS SHOWN AS “FORMULAS”. EVERY PERSON OF A RUDIMENTARY KNOWLEDGE OF THE ENGLISH LANGUAGE SHOULD KNOWS THAT THAT IS INCORRECT —IT REALLY IS “FORMULAE”

Link to thisGood eye, jleeong. Thanks.

Link to thisWhat a fun sort of number for mathematics, philosophical metaphysics, and fanciers of language. Pi is an irrational number, infinitely continuing in a non-repetitive sequence that we must approximate, yet it is a DEFINITE number (the ratio of a circles circumference to its diameter).

Link to thisOH! So THAT’S why most pies are circles!

Right?

Link to thisPi is a line plotted to overlap itself infinitely. How is this irrational?

Link to thisPIE ARE SQUARE

Link to thisInteresting article on approximating Pi with fractions:

Link to thishttp://blog.wolfram.com/2011/06/30/all-rational-approximations-of-pi-are-useless/

The cube root of 31 is a easily remembered approximation for Pi accurate to 67ppm. Not as accurate as 355/113 but more easily remembered if you have a memory like mine!

Link to this