What is the best approximation of Pi?

When I was younger I was taught to approximate Pi by 22/7.

But why 22/7? Maybe there is some better way?

I wrote a little program to explore this question by considering all possible numerators and denominators and ranking them by their percent deviation from the "true value" of Pi.

#!/usr/bin/env python3
import math

#Let's explore all three digits numerators and divisors
#Obviously, numerator must be > denominator
candidates = []
for numerator in range(1,999):
  for denominator in range(1,numerator):
    approx      = float(numerator)/float(denominator)
    diff        = abs((approx-math.pi)/math.pi)*100
    candidates += [(approx,numerator,denominator,diff)]

#Sort candidates by their percent difference from the "true" value of pi. "True"
#is in quotation marks here because math.pi is itself an approximation, albeit a
#good one, of the true value of pi.
candidates.sort(key=lambda x: x[3])

#List the sorted candidates
print("Rank Num / Den = Approx ~Diff")
for rank,candidate in enumerate(candidates):
  print("{0:>4d} {2:>3d} / {3:>3d} = {1:>.10f} ~{4:>.10f}".format(rank,candidate[0],candidate[1],candidate[2],candidate[3]))

Running the program produces the following list:

Rank Num / Den = Approx ~Diff
   0  22 /   7 = 3.1428571429 ~0.0402499435
   1  44 /  14 = 3.1428571429 ~0.0402499435
   2  66 /  21 = 3.1428571429 ~0.0402499435
   3  88 /  28 = 3.1428571429 ~0.0402499435
   4  91 /  29 = 3.1379310345 ~0.1165529561
   5  69 /  22 = 3.1363636364 ~0.1664447878
   6  85 /  27 = 3.1481481481 ~0.2086678727
   7  47 /  15 = 3.1333333333 ~0.2629023291
   8  94 /  30 = 3.1333333333 ~0.2629023291
   9  63 /  20 = 3.1500000000 ~0.2676141479
  10  72 /  23 = 3.1304347826 ~0.3551660642

So 22/7 is the best value of Pi for two-digit numbers.

How does it do with three-digit numbers? A simple modification (for numerator in range(1,999):) to the program suffices to tell us: it ranks 107th.

Rank Num / Den = Approx ~Diff
   0 355 / 113 = 3.1415929204 ~0.0000084914
   1 710 / 226 = 3.1415929204 ~0.0000084914
   2 732 / 233 = 3.1416309013 ~0.0012174620
   3 688 / 219 = 3.1415525114 ~0.0012777651
   4 377 / 120 = 3.1416666667 ~0.0023559094
   5 754 / 240 = 3.1416666667 ~0.0023559094
   6 333 / 106 = 3.1415094340 ~0.0026489630
   7 666 / 212 = 3.1415094340 ~0.0026489630
   8 776 / 247 = 3.1417004049 ~0.0034298294
   9 977 / 311 = 3.1414790997 ~0.0036145333
  10 644 / 205 = 3.1414634146 ~0.0041138037
  11 399 / 127 = 3.1417322835 ~0.0044445570
  12 798 / 254 = 3.1417322835 ~0.0044445570
  13 955 / 304 = 3.1414473684 ~0.0046245706
  14 820 / 261 = 3.1417624521 ~0.0054048547
  15 311 /  99 = 3.1414141414 ~0.0056822190
  16 622 / 198 = 3.1414141414 ~0.0056822190
  17 933 / 297 = 3.1414141414 ~0.0056822190
  18 421 / 134 = 3.1417910448 ~0.0063149876
  19 842 / 268 = 3.1417910448 ~0.0063149876
  20 911 / 290 = 3.1413793103 ~0.0067909264
  21 864 / 275 = 3.1418181818 ~0.0071787865
  22 600 / 191 = 3.1413612565 ~0.0073655967
  23 889 / 283 = 3.1413427562 ~0.0079544815
  24 443 / 141 = 3.1418439716 ~0.0079997017
  25 886 / 282 = 3.1418439716 ~0.0079997017
  26 908 / 289 = 3.1418685121 ~0.0087808494
  27 289 /  92 = 3.1413043478 ~0.0091770575
  28 578 / 184 = 3.1413043478 ~0.0091770575
  29 867 / 276 = 3.1413043478 ~0.0091770575
  30 465 / 148 = 3.1418918919 ~0.0095250510
  31 930 / 296 = 3.1418918919 ~0.0095250510
  32 952 / 303 = 3.1419141914 ~0.0102348670
  33 845 / 269 = 3.1412639405 ~0.0104632620
  34 487 / 155 = 3.1419354839 ~0.0109126268
  35 974 / 310 = 3.1419354839 ~0.0109126268
  36 556 / 177 = 3.1412429379 ~0.0111317976
  37 996 / 317 = 3.1419558360 ~0.0115604540
  38 823 / 262 = 3.1412213740 ~0.0118181949
  39 509 / 162 = 3.1419753086 ~0.0121802886
  40 267 /  85 = 3.1411764706 ~0.0132475164
  41 534 / 170 = 3.1411764706 ~0.0132475164
  42 801 / 255 = 3.1411764706 ~0.0132475164
  43 531 / 169 = 3.1420118343 ~0.0133429370
  44 553 / 176 = 3.1420454545 ~0.0144131021
  45 779 / 248 = 3.1411290323 ~0.0147575253
  46 575 / 183 = 3.1420765027 ~0.0154013965
  47 512 / 163 = 3.1411042945 ~0.0155449533
  48 597 / 190 = 3.1421052632 ~0.0163168693
  49 757 / 241 = 3.1410788382 ~0.0163552526
  50 619 / 197 = 3.1421319797 ~0.0171672831
  51 641 / 204 = 3.1421568627 ~0.0179593352
  52 245 /  78 = 3.1410256410 ~0.0180485705
  53 490 / 156 = 3.1410256410 ~0.0180485705
  54 735 / 234 = 3.1410256410 ~0.0180485705
  55 980 / 312 = 3.1410256410 ~0.0180485705
  56 663 / 211 = 3.1421800948 ~0.0186988341
  57 958 / 305 = 3.1409836066 ~0.0193865692
  58 685 / 218 = 3.1422018349 ~0.0193908422
  59 713 / 227 = 3.1409691630 ~0.0198463220
  60 707 / 225 = 3.1422222222 ~0.0200397920
  61 729 / 232 = 3.1422413793 ~0.0206495810
  62 468 / 149 = 3.1409395973 ~0.0207874268
  63 936 / 298 = 3.1409395973 ~0.0207874268
  64 751 / 239 = 3.1422594142 ~0.0212236502
  65 691 / 220 = 3.1409090909 ~0.0217584759
  66 773 / 246 = 3.1422764228 ~0.0217650488
  67 914 / 291 = 3.1408934708 ~0.0222556797
  68 795 / 253 = 3.1422924901 ~0.0222764886
  69 817 / 260 = 3.1423076923 ~0.0227603893
  70 839 / 267 = 3.1423220974 ~0.0232189169
  71 861 / 274 = 3.1423357664 ~0.0236540161
  72 223 /  71 = 3.1408450704 ~0.0237963113
  73 446 / 142 = 3.1408450704 ~0.0237963113
  74 669 / 213 = 3.1408450704 ~0.0237963113
  75 892 / 284 = 3.1408450704 ~0.0237963113
  76 883 / 281 = 3.1423487544 ~0.0240674378
  77 905 / 288 = 3.1423611111 ~0.0244607626
  78 927 / 295 = 3.1423728814 ~0.0248354211
  79 949 / 302 = 3.1423841060 ~0.0251927114
  80 870 / 277 = 3.1407942238 ~0.0254148087
  81 971 / 309 = 3.1423948220 ~0.0255338137
  82 993 / 316 = 3.1424050633 ~0.0258598040
  83 647 / 206 = 3.1407766990 ~0.0259726403
  84 424 / 135 = 3.1407407407 ~0.0271172282
  85 848 / 270 = 3.1407407407 ~0.0271172282
  86 625 / 199 = 3.1407035176 ~0.0283020780
  87 826 / 263 = 3.1406844106 ~0.0289102708
  88 201 /  64 = 3.1406250000 ~0.0308013704
  89 402 / 128 = 3.1406250000 ~0.0308013704
  90 603 / 192 = 3.1406250000 ~0.0308013704
  91 804 / 256 = 3.1406250000 ~0.0308013704
  92 983 / 313 = 3.1405750799 ~0.0323903774
  93 782 / 249 = 3.1405622490 ~0.0327987969
  94 581 / 185 = 3.1405405405 ~0.0334897985
  95 961 / 306 = 3.1405228758 ~0.0340520841
  96 380 / 121 = 3.1404958678 ~0.0349117770
  97 760 / 242 = 3.1404958678 ~0.0349117770
  98 939 / 299 = 3.1404682274 ~0.0357915965
  99 559 / 178 = 3.1404494382 ~0.0363896760
 100 738 / 235 = 3.1404255319 ~0.0371506367
 101 917 / 292 = 3.1404109589 ~0.0376145101
 102 179 /  57 = 3.1403508772 ~0.0395269704
 103 358 / 114 = 3.1403508772 ~0.0395269704
 104 537 / 171 = 3.1403508772 ~0.0395269704
 105 716 / 228 = 3.1403508772 ~0.0395269704
 106 895 / 285 = 3.1403508772 ~0.0395269704
 107  22 /   7 = 3.1428571429 ~0.0402499435

$22/7$ ranks 107th on this list.

Are any of the values above easy to remember? The following strike me as candidates:

 0 355 / 113 = 3.1415929204 ~0.0000084914
 7 666 / 212 = 3.1415094340 ~0.0026489630
15 311 /  99 = 3.1414141414 ~0.0056822190
20 911 / 290 = 3.1413793103 ~0.0067909264

None of them seem as easy to remember as 22/7, though they are more accurate.

One use of 22/7 is for doing mental math. Therefore, it's worth asking whether any of these easily-memorable three-digit candidates have convenient mathematical properties for mental math. Let's factor them:

355 / 113 = (5*71)   / (113)
666 / 212 = (3*3*37) / (2*53)
311 / 99  = (311)    / (3*3*11)
911 / 290 = (911)    / (2*5*29)

None of those factors seem conducive to mental math.

In summary, it seems as though 22/7 is the best fractional estimate of Pi in terms of ease of memorization and use for mental math.

What if you don't use a fractional estimate and prefer, instead, a decimal estimate? The Pi button's your best bet, but, barring that, how do various decimal approximations of Pi stack up?

We add the following chunk to the program after the first candidate-generating chunk:

for dec_approx in [3,3.1,3.14,3.141,3.1415,3.14159,3.141592,3.1415926]:
  diff        = abs((dec_approx-math.pi)/math.pi)*100
  candidates += [(dec_approx,0,0,diff)]

The results are as follows:

Rank Num / Den = Approx ~Diff
   0   0 /   0 = 3.1415926000 ~0.0000017058
   1 355 / 113 = 3.1415929204 ~0.0000084914
   2 710 / 226 = 3.1415929204 ~0.0000084914
   3   0 /   0 = 3.1415920000 ~0.0000208044
   4   0 /   0 = 3.1415900000 ~0.0000844664
   5 732 / 233 = 3.1416309013 ~0.0012174620
   6 688 / 219 = 3.1415525114 ~0.0012777651
   7 377 / 120 = 3.1416666667 ~0.0023559094
   8 754 / 240 = 3.1416666667 ~0.0023559094
   9 333 / 106 = 3.1415094340 ~0.0026489630
  10 666 / 212 = 3.1415094340 ~0.0026489630
  11   0 /   0 = 3.1415000000 ~0.0029492554
  12 776 / 247 = 3.1417004049 ~0.0034298294
  13 977 / 311 = 3.1414790997 ~0.0036145333
  14 644 / 205 = 3.1414634146 ~0.0041138037
  15 399 / 127 = 3.1417322835 ~0.0044445570
  16 798 / 254 = 3.1417322835 ~0.0044445570
  17 955 / 304 = 3.1414473684 ~0.0046245706
  18 820 / 261 = 3.1417624521 ~0.0054048547
  19 311 /  99 = 3.1414141414 ~0.0056822190
  20 622 / 198 = 3.1414141414 ~0.0056822190
  21 933 / 297 = 3.1414141414 ~0.0056822190
  22 421 / 134 = 3.1417910448 ~0.0063149876
  23 842 / 268 = 3.1417910448 ~0.0063149876
  24 911 / 290 = 3.1413793103 ~0.0067909264
  25 864 / 275 = 3.1418181818 ~0.0071787865
  26 600 / 191 = 3.1413612565 ~0.0073655967
  27 889 / 283 = 3.1413427562 ~0.0079544815
  28 443 / 141 = 3.1418439716 ~0.0079997017
  29 886 / 282 = 3.1418439716 ~0.0079997017
  30 908 / 289 = 3.1418685121 ~0.0087808494
  31 289 /  92 = 3.1413043478 ~0.0091770575
  32 578 / 184 = 3.1413043478 ~0.0091770575
  33 867 / 276 = 3.1413043478 ~0.0091770575
  34 465 / 148 = 3.1418918919 ~0.0095250510
  35 930 / 296 = 3.1418918919 ~0.0095250510
  36 952 / 303 = 3.1419141914 ~0.0102348670
  37 845 / 269 = 3.1412639405 ~0.0104632620
  38 487 / 155 = 3.1419354839 ~0.0109126268
  39 974 / 310 = 3.1419354839 ~0.0109126268
  40 556 / 177 = 3.1412429379 ~0.0111317976
  41 996 / 317 = 3.1419558360 ~0.0115604540
  42 823 / 262 = 3.1412213740 ~0.0118181949
  43 509 / 162 = 3.1419753086 ~0.0121802886
  44 267 /  85 = 3.1411764706 ~0.0132475164
  45 534 / 170 = 3.1411764706 ~0.0132475164
  46 801 / 255 = 3.1411764706 ~0.0132475164
  47 531 / 169 = 3.1420118343 ~0.0133429370
  48 553 / 176 = 3.1420454545 ~0.0144131021
  49 779 / 248 = 3.1411290323 ~0.0147575253
  50 575 / 183 = 3.1420765027 ~0.0154013965
  51 512 / 163 = 3.1411042945 ~0.0155449533
  52 597 / 190 = 3.1421052632 ~0.0163168693
  53 757 / 241 = 3.1410788382 ~0.0163552526
  54 619 / 197 = 3.1421319797 ~0.0171672831
  55 641 / 204 = 3.1421568627 ~0.0179593352
  56 245 /  78 = 3.1410256410 ~0.0180485705
  57 490 / 156 = 3.1410256410 ~0.0180485705
  58 735 / 234 = 3.1410256410 ~0.0180485705
  59 980 / 312 = 3.1410256410 ~0.0180485705
  60 663 / 211 = 3.1421800948 ~0.0186988341
  61   0 /   0 = 3.1410000000 ~0.0188647497
  62 958 / 305 = 3.1409836066 ~0.0193865692
  63 685 / 218 = 3.1422018349 ~0.0193908422
  64 713 / 227 = 3.1409691630 ~0.0198463220
  65 707 / 225 = 3.1422222222 ~0.0200397920
  66 729 / 232 = 3.1422413793 ~0.0206495810
  67 468 / 149 = 3.1409395973 ~0.0207874268
  68 936 / 298 = 3.1409395973 ~0.0207874268
  69 751 / 239 = 3.1422594142 ~0.0212236502
  70 691 / 220 = 3.1409090909 ~0.0217584759
  71 773 / 246 = 3.1422764228 ~0.0217650488
  72 914 / 291 = 3.1408934708 ~0.0222556797
  73 795 / 253 = 3.1422924901 ~0.0222764886
  74 817 / 260 = 3.1423076923 ~0.0227603893
  75 839 / 267 = 3.1423220974 ~0.0232189169
  76 861 / 274 = 3.1423357664 ~0.0236540161
  77 223 /  71 = 3.1408450704 ~0.0237963113
  78 446 / 142 = 3.1408450704 ~0.0237963113
  79 669 / 213 = 3.1408450704 ~0.0237963113
  80 892 / 284 = 3.1408450704 ~0.0237963113
  81 883 / 281 = 3.1423487544 ~0.0240674378
  82 905 / 288 = 3.1423611111 ~0.0244607626
  83 927 / 295 = 3.1423728814 ~0.0248354211
  84 949 / 302 = 3.1423841060 ~0.0251927114
  85 870 / 277 = 3.1407942238 ~0.0254148087
  86 971 / 309 = 3.1423948220 ~0.0255338137
  87 993 / 316 = 3.1424050633 ~0.0258598040
  88 647 / 206 = 3.1407766990 ~0.0259726403
  89 424 / 135 = 3.1407407407 ~0.0271172282
  90 848 / 270 = 3.1407407407 ~0.0271172282
  91 625 / 199 = 3.1407035176 ~0.0283020780
  92 826 / 263 = 3.1406844106 ~0.0289102708
  93 201 /  64 = 3.1406250000 ~0.0308013704
  94 402 / 128 = 3.1406250000 ~0.0308013704
  95 603 / 192 = 3.1406250000 ~0.0308013704
  96 804 / 256 = 3.1406250000 ~0.0308013704
  97 983 / 313 = 3.1405750799 ~0.0323903774
  98 782 / 249 = 3.1405622490 ~0.0327987969
  99 581 / 185 = 3.1405405405 ~0.0334897985
 100 961 / 306 = 3.1405228758 ~0.0340520841
 101 380 / 121 = 3.1404958678 ~0.0349117770
 102 760 / 242 = 3.1404958678 ~0.0349117770
 103 939 / 299 = 3.1404682274 ~0.0357915965
 104 559 / 178 = 3.1404494382 ~0.0363896760
 105 738 / 235 = 3.1404255319 ~0.0371506367
 106 917 / 292 = 3.1404109589 ~0.0376145101
 107 179 /  57 = 3.1403508772 ~0.0395269704
 108 358 / 114 = 3.1403508772 ~0.0395269704
 109 537 / 171 = 3.1403508772 ~0.0395269704
 110 716 / 228 = 3.1403508772 ~0.0395269704
 111 895 / 285 = 3.1403508772 ~0.0395269704
 112  22 /   7 = 3.1428571429 ~0.0402499435

22/7 ranks 112th on this list.

Let's pull out the interesting values:

Rank Num / Den = Approx ~Diff
  0           = 3.1415926000 ~0.0000017058
  1 355 / 113 = 3.1415929204 ~0.0000084914
  3           = 3.1415920000 ~0.0000208044
  4           = 3.1415900000 ~0.0000844664
 10 666 / 212 = 3.1415094340 ~0.0026489630
 11           = 3.1415000000 ~0.0029492554
 19 311 /  99 = 3.1414141414 ~0.0056822190
 24 911 / 290 = 3.1413793103 ~0.0067909264
 61           = 3.1410000000 ~0.0188647497
112  22 /   7 = 3.1428571429 ~0.0402499435

So, remembering 3.141 is about twice as good as remembering 22/7. This number seems pretty memorable, so 22/7 is only good if you are using it because you like playing with fractions.

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