I've heard that $\pi$ is usually approximated as 3.14, but it can also be approximated as 22/7, which is equal to 3.142857142857142857.... Guess what? $\pi$ can also be approximated as 355/113, which is equal to 3.1415929203539823008849557.... There are 112 numbers after the decimal, which then start repeating. Anyway, let's cut to the chase. Why is $\pi$ usually approximated as 3.14 or 22/7? Maybe they're close to the actual result? Anything else about it? I do know that these can be used to find the circumference or area of a circle.
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$\begingroup$Say you're trying to approximate a number by rational numbers $p/q$. Usually, the bigger $q$ is, the better your chance of approximating the number closely. On the other hand, the smaller $q$ is, the simpler the approximation.
In the case of $\pi$, if you want to have a better approximation than $22/7$, you have to go all the way up to $q = 57$. (See this.) So $22/7$ is a remarkably accurate approximation, considering how low its denominator is.
$\endgroup$ 3 $\begingroup$3.14 is two decimal places. That's the only justification I can give. When doing hand calculations (or sliderule calculations), carrying out more than three significant digits is cumbersome. Remember this: with $N$ digits in a multiplication, you have $N^2$ digits in the final answer before you can round again. In the world of engineering, often you can't (afford to) manufacture something with insane tolerances. Basically: precision=money. So carrying out more digits might not make sense. For another example, imagine you are estimating the area of grass in a circular front yard (landscape designers, please don't troll. ;)) in order to put down fertilizer. Going from $3.14\rightarrow3.1416$ is two more digits, but are you seriously going to measure pounds of fertilizer down to the $\frac{1}{10,000}$ of a pound? That's like a gallon of water $\pm<\frac{1}{10}$ of a teaspoon.
22/7 is easier to remember. It's $\frac{21+1}{7}$.
Basically, this goes back to the days of slide-rules and hand calculations. As @user_of_math pointed out, we don't need to truncate today with computers as fast as they are, but there is now the trap of artificially high precision.
$\endgroup$ 4 $\begingroup$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 ~0.3551660642So 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 ~0.0067909264None 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 ~0.040249943522/7 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 ~0.0402499435So, remembering 3.141 is about twice as good as remembering 22/7. Since 3.141 is pretty memorable, 22/7 is only good if you are using it because you like playing with fractions.
$\endgroup$ $\begingroup$355/113 is actually easy to remember (113355, written bottom to top), and is the best approximation with a denominator less than 5 digits, accurate to six decimal digits. I always wonder why we don't learn that value, being both remarkably accurate and easy to memorize.
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