Re: [math-fun] a numerical approximation close to 7 digit pi
----- 355/113 is 6.57 decimal digits ----- Please explain how the / is .57 digits. (More seriously: Where does the 6.57 come from?) —Dan
Simon is scoring the accuracy of the approximation: log_10 |pi-355/113| is -6.57... Scoring the 'complexity' or 'information' of the approximating expression seems to be a matter of taste. Rich ----------- Quoting Dan Asimov <dasimov@earthlink.net>:
----- 355/113 is 6.57 decimal digits -----
Please explain how the / is .57 digits.
(More seriously: Where does the 6.57 come from?)
?Dan
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Hello mr Asimov, 6.57 comes from Pi-355/113 = 0.266764189e-6 and log(0.266764189)/log(10) = 6.57 in absolute value, so 355/113 is 6.57 has 6.57 exact decimals compared to Pi. best regards, simon Le 2018-02-17 à 05:13, Dan Asimov a écrit :
----- 355/113 is 6.57 decimal digits -----
Please explain how the / is .57 digits.
(More seriously: Where does the 6.57 come from?)
—Dan
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My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494. -- Gene
There is also (77729/254)^(1/5) = 3.1415926541114871, with log10 of relative error -9.78 On Sat, Feb 17, 2018 at 11:14 AM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494.
-- Gene
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Just as a rule of thumb, the number of digits in the arbitrary constants on the left side ought to be smaller than the negative of the log10 of the relative error. So Eugene Salamin's example is more interesting than James Buddenhagen's, because 8 < 9.494, but 10 > 9.78. By this rule, 355/113 is barely interesting, because 6 < 6.57. On Sat, Feb 17, 2018 at 4:03 PM, James Buddenhagen <jbuddenh@gmail.com> wrote:
There is also (77729/254)^(1/5) = 3.1415926541114871, with log10 of relative error -9.78
On Sat, Feb 17, 2018 at 11:14 AM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494.
-- Gene
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I sometimes think we are spoiled by modern computational accuracy. If you draw a circle using 355/113 instead of pi, and the circumference is 1 inch off, the circle is over 59 miles in diameter. I suspect New York City would easily fit inside that circle.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Allan Wechsler Sent: Saturday, February 17, 2018 5:30 PM To: math-fun Subject: Re: [math-fun] a numerical approximation close to 7 digit pi
Just as a rule of thumb, the number of digits in the arbitrary constants on the left side ought to be smaller than the negative of the log10 of the relative error. So Eugene Salamin's example is more interesting than James Buddenhagen's, because 8 < 9.494, but 10 > 9.78. By this rule, 355/113 is barely interesting, because 6 < 6.57.
On Sat, Feb 17, 2018 at 4:03 PM, James Buddenhagen <jbuddenh@gmail.com> wrote:
There is also (77729/254)^(1/5) = 3.1415926541114871, with log10 of relative error -9.78
On Sat, Feb 17, 2018 at 11:14 AM, Eugene Salamin via math-fun < math-fun@mailman.xmission.com> wrote:
My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494.
-- Gene
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On 17/02/2018 17:14, Eugene Salamin via math-fun wrote:
My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494.
I was at a mathematics conference once, and at the start we were told that the combination to get into some room was 2143. "Easy to remember", I said. "Nearest integer to 22 pi^4". Everyone within earshot looked at me as if I had two heads. I was most disappointed. -- g
Here is a nice one too, 689/396/log(689/396) = 3.141592595088355621194, The relative error is 7.232. Simon Plouffe Le 2018-02-19 à 00:59, Gareth McCaughan a écrit :
On 17/02/2018 17:14, Eugene Salamin via math-fun wrote:
My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494.
I was at a mathematics conference once, and at the start we were told that the combination to get into some room was 2143. "Easy to remember", I said. "Nearest integer to 22 pi^4". Everyone within earshot looked at me as if I had two heads. I was most disappointed.
Here's an even better approximation. 4 arctan 1. It's exact. -- Gene On Sunday, February 18, 2018, 7:19:16 PM PST, Simon Plouffe <simon.plouffe@gmail.com> wrote: Here is a nice one too, 689/396/log(689/396) = 3.141592595088355621194, The relative error is 7.232. Simon Plouffe Le 2018-02-19 à 00:59, Gareth McCaughan a écrit :
On 17/02/2018 17:14, Eugene Salamin via math-fun wrote:
My favorite is (2143/22)^(1/4) = 3.1415926525826463. Log10 of relative error is -9.494.
I was at a mathematics conference once, and at the start we were told that the combination to get into some room was 2143. "Easy to remember", I said. "Nearest integer to 22 pi^4". Everyone within earshot looked at me as if I had two heads. I was most disappointed.
participants (8)
-
Allan Wechsler -
Dan Asimov -
David Wilson -
Eugene Salamin -
Gareth McCaughan -
James Buddenhagen -
rcs＠xmission.com -
Simon Plouffe