All Rational Approximations of Pi Are Useless

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

by ColinWright • 14 years ago

185 110 14 years ago

blog.wolfram.com

IsTom• 14 years ago

I think the reason for this is Thue–Siegel–Roth theorem that says there are only finitely many "good" rational approximations of pi (or any other irrational number).

cperciva• 14 years ago

The Thue–Siegel–Roth theorem deals specifically with algebraic numbers, of which pi is not one.

afrozenator• 14 years ago

Upto 10 digits, pi is sqrt(sqrt(2143 / 22)) .. and that is 9 characters as per his definition or 8 if you allow ()^(1/4) as an elementary operation.

3.141592652 vs pi ~= 3.141592653

protomyth2• 14 years ago

He likely bets the person who was taught 22/7 as an approximation will not be able to do the sqrt of a sqrt of a fraction in their head or on paper. If they have a calculator they might as well use the dedicated pi key.

finnw2• 14 years ago

3.1415927 is really just a shorthand for 31415927/10000000, so shouldn't it count as 17 characters rather than 9?

guard-of-terra2• 14 years ago

No it isn't. It isn't "a shorthand". It is a notation. Of those two notations, no one is "a shorthand" of another.

beefman2• 14 years ago

I think the point is that McLoone is using two different notations and measuring an artifact of this. There's nothing special about base-10 denominators.

nandemo• 14 years ago

Still, it is a rational approximation of pi.

mey• 14 years ago

Ok, I'm confused, what exactly is the point here? If you don't care, 3.14 is sufficient, if you do care then you use π, and if you really care you use τ[1].

[1](http://tauday.com/tau-manifesto)

(And if you really really really care, you use a different font then the one HN defaults too.)

protomyth2• 14 years ago

The point is that it is easier to remember X digits of PI than a fraction, because the fraction will have more digits in total than X.

tedunangst2• 14 years ago

The point is that you should just memorize whatever precision decimal you need. There's no shortcut like 22/7 that will magically give you more accuracy and be easy to memorize.

tailrecursion6• 14 years ago

(pi - 22/7) < (pi - 3.14). 22/7 is closer to pi than 3.14.

jj00nathan• 14 years ago

Did you read the article? He explains very precisely what he means and what his conclusion is. "Rational" in the context of the title means "one integer divided by another to which the first is relatively prime", and the word "useless" indicates that there is no gain in accuracy/complexity in comparison to the decimal notation for a rational number (where you express a rational number as an integer divided by an implicit power of ten).

raverbashing• 14 years ago

Agreed

22/7 is the worse. Too much trouble for too little benefit

If you need the value of pi to do a hand calculation, 3.14 is more than enough

And if you need to "produce" pi just remember pi/4 = 1 - 1/3 + 1/5 - 1/7... (there are formulas that are better, sure, but less memorizable)

psykotic• 14 years ago

>And if you need to "produce" pi just remember pi/4 = 1 - 1/3 + 1/5 - 1/7...

This converges too slowly to be practically useful.

If you group the consecutive terms in pairs you get that the nth pair sums to 1/(2n + 1) - 1/(2n + 3) = (2n+3 - (2n+1))/(2n+1)(2n+3) = 2/(2n+1)(2n+3) = Theta(1/n^2). Thus it has the same asymptotic growth order as sum 1/n^2. That has has monotone terms and so it's easy to estimate how many terms we need to get k correct fractional digits by solving 10^(-k) = 1/n^2, giving n = 10^(k/2).

This is off by a big constant factor but it gives you the right idea that you need an exponential number of terms relative to the desired number of significant digits. From Wikipedia: "After 500,000 terms, it produces only five correct decimal digits of pi."

So, this series for pi has only theoretical relevance.

dllthomas3• 14 years ago

22/7 is way better than 3.14 for finding the area of a circle with radius sqrt(7).

tzs3• 14 years ago

I think blog.wolfram.com is probably the best company blog I've seen, from a marketing perspective. A large number of the entries are basically of the form "Here is an interesting problem, and here's how I solved it with Wolfram products". They generally let the problem have the spotlight rather than focus on the Wolfram products, so it doesn't feel like you are getting pitched.

Here's a related problem, but for e: using each digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 at most once each, and only the operators +, -, x, /, and ^ (exponentiation), and parenthesis for grouping, how close can you get to e? Digits may not be concatenated--for instance, you cannot get a 23 by simply placing the 2 next to the 3.

My best, after about 15 minutes of fiddling, were:

   (3x(4x7+1))/2^5 = 2.71875 ≈ e + 0.000468172

and

    2x(9x6-1)/(3x(8+5)) = 2.717948... ≈ e - 0.000333111

but it turns out you can do FAR better.

coliveira• 14 years ago

This has been Wolfram's strategy since the beginning. They have their own journal, books, user groups, all showing how to solve problems with mathematica.

Jabbles2• 14 years ago

((8^(5/4))^(((2/(0-(7+1)))^9)^(3-6))) = 2.718281848685499 ~ e +/- 2.0226453845140213e-08

That was fun. Although I'm having trouble confirming the answer... - Also I'm meant to be packing.

http://play.golang.org/p/G_Y5SblSuv for some brute force eval :)

Edit: This could be a nice demonstration for a genetic algorithm: A clearly defined fitness function yet an unknown (unknowable?) goal, and a distinct representation of the genes. I'm not sure how hereditary the fitness is though.

Further edit: Just 4 digits: (6-(8^(4/7))) 2.718658575969448 0.00037674751040306376 - I hope this is a) correct, b) interesting to someone else.

tzs3• 14 years ago

You beat me by about 5 orders of magnitude, but it is possible to do better, by another 200000 orders of magnitude or so! :-)

Best I've seen is given in this discussion: http://www.reddit.com/r/math/comments/zakqh/using_the_number...

Dn_Ab3• 14 years ago

I have one of those, a symbolic regressor (not genetic) lying around in a broken state. The last working version gave:

  short: 2981^ (1/8)
  less short: (2574 + 4903 ^ (1/4))/950

I used: http://apod.nasa.gov/htmltest/gifcity/e.1mil

Perhaps I am misunderstanding your output but it is only correct up on till: 2.7182818

reeses3• 14 years ago

I can do it with only 9.

anthonyb2• 14 years ago

FlipH is not a legal operator.

ps2000• 14 years ago

Yeah agree, they know how to market their stuff. (Maybe Alpha is a marketing instrument after all? ;))

However I think for this problem, and many others, mastering a general purpose programming language is much more efficient. After all you can pipeline symbolic expressions to Mathematica - or Maple >:) - and you have the best of both worlds. Or you just use Ruby or another highly expressive language - the syntax of simple symbolic expressions is basically the same.

Too• 14 years ago

What if you are on embedded hardware and/or can't use floats for some reason?

VLM• 14 years ago

Use integers, calculate in pennies. So Pi=314 if your internal math is in hundredths or "pennies". Or if you use millimeters instead of meters, pi=3141 What we did 30 years ago on desktops is what embedded hardware still does today... The endless wheel of IT eternally rotates the same concepts back to the top, if you wait long enough.

Another classic is the old hard science error analysis. Lets say you're squaring the radius and multiplying by pi, turns out you need to measure the radius much more accurately than you measure pi, so pi=4 might not be the limiting factor if R is a 8 bit A/D converter and you're not taking full advantage of the entire 8 bit range (so its really a 4 bit a/d or whatever)

Another is systemic effects. Some weird hydraulic PLC thing I was messing with probably 20 years ago basically needed the ratios of areas of circles, and it turns out that any approximation of pi divided by itself always equals 1. The puzzler is for diagnostic purposes they used pi=4 so the numbers kinda made sense in the debugger before the ratio was calculated. I must have thought about that 4 for an hour trying to reverse engineer what they were trying to do before I realized that "4" was their pi approximation and it didn't matter anyway.

tailrecursion6• 14 years ago

If we're on embedded hardware, or can't use floats for some reason, we use 22/7. It's called fixed point arithmetic. Most CPUs have an integer instruction that multiplies into a double-precision integer result, and an instruction that takes a double-precision integer dividend.

So to scale by pi, we multiply by 355, then divide by 113.

a1k0n• 14 years ago

Is that any better than just multiplying by pi * 2^n and then shifting right by n?

cs702• 14 years ago

Very neat: naive attempts to memorize Pi with rational-number shortcuts (i.e., fractions with integers in the nominator and denominator, such as 22/7 and 355/113) seem pointless, because getting more decimal digits of Pi right requires that one memorize a correspondingly larger number of digits in the numerator and/or denominator, defeating the purpose of these native attempts.

PS. The author is offering a prize to the reader who finds the rational number which gets the most decimals of Pi right for every digit of such rational number that has to be memorized. (Note that only rational numbers are allowed -- that is, fractions with integers in the numerator and denominator. Using formulas or numbers that are not rational is not allowed in the competition.)

kalid• 14 years ago

I might be different, but 355/113 is very easy to remember and is not 7 unique segments of information. I just think "double the odds"

113355

We know we want a fraction, not a single number, so split down the middle:

113/355

And we know pi won't be less than 1, so flip it: 355/113. Knowing that the digits sequences are doubled lets you do some cheap, mental run length encoding. In a case where you need a hand calculation, spending 3 seconds to re-derive the sequence seems tolerable.

3rd3• 14 years ago

Unwillingly, I just memorized a Pi approximation.

Evbn4• 14 years ago

Required information for your approach:

Repeat the digits

Use odd digits

Use increasing sequence

Use three digits

Split the result in half and divide

Pi is greater than 1

But since humans have associative memory, these are easier to learn than a seemingly arbitrary digits sequence.

jfoutz2• 14 years ago

Rational approximations are good for finding reasonable answers without a calculator. My contrived example is the area of a circle, r=7. i can do 22 * 7 in my head and get 154, which is pretty darn close to 153.86. 7 is awkward (for me anyway) but it does offer opportunities to cancel.

Jach2• 14 years ago

How fast can you do 22 * 7 in your head? I get a lot of mileage out of simply approximating pi=3, and optionally adding in a factor of 10% later. For your example, I know 7^2=49 immediately, which I round to 50, 50 * 3=150 which is fairly close to 153.938. (Use more digits of pi!) I can quickly improve my estimate by adding 4.9. (Edit: It's also fairly trivial to get even better estimates quickly from here, but at this point it's probably faster to grab a cell phone / [favorite language] repl. From 154.9, subtract the additional 3 gained from using 50 instead of 49, now 151.9, add 4.9/2~=2.4, now 154.3, subtract 4.9 * 10% ~= .5, 153.8.)

kdavis6• 14 years ago

This seems slightly silly if one is allowed to use, as Ed Pegg suggested, log and square root.

If log and square root are allowed, then the obvious solution is log(-1)/(sqrt(-1)*log(e)) which is accurate to an infinite number of digits.

scott_s6• 14 years ago

e is not rational.

kdavis6• 14 years ago

ln(-1)/sqrt(-1)

ColinWright3• 14 years ago

I'm confused - he's addressing the point made in the article by Ed Pegg that things are interseting if you allow log and sqrt. Surely that means he's allowed to use log and sqrt, and in particular, to use them in rational expressions.

So I don't really understand what your point is.

dllthomas3• 14 years ago

I don't like how he's measuring accuracy, here. Something that produces (3.149) is treated as closer than (3.139).

22/7 looks marginally better, if we compare actual error. It's the same number of characters as 3.14, but about 20% less error. 355/113, meanwhile, is not only better than 3.14159 (same number of characters) but actually even better than 3.141592 (about 60% less error).

It's also easier to remember, due to the repeated digits.

tailrecursion6• 14 years ago

Yes, Jon's metric for measuring the value of an approximation is wrong and so is his conclusion.

monochromatic• 14 years ago

Why is the conclusion wrong? It still seems like the trend should be the same.

frobozz• 14 years ago

It is easier to mentally calculate the approximate circumference of a circle whose radius is a multiple of 7 using 22/7, than it is using 3.14(...)

Admittedly, that's a pretty niche use, but a use, nonetheless.

JumpCrisscross• 14 years ago

3 X 7 = 21 and 7 X .14 is 7 X 10 X 1.5 - .07 with the appropriate decimal shift. Now I know it's something approaching 22, a detail 22/7 denies.

frozenport• 14 years ago

Humbug! In my day the TI DSPs used a counting system that was based on PI. So 1.0 was PI, 0.5 was PI/2.

mistercow• 14 years ago

There are other reasons, though, that a rational approximation can be useful. If you're doing mental arithmetic, for example, multiplying by a fraction can be easier than multiplying by a decimal, but this depends on the specific numerator and denominator. In this way, 22/7 fails horribly, because multiplying by 22 and dividing by 7 are not particularly easy operations.

For example, 100/32 is a less accurate representation than 22/7, but it is far easier to multiply, since you can do so with only two mental registers. In fact it also takes fewer registers to multiply than 3.1 (which is also less accurate than 100/32). However, it does require more mental operations than 3.1; 100/32 takes five halvings and a decimal shift, while 3.1 requires multiplication-by-3, a decimal shift, and an addition.

What would be really cool would be an analysis similar to the one in the article that used a model of mental computation to find the best tradeoff for working numbers in your head.

That said, for mental calculations you're probably better off just pretending that pi = 3, unless you're just trying to impress someone.

asmithmd1• 14 years ago

using a rational version is also easier when using a slide rule

evincarofautumn• 14 years ago

Slide rules usually have markings for common constants. And you can always add your own, if you’re careful enough.

jscipione• 14 years ago

The reason that 22/7 is used to approximate pi is because it is convenient to calculate using a slide rule.

Someone2• 14 years ago

I would think most slide rules have a marker for pi that make this even easier.

lutusp5• 14 years ago

Given how slide rules work, there would have to be a great number of marks, for the same reason that there cannot be only one unique ratio of integers that approximates Pi.

dollarpizza• 14 years ago

Well the trick is not to "memorize" the rational approximation, but to derive it (or rather, the CF approximation to it) directly from first principles, i.e. using the analytic properties of pi itself.

btilly• 14 years ago

This is expected behavior. The best fractions are tied to the http://en.wikipedia.org/wiki/Continued_fraction representation of pi. The ones which have a chance of giving you several "free" digits are going to be tied to large terms in the representation. Glancing at http://oeis.org/A001203/b001203.txt gives you a sense that large terms are kind of rare. http://mathworld.wolfram.com/Gauss-KuzminDistribution.html quantifies how rare they are.

Incidentally if you're looking for a good fractional approximation to pi and e you'll have a lot of work, but for sqrt(2) it is easy because the continued fraction representation is 1 followed by 2, 2, 2, .... Thus sqrt(2) is 1 + 1/(2 + 1/(2 + 1/(2 + ...))). Very easy pattern to remember, and can give a rational approximation as precise as you possibly want.

derekp7• 14 years ago

This also makes sense from data compression theory. If the digits of an irrational / transcendental number share some of the properties of a random string, then you shouldn't be able to compress it. And finding a fractional representation with fewer total digits is a form of data compression.

JackC• 14 years ago

Nitpick: data compression theory says you can't have a general-purpose algorithm that on average compresses random strings. The best any algorithm can do is make some strings shorter and some strings longer, which is why compression is only useful on strings with known properties. But given a particular random finite string (such as N digits of pi) you can very likely (certainly?) find an algorithm that compresses it -- which is why people are able to offer a number of compression algorithms for approximations of pi in the post.

politician• 14 years ago

Is the symbol for pi considered "compression"? Why or why not?

bo1024• 14 years ago

I dunno, I think sum_{i=0}^{\infty} (-1)^i 4/(2i+1) is a pretty good compression for an infinite string.

nealabq• 14 years ago

π=◯/―

tailrecursion6• 14 years ago

22/7 is a better approximation of pi than 3.14 because 22/7 is closer to pi than 3.14. The hypothesis of the article is wrong.

I believe Colin [EDIT: I meant Jon McLoone, the author of the article. Sorry Colin!] is counting only full digit matches. It is this metric that is useless.

ColinWright3• 14 years ago

Are you confusing the person who actually wrote the article with me, who simply submitted it here?

If so, perhaps you can explain why people do that so often. I see it quite frequently, and am baffled by it.

tailrecursion6• 14 years ago

Yes that was my mistake. Sorry about that. It must happen because of the byline above the box where one types in the top-level comment. Thanks for mentioning that it's a frequent error, that makes me feel better.

Xcelerate• 14 years ago

I stumbled across this interesting paper about the subject at some point: http://cogprints.org/3667/1/APRI-PH-2004-12b.pdf I know a lot of you will gloss over this, but I think you will be surprised at how interesting it is if you read it.

Basically, it tries to see if mathematical equations have meaning by determining how well they "compress" the results. For instance, the author says the equation eπ−π=19.9990999... is compressible (the equation generates more bits of π than it takes up itself), and thus it is likely that there is some mathematical reason for this -- it's not just a coincident. On the other hand, 314100 gives an approximation to π but does not compress its representation, so there is nothing intriguing about this formula.

Evbn4• 14 years ago

How do you define e succinctly enough to make that representation efficient?

lutusp5• 14 years ago

My favorite remark from the article: "Anything that I judge to be outside of the spirit of the competition will be disqualified—that includes the use of programs, integrals, sums, inverse trig functions, π-related values of special functions, or π-related constants (such as π).)"

Can't be too careful -- some breathtakingly literal-minded soul might submit π as a candidate for π.

insaneirish• 14 years ago

As someone who has memorized pi to 74 decimal places, I concur.

seles• 14 years ago

Setting MaxExtraPrecision to "to use as much automation as it needs to resolve numerical values" looks very interesting. It seems like it is being lazy, because the amount of precision needed isn't known until after the result is used (in this case the function that counts number of correct digits).

Any idea how this is implemented in mathematica?

lmm• 14 years ago

I don't know about mathematica specificly, but the standard way is lazily, as you say. You pass around a lazy list of digits (i.e. a partial list and a function that knows how to compute the next digit and the next function); it's trivial to add or subtract two such things, multiplication and division are harder but not too bad, and once you've got the idea square root and other more complex operations are pretty straightforward.

rsaarelm• 14 years ago

Some people really like their 22/7: http://www.psychologytoday.com/blog/freedom-learn/201003/whe...

Though come think of it, is this a thing particular to US education? I don't remember ever encountering this in elementary school. I think the early math lessons were just set up never to require irrational numbers, and from 7th grade on (this was somewhere around 1993) we used calculators which gave us PI and trigonometric functions.

pfortuny2• 14 years ago

Sorry to nag but the 'goodness' of 22/7 is due to the size of the denominator, not to the number of digits you have to memorize (actually 2, not 3, but anyway). You get less than a .001 relative error with a denominator less than 10, that is why it is a good approximation: nothing to do with memory.

The thing is: as pi is transcendental, there are very very good rational approximations in that sense (this is an old theorem due to Liouville): http://mathworld.wolfram.com/LiouvillesApproximationTheorem...., there is no 'memorizing' going on there.

lutusp5• 14 years ago

> The thing is: as pi is transcendental, there are very very good rational approximations in that sense (this is an old theorem due to Liouville) ...

Expressed another way, for every estimate of Pi's value, however large, there are two integers that, expressed as a ratio, will produce the same result.

pfortuny2• 14 years ago

No, no, not at all: the meaning of the theorem is that there are amazingly accurate approximations for small sized denominators, that is.

All Rational Approximations of Pi Are Useless

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