Talk:Approximations of π

Wallis product?

Should the Wallis product be mentioned on this page or is it too obscure? It does seem that some books use it at leas as an example; see http://math.stackexchange.com/questions/1097633/how-to-show-frac-pi4-frac2-cdot4-cdot4-cdot6-cdot6-cdot8-dotsm3-cdot — Preceding unsigned comment added by 86.121.137.79 (talk) 15:50, 9 January 2015 (UTC)

Done. Wqwt (talk) 20:37, 10 January 2024 (UTC)

Power roots

I recently discovered that ${\sqrt[{3}]{31}}\approx 3.14138$ is another approximation of π, better than 3.14 or 22/7 but not as good as 355/113. Looking further, I have discovered that ${\sqrt[{5}]{306}}$, ${\sqrt[{8}]{9489}}$ and ${\sqrt[{9}]{29809}}$ are progressively closer approximations.

A quick search of OEIS reveals OEIS: A080022 and OEIS: A002160. But A080022 is just the closest integer to $\pi ^{n}$, hence the integer whose $n$th root is closest to π, without consideration for whether taking the root gets you closer to π than any previous entry in the sequence. And A002160 isn't quite the same: not only does it count some powers twice (${\sqrt[{8}]{9488}}$ is closer to π than any previous, but ${\sqrt[{8}]{9489}}$ is closer still), but it is based on proximity of the logarithm to an integer, which isn't the same as proximity of the power root to π even if it seems to be giving the same results!

What I'm wondering is: Have power root approximations of π been studied enough to warrant inclusion here? — Smjg (talk) 16:07, 6 June 2020 (UTC)

Just realised "the closest integer to $\pi ^{n}$, hence the integer whose $n$th root is closest to π" is a flawed argument. But this doesn't affect my question. — Smjg (talk) 00:30, 8 June 2020 (UTC)

20th and 21st centuries

"In November 2002, Yasumasa Kanada and a team of 9 others used the Hitachi SR8000 ... to calculate π to roughly 1.24 trillion digits ... In October 2005, they claimed to have calculated it to 1.24 trillion places." What's the difference? The page on Kanada and the reference confirm the 1.2411 trillion places in Nov. 2002 - so is the claimed figure for Oct. 2005 incorrect? Shouldn't it be higher? Prisoner of Zenda (talk) 21:19, 29 September 2021 (UTC)

(1) You make a very valid point. I have looked at the Oct.2005 reference and find that it reports the 2002 result.
Therefore I have updated the text, removing the repetition but leaving the reference/link intact.
jw (talk) 19:56, 30 September 2021 (UTC)

(2) Note that I considered removing the 2005 reference entirely (i.e. the archived version; the original link is dead) but I left it intact as it contains some interesting information regarding the results and methods used.
I submit that it would be a good idea to find a valid reference nearer the 2002 date.
jw (talk) 19:56, 30 September 2021 (UTC)

Fractional approximations

Here is a list of fractions giving approximations of pi with increasing denominators and increasing precision:
fraction = approximation (error) [number of exact digits]
3 / 1 = 3.000 (4.507% err) [1] <<<
13 / 4 = 3.250 (3.451% err) [1]
16 / 5 = 3.200 (1.859% err) [1]
19 / 6 = 3.166667 (0.798% err) [2]
22 / 7 = 3.142857 (0.04025% err) [3] <<
179 / 57 = 3.140350 (0.03953% err) [3]
201 / 64 = 3.140625 (0.03080% err) [3]
223 / 71 = 3.140845 (0.02380% err) [3]
245 / 78 = 3.141025 (0.01805% err) [4]
267 / 85 = 3.141176 (0.01325% err) [4]
289 / 92 = 3.141304 (0.00918% err) [4]
311 / 99 = 3.141414 (0.00568% err) [4]
333 / 106 = 3.141509 (0.00265% err) [5]
355 / 113 = 3.141592920 (0.0000084914% err) [7] <<<<<
52163 / 16604 = 3.141592387 (0.0000084738% err) [7]
52518 / 16717 = 3.141592390 (0.0000083592% err) [7]
52873 / 16830 = 3.141592394 (0.0000082460% err) [7]
53228 / 16943 = 3.141592398 (0.0000081344% err) [7]
53583 / 17056 = 3.141592401 (0.0000080242% err) [7]
53938 / 17169 = 3.141592404 (0.0000079155% err) [7]
54293 / 17282 = 3.141592408 (0.0000078083% err) [7]
54648 / 17395 = 3.141592411 (0.0000077024% err) [7]
55003 / 17508 = 3.141592414 (0.0000075979% err) [7]
55358 / 17621 = 3.141592418 (0.0000074947% err) [7]
55713 / 17734 = 3.141592421 (0.0000073928% err) [7]
56068 / 17847 = 3.141592424 (0.0000072922% err) [7]
56423 / 17960 = 3.141592427 (0.0000071929% err) [7]
56778 / 18073 = 3.141592430 (0.0000070949% err) [7]
57133 / 18186 = 3.141592433 (0.0000069980% err) [7]
57488 / 18299 = 3.141592436 (0.0000069024% err) [7]
57843 / 18412 = 3.141592439 (0.0000068079% err) [7]
58198 / 18525 = 3.141592442 (0.0000067146% err) [7]
58553 / 18638 = 3.141592445 (0.0000066224% err) [7]
58908 / 18751 = 3.141592448 (0.0000065313% err) [7]
59263 / 18864 = 3.141592451 (0.0000064413% err) [7]
59618 / 18977 = 3.141592454 (0.0000063524% err) [7]
59973 / 19090 = 3.141592456 (0.0000062645% err) [7]
60328 / 19203 = 3.141592459 (0.0000061777% err) [7]
60683 / 19316 = 3.141592462 (0.0000060919% err) [7]
61038 / 19429 = 3.141592464 (0.0000060071% err) [7]
61393 / 19542 = 3.141592467 (0.0000059232% err) [7]
61748 / 19655 = 3.141592470 (0.0000058404% err) [7]
62103 / 19768 = 3.141592472 (0.0000057584% err) [7]
62458 / 19881 = 3.141592475 (0.0000056774% err) [7]
62813 / 19994 = 3.141592477 (0.0000055974% err) [7]
63168 / 20107 = 3.141592480 (0.0000055182% err) [7]
63523 / 20220 = 3.141592482 (0.0000054399% err) [7]
63878 / 20333 = 3.141592485 (0.0000053625% err) [7]
64233 / 20446 = 3.141592487 (0.0000052859% err) [7]
64588 / 20559 = 3.141592489 (0.0000052102% err) [7]
64943 / 20672 = 3.141592492 (0.0000051353% err) [7]
65298 / 20785 = 3.141592494 (0.0000050612% err) [7]
65653 / 20898 = 3.141592496 (0.0000049879% err) [7]
66008 / 21011 = 3.141592499 (0.0000049154% err) [7]
66363 / 21124 = 3.141592501 (0.0000048437% err) [7]
66718 / 21237 = 3.141592503 (0.0000047728% err) [7]
67073 / 21350 = 3.141592505 (0.0000047026% err) [7]
67428 / 21463 = 3.141592508 (0.0000046331% err) [7]
67783 / 21576 = 3.141592510 (0.0000045643% err) [7]
68138 / 21689 = 3.141592512 (0.0000044963% err) [7]
68493 / 21802 = 3.141592514 (0.0000044290% err) [7]
68848 / 21915 = 3.141592516 (0.0000043624% err) [7]
69203 / 22028 = 3.141592518 (0.0000042965% err) [7]
69558 / 22141 = 3.141592520 (0.0000042312% err) [7]
69913 / 22254 = 3.141592522 (0.0000041666% err) [7]
70268 / 22367 = 3.141592524 (0.0000041026% err) [7]
70623 / 22480 = 3.141592526 (0.0000040393% err) [7]
70978 / 22593 = 3.141592528 (0.0000039767% err) [7]
71333 / 22706 = 3.141592530 (0.0000039146% err) [7]
71688 / 22819 = 3.141592532 (0.0000038532% err) [7]
72043 / 22932 = 3.141592534 (0.0000037923% err) [7]
72398 / 23045 = 3.141592536 (0.0000037321% err) [7]
72753 / 23158 = 3.141592538 (0.0000036725% err) [7]
73108 / 23271 = 3.141592540 (0.0000036134% err) [7]
73463 / 23384 = 3.141592541 (0.0000035549% err) [7]
73818 / 23497 = 3.141592543 (0.0000034970% err) [7]
74173 / 23610 = 3.141592545 (0.0000034396% err) [7]
74528 / 23723 = 3.141592547 (0.0000033828% err) [7]
74883 / 23836 = 3.141592549 (0.0000033265% err) [7]
75238 / 23949 = 3.141592550 (0.0000032707% err) [7]
75593 / 24062 = 3.141592552 (0.0000032155% err) [7]
75948 / 24175 = 3.141592554 (0.0000031608% err) [7]
76303 / 24288 = 3.141592555 (0.0000031065% err) [7]
76658 / 24401 = 3.141592557 (0.0000030528% err) [7]
77013 / 24514 = 3.141592559 (0.0000029996% err) [7]
77368 / 24627 = 3.141592561 (0.0000029469% err) [7]
77723 / 24740 = 3.141592562 (0.0000028947% err) [7]
78078 / 24853 = 3.141592564 (0.0000028429% err) [7]
78433 / 24966 = 3.141592565 (0.0000027916% err) [7]
78788 / 25079 = 3.141592567 (0.0000027407% err) [7]
79143 / 25192 = 3.141592569 (0.0000026904% err) [7]
79498 / 25305 = 3.141592570 (0.0000026404% err) [7]
79853 / 25418 = 3.141592572 (0.0000025909% err) [7]
80208 / 25531 = 3.141592573 (0.0000025419% err) [7]
80563 / 25644 = 3.141592575 (0.0000024933% err) [7]
80918 / 25757 = 3.141592576 (0.0000024451% err) [7]
81273 / 25870 = 3.141592578 (0.0000023973% err) [7]
81628 / 25983 = 3.141592579 (0.0000023500% err) [7]
81983 / 26096 = 3.141592581 (0.0000023030% err) [7]
82338 / 26209 = 3.141592582 (0.0000022565% err) [7]
82693 / 26322 = 3.141592584 (0.0000022103% err) [7]
83048 / 26435 = 3.141592585 (0.0000021646% err) [7]
83403 / 26548 = 3.141592587 (0.0000021192% err) [7]
83758 / 26661 = 3.141592588 (0.0000020743% err) [7]
84113 / 26774 = 3.141592589 (0.0000020297% err) [7]
84468 / 26887 = 3.141592591 (0.0000019854% err) [7]
84823 / 27000 = 3.141592592 (0.0000019416% err) [7]
85178 / 27113 = 3.141592593 (0.0000018981% err) [7]
85533 / 27226 = 3.141592595 (0.0000018550% err) [7]
85888 / 27339 = 3.141592596 (0.0000018122% err) [7]
86243 / 27452 = 3.141592597 (0.0000017698% err) [7]
86598 / 27565 = 3.141592599 (0.0000017278% err) [7]
86953 / 27678 = 3.1415926006 (0.0000016860% err) [8]
87308 / 27791 = 3.1415926019 (0.0000016447% err) [8]
87663 / 27904 = 3.1415926032 (0.0000016036% err) [8]
88018 / 28017 = 3.1415926044 (0.0000015629% err) [8]
88373 / 28130 = 3.1415926057 (0.0000015225% err) [8]
88728 / 28243 = 3.1415926070 (0.0000014824% err) [8]
89083 / 28356 = 3.1415926082 (0.0000014427% err) [8]
89438 / 28469 = 3.1415926095 (0.0000014033% err) [8]
89793 / 28582 = 3.1415926107 (0.0000013641% err) [8]
90148 / 28695 = 3.1415926119 (0.0000013253% err) [8]
90503 / 28808 = 3.1415926131 (0.0000012868% err) [8]
90858 / 28921 = 3.1415926143 (0.0000012486% err) [8]
91213 / 29034 = 3.1415926155 (0.0000012107% err) [8]
91568 / 29147 = 3.1415926167 (0.0000011731% err) [8]
91923 / 29260 = 3.1415926179 (0.0000011358% err) [8]
92278 / 29373 = 3.1415926190 (0.0000010987% err) [8]
92633 / 29486 = 3.1415926202 (0.0000010620% err) [8]
92988 / 29599 = 3.1415926213 (0.0000010255% err) [8]
93343 / 29712 = 3.1415926225 (0.0000009893% err) [8]
93698 / 29825 = 3.1415926236 (0.0000009534% err) [8]
94053 / 29938 = 3.1415926247 (0.0000009177% err) [8]
94408 / 30051 = 3.1415926258 (0.0000008824% err) [8]
94763 / 30164 = 3.1415926269 (0.0000008473% err) [8]
95118 / 30277 = 3.1415926280 (0.0000008124% err) [8]
95473 / 30390 = 3.1415926291 (0.0000007778% err) [8]
95828 / 30503 = 3.1415926302 (0.0000007435% err) [8]
96183 / 30616 = 3.1415926313 (0.0000007094% err) [8]
96538 / 30729 = 3.1415926323 (0.0000006755% err) [8]
96893 / 30842 = 3.1415926334 (0.0000006420% err) [8]
97248 / 30955 = 3.1415926344 (0.0000006086% err) [8]
97603 / 31068 = 3.1415926355 (0.0000005755% err) [8]
97958 / 31181 = 3.1415926365 (0.0000005427% err) [8]
98313 / 31294 = 3.1415926375 (0.0000005100% err) [8]
98668 / 31407 = 3.1415926385 (0.0000004777% err) [8]
99023 / 31520 = 3.1415926395 (0.0000004455% err) [8]
99378 / 31633 = 3.1415926406 (0.0000004136% err) [8]
99733 / 31746 = 3.1415926415 (0.0000003819% err) [8]
100088 / 31859 = 3.1415926425 (0.0000003504% err) [8]
100443 / 31972 = 3.1415926435 (0.0000003192% err) [8]
100798 / 32085 = 3.1415926445 (0.0000002881% err) [8]
101153 / 32198 = 3.1415926455 (0.0000002573% err) [8]
101508 / 32311 = 3.1415926464 (0.0000002267% err) [8]
101863 / 32424 = 3.1415926474 (0.0000001963% err) [8]
102218 / 32537 = 3.1415926483 (0.0000001661% err) [8]
102573 / 32650 = 3.1415926493 (0.0000001362% err) [8]
102928 / 32763 = 3.14159265025 (0.00000010644% err) [9]
103283 / 32876 = 3.14159265117 (0.00000007689% err) [9]
103638 / 32989 = 3.14159265210 (0.00000004754% err) [9]
103993 / 33102 = 3.14159265301 (0.00000001839% err) [10]
104348 / 33215 = 3.14159265392 (0.00000001055% err) [10] <
208341 / 66317 = 3.14159265347 (0.000000003894% err) [10]
312689 / 99532 = 3.14159265362 (0.0000000009276% err) [10] <
833719 / 265381 = 3.141592653581 (0.0000000002774% err) [12]
1146408 / 364913 = 3.1415926535914 (0.00000000005127% err) [11] <
3126535 / 995207 = 3.1415926535886 (0.00000000003637% err) [12]
4272943 / 1360120 = 3.1415926535894 (0.000000000012863% err) [13]
5419351 / 1725033 = 3.14159265358981 (0.0000000000007068% err) [13] <<
The entries with the '<' signs are particularly interesting because of the ratio of added precision over increase of denominator.
Aside from some mathematical trivia, generally a good use of approximation of pi would be for the memorization of a smaller number of digits than the approximation can give. For this, only 355/113 is useful.
Another use is integer math. For example, if you use integer math with 32 bit numbers to calculate the circumference of an object, and the maximum diameter of that object is 130000 units, then the max denominator would be, 2^32/130000 = 33038. Then the best approximate fraction you can use, would be 103638/32989.
Currently the article mentions 125648/39995 as a fraction that produces 8 correct digits. This is not wrong, but it's not useful. There are at least 45 better fractions that do the same, and use smaller denominators. And half of then are more accurate. So I am replacing 125648/39995 with 99733/31746 which is more accurate and needs less digits. Dhrm77 (talk) 16:15, 18 August 2022 (UTC)

...if I may..I find it difficult to memorize any of those fractions after 355/311.

however, I'd like to go the other way and suggest the following approximation:

( 355 -3015E-8 ) / 113

which yields Pi accurately to 10 decimal places..if you need that much accuracy, but

trying to find it is always fun.

my calculator shows the result to be 3.1415926535 (39623)

qed Criticatlarge (talk) 02:58, 15 February 2023 (UTC)

22/7 is definitely ancient

"Approximations" of pi are mostly best geometry, rational exhaustion.

Here that's "at least as mucn precision, ..., as 7 digits of pi". 97.113.48.144 (talk) 05:24, 14 October 2022 (UTC)

approximations based on ${\sqrt {2}}$ and ${\sqrt {3}}$

Based on a recent addition, this approximation: ${\sqrt {2}}+{\sqrt {3}}+{\frac {{\sqrt {2}}-{\sqrt {3}}-18}{3921}}=3.141592644\ 0^{+}$ is accurate to 8 digits. But I don't think it's worth adding to the article. Dhrm77 (talk) 11:12, 30 May 2023 (UTC)

Babylonian and Egypt Pi?

"one Old Babylonian mathematical tablet excavated near Susa in 1936 (dated to between the 19th and 17th centuries BCE) gives a better approximation of π as ²⁵⁄₈ = 3.125, about 0.528% below the exact value.

At about the same time, the Egyptian Rhind Mathematical Papyrus (dated to the Second Intermediate Period, c. 1600 BCE, although stated to be a copy of an older, Middle Kingdom text) implies an approximation of π as ²⁵⁶⁄₈₁ ≈ 3.16 (accurate to 0.6 percent) by calculating the area of a circle via approximation with the octagon."

The problem is that neither of those cultures had yet a concept of pi as either circumference/diameter or as area/(radius^2).

For the babylonians, they have a tablet that basically says that the circumference of a circle is 25/24 multiplied by the perimeter of the inscribed regular hexagon. So if the circle has diameter=1, the side of the hexagon is 0.5 and the perimeter of the hexagon is 3 so the circumference of the circle would be 25/24*3=25/8=3 1/8. So this is a formula for circumference of a circle, basically 25/8 * diameter, so it is not totally wrong to say 'by implication treats pi as 25/8".

But for Egypt, this is much more of a stretch. They have a formula for the area of a circle which is A=(D-D/9)^2. It is a great formula, but to say "treats pi as 256/81" is really not accurate. While it is true that this formula could be written as A=(2r-2r/9)^2=(16r/9)^2=256/81*r^2 it is not accurate to say that it treated pi as 256/81.

I think it would be better to just say that these cultures had formulas for circumference and area which are equivalent to the formulas C=(25/8)D and A=(256/81)r^2 so it is like they had values for pi, but it wasn't like they were using the formulas C=pi*D and A=pi*r^2 and they were trying to use the best approximation of pi they could think of.

Might there be a simple way to edit this so that it is more accurate and does not claim that these cultures were aware there there was this constant pi, but not to make it too complicated to explain? Nymathteacher (talk) 20:59, 22 August 2023 (UTC)

Borwein's approximation

Ramanujan's approximation in his 1914 paper:

$\pi \approx \pi _{1}(n)={\frac {3}{(1-R){\sqrt {n}}}},\quad R={\frac {1-{\frac {3}{\pi {\sqrt {n}}}}-24\sum _{r=1}^{\infty }{\frac {r}{\exp(2\pi r{\sqrt {n}})-1}}}{1-24\sum _{r=1}^{\infty }{\frac {2r-1}{\exp(\pi (2r-1){\sqrt {n}})+1}}}}$

is valid when n is odd. For example,

$\pi _{1}(25)={\frac {9}{5}}+{\sqrt {\frac {9}{5}}}$

is a simple approximation, but

$\pi _{1}(58)={\frac {1037785473+70101072{\sqrt {2}}+192518946{\sqrt {29}}+311451846{\sqrt {58}}}{1446914567}}$

is complicated. The Borwein's brothers mention the following approximation in their book.

$\pi _{4}(58)={\frac {66{\sqrt {2}}}{33{\sqrt {29}}-148}}$

where

$\pi _{4}(n)={\frac {6}{(1-S){\sqrt {n}}}},\quad S={\frac {1-{\frac {6}{\pi {\sqrt {n}}}}-24\sum _{r=1}^{\infty }{\frac {r}{\exp(\pi r{\sqrt {n}})-1}}}{1+24\sum _{r=1}^{\infty }{\frac {2r-1}{\exp(\pi (2r-1){\sqrt {n}})-1}}}}$

is valid when n is even. I added this approximation to the article. Nei.jp (talk) 21:59, 21 October 2023 (UTC)

Miscellaneous approximations

This section has become a magnet for the insertion of ad hoc approximations of a few decimal places that anyone can dream up. I recommend removing the section altogether per WP:SPAMBAIT.—Anita5192 (talk) 22:14, 21 March 2024 (UTC)