March 14, 2010

The Mystery of 355/113

Today's date is a good excuse to memorize a few more digits of 3.1415926535897932384626433832795....

And yet decimal approximations to pi are an artifact of our ten-fingered anatomy. Fractional approximations to pi are more satisfying, and they promise to teach us something more universal about pi.

We all know that 22/7 is a very good approximation to pi. But this well-known fraction is is actually 1/791 larger than a slightly less-well-known but much more mysterious rational approximation for pi:


Remembering 355/113

The fraction 355/113 is incredibly close to pi, within a third of a millionth of the exact value. This level of accuracy is far beyond its rights as a fraction with such a small denominator, and it causes various oddities elsewhere in math. For example, use any scientific calculator to compute cos(355) in radians. The oddball result is due to the freakish closeness of 355/113 to pi.

A cute mnemonic makes it easy for our base-10 species to remember this useful fraction. Write down the first three odd numbers twice: 1 1 3 3 5 5. Then divide the decimal number represented by the last 3 digits by the decimal number given by the first three digits.

The mystery is: why is this fraction so close to pi? The deeper you look, the more unique and unexplained 355/113 appears to be.

The Unusualness of 355/113

In a 2000 lecture on rational approximations of pi, Fritz Beukers defines the "quality" of a rational approximation p/q as a number M such that

In words, the Beuker's quality M is the ratio between the number of digits of precision by the number of digits of the denominator.

It is no surprise when we find a fraction that approximates pi with M around 1: for any q there will be a p/q within 1/q of any value. But it is rare to find M larger than 2. The best two approximations for pi we have seen are:


These approximations both have quality M > 3, which is unusually good. Both these fractions provide an approximation that have a precision with about triple what you would have any right to expect for their small denominators.

Other Good Rational Approximations for Pi

The fraction 355/113 overestimates pi by less than .

It is not easy to approximate pi as economically as 355/113, but you can certainly try. If you have a good enough memory to remember the number 3748629, then you might find it handy to know that the following difference is even closer to pi:

In other words, subtracting 1/3748259 from 355/113 will provide an approximation to pi that is well within 10-14 of the true value. By subtracting a couple fractions you are near the 10-15 limits of IEEE 754 double-precision arithmetic. This new fraction is an excellent and economical way to approximate pi as the difference between two fractions.

On the other hand, this new fraction is not as remarkable as 355/113:

While M > 1 tells us that the precision is better than a random denominator would give us, M is still much lower than the quality of 355/113.

Looking for 355/113 in Wallis's Rational Expression

Close coincidences in math are usually a hint of something deeper. So let us take a look at some computable sequences of rational numbers that converge to pi to see if 355/113 appears on some explainable path to pi.

There are several beautifully computable and well-known sequences of rational numbers that converge to pi. In 1655, shortly before the dawn of calculus, English mathematician John Wallis painstakingly computed an extended rational expression for the area of a circle to come up with one of the most memorable rational expressions for pi:

The partial expansions of this infinite fraction are:

This series converges very slowly despite having huge denominators, and our efficient fraction 355/113 is nowhere in sight.

Looking for 355/113 in Taylor Series

While developing modern calculus, Gregory and Leibniz systematicaly computed what we call the Taylor expansion for arctan. That formula leads to another pretty rational expansion for pi:

Partial expansions give us the converging sequence of rationals:

Unfortunately this sequence again converges slowly, and 355/113 does not appear.

Looking for 355/113 in Gauss's continued fraction

Gauss developed a clever generalized continued fraction for arctan that gives us a faster-converging expression:

This expands to a sequence of rationals that converges much more quickly than the previous examples:

Ah, hah! Here we can see that Gauss's expansion of arctan has justified the rational approximation 22/7, which appears as the third approximation in the sequence. This seems to be a side-effect of the efficiency of Gauss's sequence of rational numbers, which gets much closer to pi for a denominator of any given size.

But 355/113 is still beyond the reach of this sequence. It does not appear here.

Looking for 355/113 in Lange's Sequence

In 1999 Lange derived another elegant expression for pi based on a continued fraction for arcsin:

The sequence of partial fractions begins:

This is another fast-converging sequence that includes 22/7. But again, 355/113 does not appear.

The quest is not over; Mathematicians are still hunting for faster-converging sequences of rational pi approximations. Beukers's lecture on this topic is worth a read. But 355/113 does not appear naturally in any these sequences that can be used to derive pi.

Where Does 355/113 Come From?

So, then, where does 355/113 come from? Is its nearness to pi a mere coincidence? A mathematical accident? A freak of nature?

It may be.

Or it might be a hint that there exists some as-yet undiscovered sequence of rationals that converge much faster towards pi, for which the highly precise 355/113 is just one remarkable number among many.

Posted by David at March 14, 2010 04:03 AM

is their any other fraction that is very close to pi...

Posted by: pi master at October 1, 2010 08:36 PM

These are very interesting thoughts indeed, but isn't seeking for anything special in 355/113 like sticking too much
to numbers themselves and the semiotic properties of the decimal system?

What would all of this look like, if we had sixteen fingers and would be counting hexadecimal like natural?

Please redo the math in HEX for me! ;)

For a start, 22/7 in HEX would be 16/7 and
355/113 would be 163/71.. but I can't do fractions in hex, so what would be the value of pi then?


Posted by: at October 24, 2010 09:13 AM

Hi David,
look at this continued fraction:
It looks very good, both 22/7 and 355/113 appear in the approximations sequence...

Posted by: GuidoB at December 9, 2010 06:02 PM

Guido - the continued fraction sequence 3 7 15 1 292 1 1 1 2 1... is the continued fraction analogy to the decimal sequence 3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 etc: they are both guaranteed-to-exist sequences of successively better approximations for pi that come from the real value of pi. Neither is a way to derive or compute pi if you don't already have the exact value.

In other words, if you were to want to compute the continued fraction sequence, you would have to begin by computing a near-exact value of pi, and then work back to the continued fraction approximations.

What would be remarkable if there was some direct way to derive the sequence 3 7 15 1 292 1 1 1 2 1 starting from just the counting numbers, without effectively computing pi some other way first. So far, no natural sequence that converges to pi so economically is known.

The particular mystery of 113/355 is the same as asking "why is the number 292 in the continued fraction sequence so unusually large?"

Posted by: David at December 20, 2010 06:53 AM

Greetings David,

Once upon a time some French lady gave me a GREAT present on PI day. What was the present? Well it was indeed PI ~= 355/113 with a reference back to my findings of prime numbers in the holy book of Islam, The Quran.

The Quran has 114 chapters the first of which is called The Opener (The Key) while the remaining 113 are The Message (Straight Path to GOD).

Within the Message I found a chapter that points to 2012 (well 1433 in Islamic Calendar to be precise) but I couldn't convince people that the number was the year 1433AH until I found that the number of words in the chapter was 355! This is the same exact number of days in an Islamic leap year which 1433AH is to be.

So in short, the 113 is the Straight Path to GOD (Diameter) and the 355 is the number of days in a revolving Islamic year (Circumference).

Now, to know that the Arabs used 22/7 while the Chinese used 355/113 only imply that the Chinese are more Muslim (Submissive to the Will of GOD) than the Arabs :)

Ali Adams
God > infinity

Posted by: Ali Adams at March 19, 2011 09:52 PM

The next fraction after 355/113 that approximates pi with similar precision is 52163/16604. Averaging the two gives you 3.14159265386526 -- precise to the 9th decimal. The equivalent fraction is (gasp!) 11788839/3752504.

Posted by: Joe at April 27, 2011 01:07 AM

use this method:
3/1 = 3
4/1 = 4
add the numbers up
7/2 = 3.5
11/3 = 3.3
i did this and it DID contain
and 355/113.
i do this when i have free time :D

Posted by: pimasterfromhk at November 8, 2011 08:22 AM

pimasterfromhk needs to explain himself especially since 11/3 = 3.666... Now 10/3 = 3.33... So if you start with 4/1 = 4 and 3/1 = 3 you get the first two new terms 7/2 and 10/3, but this sequence converges to 3.38196...

Posted by: Chris Staffa at February 29, 2012 03:42 PM

I know this comes in a bit late, but:

3/1 7/2
3/1 10/3
3/1 13/4
3/1 16/5
3/1 19/6
3/1 22/7
3/1 25/8

25/8 47/15
47/15 69/22
69/22 91/29
91/29 113/36
113/36 135/43
135/43 157/50
157/50 179/57
179/57 201/64
201/64 223/71
223/71 245/78
245/78 267/85
267/85 289/92
289/92 311/99
311/99 333/106
333/106 355/113

Posted by: flatfly at July 24, 2012 10:20 AM

mmm... your blog doesn't accept greater than and less than signs, so these are missing from my previous post... ok will repost later

Posted by: flatfly at July 24, 2012 10:22 AM

Wallis's Rational Expression starts by repeating each denominator twice.

If you tack on a 1/1 in front of it, so the 1 is also repeated twice, then the denominators are 1,1,3,3,5,5, and etc.

1,1,3,3,5,5 is the "cute mnemonic" you told us to write down and remember above, and according to A.A., also the straight path to GOD and the number of days in an Islamic leap year. Clearly Wallis was reading the holy book, but left out the one true one to throw us off.

Posted by: Skeuomorph at July 24, 2012 10:35 AM

Try this ruby code:

def strength(x, y)
diff = Math::PI - ((1.0 * x)/y)
return - Math::log(diff.abs) / Math::log(y)

strength(22, 7)
strength(355, 112)
strength(1330631944, 423553267)

Posted by: Hugh Brown at July 24, 2012 11:19 AM

Argh. Somehow, I cut and pasted a typo into the code.

def strength(x, y)
diff = Math::PI - (x.to_f / y)
return - Math::log(diff.abs) / Math::log(y)

strength(22, 7)
strength(355, 113)
strength(1330631944, 423553267)

Posted by: Hugh Brown at July 24, 2012 11:25 AM

@Pi Master

Yes there are:

Posted by: TheSisb at July 24, 2012 01:32 PM

There is another fraction given by fourth root of (9^2 + 19^2/22)

Posted by: at July 24, 2012 02:03 PM

It has been said that the Old Testament (1 Kings 7:23) claims pi is equal to exactly 3.

However, there is a theory that the Hebrew text hides a much more accurate value for pi:

" Rabbi Belaga presents the following explanation: The Hebrew word for line or circumference is written in the Bible as a 3 letter Hebrew word transliterated as kaveh,and whose equivalent English letters are KVH (kof, vav, hei). Yet, that word is read as a 2 letter Hebrew word whose equivalent English letters are KV.

Hebrew letters have numerical values, and the letters in question have values kof = 100, vav = 6, and hei = 5. So KVH = 100 + 6 + 5 = 111, and KV = 100 + 6 = 106. The ratio of KVH to KV is 111/106, which when multiplied by the value of 3 that was implied by 1 Kings 7:23, gives 3.141509 (rounded), which is again pretty close to pi."

Posted by: tao of me at July 24, 2012 03:18 PM

There are mathematical facts that are true for no reason, as Kurt Gödel demostrated in 1931. No mistery behind 355/113.

Posted by: KurtGodel at July 24, 2012 07:16 PM

This is great information. You may need to take care of little typos around the number 3748629. In next line it is 3748269 and in the next it is 3748259. Indeed the number is not easy to remember.

Posted by: Manoj R. at July 25, 2012 12:10 AM

So, if I remember the six digits of 355/113 it gives me the first six decimal places of pi. Doesn't seem like much of a saving.

Posted by: Paul Linton at July 25, 2012 01:24 AM

Kurt Godel did no such thing. He proved the existense of undecidable propositions in logical systems such as set theory.

Posted by: Gaus at July 25, 2012 06:17 AM

This is called Diophantine approximation.
p.s. Take a look at the "Liouville's result" part.

Posted by: Wandspiegel at July 25, 2012 07:26 AM

If I remember my math in college, one can find a fraction arbitrarily close to any real number. So, I don't there's anything "special" about this fraction and it's kind of silly to ask "why" and if it is a "coincidence". Is it a conincidence that a solution of x^2 = 2x is 2? Spoookkkyyy...

Posted by: Lancem at July 25, 2012 10:17 AM

Here is another way of "measuring" how coincidental this precision is.

When you pick a denominator q, the pi-approximation with that denominator will be within 1/(2q) of pi. Over all randomly chosen q, that error ought to be uniformly distributed between 0 and 1/(2q).

Express this as a precision relative to 1/(2q) -- i.e., u = |p/q - pi| / (1/(2q)), so that u is distributed as Uniform(0,1).

For 355/113, u=60e-6. If you choose one denominator at random, you'd only have 60 chances in a million of obtaining this level of precision.

But, of course, you could look at enough denominators to eventually find something this good or better. What makes this approximation cool is that the denominator is only 3 digits. So the question to ask is what the probability of there being a 3-digit denominator with a u<=60e-6, purely by chance, if u is uniform. This probability turns out to be 6%. (Given by 1 - (1- 60e-6)^1000 ).

In statistics speak, that is almost, but not quite, statistically significant.

Posted by: Lonnie at July 25, 2012 12:59 PM

I'm no math expert, but I will try to respond to a few of the issues raised in the comments.

@Lancem True, it is not remarkable to find a fraction that is close to pi, but the interesting part is finding one that is surprisingly close to pi considering it's small numerator and denominator. Usually, these have to be much larger for the fraction to be a good approximation.

@tao_of_me @Ali_Adams I'm pretty sure the Quran and Bible are long enough that you can find any numbers you want by counting various subsets of chapters, words, letters, etc. Unless there is a definitive reason why one of these counts *should* equal an important number in math, it feels like pure coincidence. But, people love to find meaning in coincidence, so it is still fun to see.

@(the guy posting about hex numbers) I like your point about decimal being arbitrary. Hexadecimal is just as good. Or any other base. However, it seems the point being made in this article is equally valid for any base. Regardless of the base, numbers with a greater value require more digits to write and vice versa. So, if a number is shorter and easier to memorize in decimal, the same is true in hexadecimal. For example, 162/71 is 355/113 in hex and it is shorter than 4F4FD508/193EE8F3, which is 1330631944/423553267 in hex.

Posted by: FishDawg at July 25, 2012 02:56 PM

The closest I have found thus far is

9999999/3183101 = 3.141590229150760000000
355/113 = 3.141592920353980000000
22/7 = 3.142857142857140000000

In Biblical theology, 7 is said to be
the number of completion as God rested
on the 7th day.

Hmm, it would be more accurate for the
number 7 to be the number of rest as God
rested from His work on the seventh day,
and the commandment to keep the Sabbath holy
which also states, "six days you shall work
and on the seventh rest".

The number 9 is a unique number in and of
itself as:

9 = Sum ( product [ ] ) = x * 9;

Distil is adding the digits of the product
together until there is a single digit.
In this case, distill will always equal 9.

The following pseudo code can produce this

product = 9 * x

While ( count( product [ ] ) > 1 )

foreach ( product as digit )

newProduct += digit


product = newProduct


Take care.

Scott A. Tovey

Posted by: Scott at July 25, 2012 06:20 PM

These are great comments and I understand almost all of them.

Or maybe that's the tequila talking....

Any case, I appreciate the comments.


Posted by: at July 25, 2012 07:49 PM

This fraction is a good approximation of pi:

31415926535897932384626433832795 / 10000000000000000000000000000000

Posted by: Isaac Tesla at July 25, 2012 09:56 PM

Archimedes, Huygens, Lambert, Legendre.
Vier Abhandlungen über die Kreismessung. Deutsh hrsg. und mit einer Übersicht über die
Geschichte des Problemes von der Quadratur des Zirkels
Published 1892 by B.G. Teubner in Leipzig .
Written in German.

pages 146-147 has a table of rational approximations of pi

113 : 355
33102 : 103993
33215 : 104348
66317 : 208341
99532 : 312689
265381 : 833719
364913 : 1146408
1360120 : 4272943
1725033 : 5419351
25510582 : 80143857
52746197 : 165707065
78256779 : 245850922
131002976 : 411557987
340262713 : 1068966896
811528438 : 2549491779
1963319607 : 6167950454
4738167652 : 14885392687
6701487259 : 21053343141
567663097408 : 1783366216531
1142027682075 : 3587785776203
1709690779483 : 5371151992734
2851718461558 : 8958937768937
107223273857129 : 336851849443403
324521540032945 : 1019514486099146 etc.

And then he points out that the last pair come to represent pi within 25 decimal places.

Lambert came up with the idea that pi is irrational. This is part of the proof, but due to my poor understanding of German, I can not figure out where he came up with this series of rational approximations.

Posted by: Bernard Greening at July 26, 2012 04:22 PM

in your section
Other Good Rational Approximations for Pi

I think one of these three is correct but you should correct the other two:

A dylsexic moment?

Posted by: scottm at July 27, 2012 12:52 PM

3141592654:1000000000 :-)

Posted by: at August 3, 2012 06:15 PM

Aquí en la tierra existen dos personajes que han dicho ser Dios. Ellos son Jehová y Krishna. Es obvio que el Creador es uno solo.
¿Cual de los dos es el mentiroso? ¿Y cuál es el verdadero? ¿O son los dos falsos? Yo sé quien es el Dios verdadero y te lo voy a decir después de...


Posted by: Tony at January 4, 2013 12:53 PM

355/113 = (2^5 + 3/11)/(2*5 + 3/11)

3/11 is the ratio of the moon's radius to the earth's.

Posted by: Trevor Streeton at April 18, 2013 07:58 AM

If you program a computer to look for 2 3-digit numbers, the ratio of which is between 223/71 and 22/7, only 4 possibilities arise.

(300 + 22*k)/(100 + 7*k) for k = 1 to 4.

(I'm still trying to figure out how the ancients could quantify the precision of any approximation via fractions). Whu 355/113 should have been chosen out of these 4 options, all better than the initial 2.

Getting mystical (numerical synchronicities)
355/113 = (2^5 + 3/11)/(2*5 + 3/11). Pretty, aint it?

Strike out all the digits found in the right hand side from the left and you are left with 1 1 2 2.
Can you see this as 11 22 or as 11 11 11 three elevens?

Now, the extra digits in 355/113 gained over the two starting values are 1592.

Here is a wierd numerical synchronicity connecting PI, 22/7 and Euler's number e where these digits turn up reversed.

The first 9 digits of e (including the integer part) can be found in the sequence 228118278 at positions 21,33, 34, 37, 49, 52, 63, 66 & 68 of PI.

Only 2 of these digits are correctly placed, the leading 2 and the 8 in the 6th place. Four swaps are necessary, the first is swapping the 2nd and 8th digits. The four swap codes, read zig-zag, form a PI symbol rotated 180 degrees.

The digits of e in these positions are
024795741. Ignoring the leading 0 and stripping the 2591 out you are left with 4774, a palindrome. Sum of the digits = 22. (I was born in 1947.)

Also 355/113 = 3*[360 - (3+7)/2]/(360 - 3*7).

To me it looks like the ratio is an artifact of the structure of number itself and not an analytically derived one.

Posted by: at May 5, 2013 02:29 PM

Enjoy a look at Johann Heinrich Lambert, 1792, at – Fritz

Posted by: Fritz Jörn at May 31, 2013 06:02 PM

Notice the first approximation of pi that is even CLOSER than 355/113 is 52163/16604 (3.14159238...). Notice that 52163/16604pi, so just by adding 355 to 52163 and 113 to 16604 we get 52518/16717, a slightly closer approximation than 52163/16604. This can keep going until 104348/33215=3.14159265392.

Posted by: Justin at November 28, 2013 09:36 PM

Happy New Year!

I have realized that 355/113 uses six digits in all, and 3.141592920354... has six decimal places correct, a very high degree of accuracy. This is most likely because in the continued fraction of pi, 355/113 occurs right before a 292! The next fraction this good is 21053343141/6701487259, followed by 2646693125139304345/842468587426513207!

Posted by: Justin at January 2, 2014 08:00 PM

283 divided by 90
6352 divided by 2023
13308 divided by 4235 = 3.14238488784 = very close to Pi too....

Posted by: jamie patton at January 4, 2014 10:52 AM

You can reach pi by using any number!
254.46900494077325 / 81 = 3.141592653589793,...

it's quite simple simply get a number multiply it by PI...

{81pi} = 254.46900494077325

next get that number now divide by the original number in this case (81)...
you can do this with any number. I love mathematics but when you really get into it its pretty absurd. all numbers can be equal to zero or infinity so what's the point for for example you can have an infinite amount of numbers between any two numbers. therefore all numbers have the possibility of having the same amount of numbers in between each number basically they cancel each other out. you could simply use just the numbers 12 and 3 but by getting really specific with decimals you could go on infinitely with just three numbers no need for 456789. you can do the same job with three numbers but human beings have a tendency to get unnecessarily elaborate. if all numbers can have an infinite value then is there anyone number any bigger or smaller than another number and I do include 0. heroes actually my favorite number it seems like it doesn't do much but its infinite, to me it's a weird coincidence it's a circle. if mathematics was really tired how it should be people when I have many problems with it. it's basically coming down the mathematicians and people that with big egos math is simple if you can count to three you can count anything. think about the simple arithmetic of

1 x 0 = 0..... okay let's say I had one card in my hand, I multiply the amount of card(s) by zero how many cards do I have.. yeah I come up with the number one so why the hell did 0 become an accepted answer. mathematics is like magic it's an illusion.

Posted by: JOSKi at August 15, 2014 04:54 PM

Not that you asked, but…
In Japanese, fractions are read in the opposite order from English: denominator _bun no_ numerator.
So 355/113 is "113 bun no 355," which is especially memorable if read digit by digit (here capitalized): "Ichi, Ichi, San bun no San, Go, Go."
('Bun' means "part[s]" and 'no' is the possessive marker.)
This system for reading fractions is borrowed from Chinese and is also used in Korean.

Meanwhile, I notice that in base 11, 355/113 comes out as 2X3/X3 (where X stands for the number after 9), which I think is pretty neat in either order.

Posted by: Jeremy Whipple at November 9, 2015 05:00 AM

I've found most approximate fractional number.
117766625/(113x331737) =3.1415926535897...

Posted by: Naga sanjay at January 12, 2016 11:38 AM

pi varies between 3.1416 to 3.164

look Aetzbar in amazon

Posted by: Aetzbar at February 5, 2016 06:30 PM

I have developed a certain continued fraction for 1/pi,which has 7/22 as one of it's convergents .Please check it out in this post

Posted by: mandelbrot at April 17, 2016 10:11 AM

I have developed a certain continued fraction for 1/pi,which has 7/22 as one of it's convergents .Please check it out in this post

Posted by: Mandela at April 17, 2016 10:12 AM

From the very first comment--

"is their any other fraction that is very close to pi..."

One that comes to mind which gives outstanding results is Srinivasa Ramanujan's excellent approximation

pi ≅ (2143/22)^(1/4).

To make this tractable with low-cost calculators, the fourth root is usually written as a double square root, i.e.,

pi ≅ ((2143/22)^(1/2))^(1/2).

[Please pardon any duplication, but I have not read this article in its entirety]

Posted by: jimmike at May 26, 2016 11:08 PM


Posted by: at July 1, 2016 01:00 AM

INTERSSANT; Schade, dass ihr die Quadratur des LdV nicht zeichnet könnt, wie in scmar beschrieben. Lesen geht hoffentlich noch. Allerdings muss man sich die paar Seiten ausdrucken. Es ist meine Version, kein Buch zu finanzieren.

Liebe Grüße, Udo

PS, Viren sind nicht drin, kann ich nicht-

Posted by: Udo Schmidt at August 12, 2016 06:24 PM