Rational Approximations to π


It is a little known story—just a footnote in the calculation of π—but during the during the 17th century in England there was considerable interest in computing decimal digits of π. The interesting problem that circulated all across the country was not to compute more digits, but who could calculate 20 digits the most rapidly. All sorts of people worked on this to see who would have the fastest rate. The two sisters of the father of William Penn, noted for his involvement in Pennsylvania, were among the leaders in this enterprise. A major topic of discussion whenever two mathematicians, professional or amateur, got together was: "What are the π rates of Penn's aunts?"


The value π is an irrational number. It cannot be represented exactly as a rational number; that is, a fraction with numerator and denominator both integers. The decimal representation of π is a never ending, never repeating string of digits.

In grade school we learned that a simple approximation for π is 22/7. This is easy to remember and does proved a resonable approximation (22/7 = 3.14285 ...). However, there are better rational approximations. The table below gives successive better ones to π.

Note that all denominators between 7 and 57 (with the appropriate numerator) all give worst approximations than 22/7. As an example, the two closest possible rational values using 8 as a denominator are 25/8 = 3.12500 and 26/8 = 3.25000. Of these two, 25/8 is closer; however, neither are as good as 22/7 (3.14285...).

Of particular note is the fraction 355/113. This will give us a value of π correct to six digits to the right of the decimal point. To get a better rational approximation, we would need to use 52163/16604 which is only slightly better.

Another rational approximation that is often used is 3.1416. This is the value of π rounded to four digits. Since the 6 has been rounded from 59, this should be a reasonably good approximation. At first glance, it does not appear like a rational value (ratio of two integers); however, because it is a terminating decimal, it is indeed a rational value. We only need to multiply and divide by the appropriate power of 10 to move the decimal place to the right of the last digit thereby turning the denominator into an integer. We get 31416/10000 = 3927/1250. This is actually a worse approximation than 355/113 with bigger numbers in the numerator and denominator (so it does not appear in the Table), but has the virtue of a reasonably good value with only four digits to remember even though the last digit has been rounded and is not the exact one in the sequence of the digits of π!


Successive Better Rational Approximations to π
 
 numerator   denominator  approximationerror correct 
digits
 
3 / 1  =  3.00000 00000 00000 . . .   − 1.42×10−10
13 / 4 = 3.25000 00000 00000 . . .+ 1.08×10−10
16 / 5 = 3.20000 00000 00000 . . .+ 5.84×10−20
19 / 6 = 3.16666 66666 66666 . . .+ 2.51×10−21
22 / 7 = 3.14285 71428 57142 . . .+ 1.26×10−32
179 / 57 = 3.14035 08771 92982 . . .− 1.24×10−32
201 / 64 = 3.14062 50000 00000 . . .− 9.68×10−42
223 / 71 = 3.14084 50704 22535 . . .− 7.48×10−42
245 / 78 = 3.14102 56410 25640 . . .− 5.67×10−43
267 / 85 = 3.14117 64705 88235 . . .− 4.16×10−43
289 / 92 = 3.14130 43478 26086 . . .− 2.88×10−43
311 / 99 = 3.14141 41414 14141 . . .− 1.79×10−43
333 / 106 = 3.14150 94339 62264 . . .− 8.32×10−54
355 / 113 = 3.14159 29203 53982 . . .+ 2.67×10−76
52163 / 16604 = 3.14159 23873 76535 . . .− 2.66×10−76
52518 / 16717 = 3.14159 23909 79242 . . .− 2.63×10−76
52873 / 16830 = 3.14159 23945 33570 . . .− 2.59×10−76
53228 / 16943 = 3.14159 23980 40488 . . .− 2.56×10−76
53583 / 17056 = 3.14159 24015 00937 . . .− 2.52×10−76
53938 / 17169 = 3.14159 24049 15836 . . .− 2.49×10−76
 
skipping 91 entries
 
86598 / 27565 = 3.14159 25993 10720 . . .− 5.43×10−86
86953 / 27678 = 3.14159 26006 21432 . . .− 5.30×10−87
87308 / 27791 = 3.14159 26019 21485 . . .− 5.17×10−87
 
skipping 42 entries
 
102573 / 32650 = 3.14159 26493 10872 . . .− 4.28×10−97
102928 / 32763 = 3.14159 26502 45704 . . .− 3.34×10−98
103283 / 32876 = 3.14159 26511 74108 . . .− 2.42×10−98
103638 / 32989 = 3.14159 26520 96153 . . .− 1.49×10−98
103993 / 33102 = 3.14159 26530 11902 . . .− 5.78×10−109
104348 / 33215 = 3.14159 26539 21421 . . .+ 3.32×10−109
208341 / 66317 = 3.14159 26534 67436 . . .− 1.22×10−109
312689 / 99532 = 3.14159 26536 18936 . . .+ 2.91×10−119
833719 / 265381 = 3.14159 26535 81077 . . .− 8.72×10−1211
1146408 / 364913 = 3.14159 26535 91403 . . .+ 1.61×10−1210
3126535 / 995207 = 3.14159 26535 88650 . . .− 1.14×10−1211
 
for comparisonπ ≈  3.14159 26535 89793 


The last three rational entries are interesting since the first of the three gives 1 more correct digit than the following entry. The actual error is less with the later one, but because the way the true value would round, the number of identical digits is one fewer.

The number of digits given in the table are, as normal, the number of digits to the right of the decimal point. If we examine the table we discover that for most of the rational approximations, one needs more digits combined in the numerator and denominator than are simply given in the number of correct decimal points. The exception is 355/113 for which, with the effort of memorizing these 6 digits, we get an approximation to π that contains 6 digits! It is less memorizing to simply learn the necessary decimal digits of π!