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Today is Pi Approximation Day. Well, at least in countries that use DD/MM date format, like mine.
As a small celebration, here’s a \(\pi\) approximation good to 16 significant digits. That’s just about the same number of digits required to write it. No big deal, except perhaps for the mnemonic part.

\[\frac{22}{7}-\frac{1}{790+\frac{55567}{66697}}\]
Hi Gerson,

Very good !

Have a good day.
Have an approximately good day!
≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈≈ !
Very nice numbers in there Gerson. You must be quite satisfied to find numbers with repeating digits and ending in both 67 and 97 as components of the approximation. It may not actually be Pi, but most of my calculators can't tell the difference. Wink
Thank you all for your comments!

Yes, I am having an approximately good day (I wish the numbers on the callendar weren't a reminder I am getting older -- 58 and counting :-)

Now, time for a little riddle.

The following appears to be a pretty bad approximation. It really is, depending on how we look at it. However, when I change only one digit or, equivalently, when I remove one of its parts, it returns a perfect 10-digit result on my HP-41C, which I am using to evaluate it. BTW, I have used the HP-41C for this one because of its 10-digit display and a seldom used useful built-in function which most of my Voyagers lack. Too many tips, but it doesn't matter :-)

Have fun!

\(\frac{\frac{26}{7}-\frac{6}{11211}}{\left (\frac{4141}{3313}\right ) ^{\frac{3}{4}}}\)

Edited to include a pair of parentheses in order to avoid ambiguity
(07-22-2019 03:14 PM)Gerson W. Barbosa Wrote: [ -> ]However, when I change only one digit or, equivalently, when I remove one of its parts, it returns a perfect 10-digit result on my HP-41C,

In addition to this change, something else has to be assumed. Sorry for the omission.
I got the simplest formula:

3

Sorry, I'm an engineer, that's enough precision most of the time...
Perfect! Just for fun I put it on my HP-38C simulator on iPhone:

01 - 2 2
02 - 2 2
03 - 31 ENTER
04 - 7 7
05 - 71 ÷
06 - 5 5
07 - 5 5
08 - 5 5
09 - 6 6
10 - 7 7
11 - 31 ENTER
12 - 6 6
13 - 6 6
14 - 6 6
15 - 9 9
16 - 7 7
17 - 71 ÷
18 - 7 7
19 - 9 9
20 - 0 0
21 - 51 +
22 - 24 71 1/x
23 - 41 −
24 - 25 7 00 GTO 00
 
Hi, all:

Happy Pi Approximation Day (i.e.: 22/7)

Use your CAS (or your math ingenuity) to compute the exact symbolic value (not numeric) of:

          Integral between 0 and 1 of:     x4(1-x4)/(1+x2) . dx

and there you are, your Pi Approximation present. Smile

Regards to all.
V.
 
(07-22-2019 10:49 PM)Valentin Albillo Wrote: [ -> ] 
...your Pi Approximation present. Smile
 

Thank you, Valentin!

Now, let’s unwrap it:

On the HP-49G, in exact mode,

'∫(0,1,X^4*(1-X)^4/(1+X^2),X)'

EVAL

—> '-((7*π-22)/7)'

→NUM

—> 1.26448927143E-3

= 22/7 - π
(07-22-2019 07:48 PM)Claudio L. Wrote: [ -> ]I got the simplest formula:

3

Sorry, I'm an engineer, that's enough precision most of the time...

As long as you construct square buildings only!
(07-22-2019 10:49 PM)Valentin Albillo Wrote: [ -> ] 
Hi, all:

Happy Pi Approximation Day (i.e.: 22/7)

Use your CAS (or your math ingenuity) to compute the exact symbolic value (not numeric) of:

          Integral between 0 and 1 of:     x4(1-x4)/(1+x2) . dx

and there you are, your Pi Approximation present. Smile

Regards to all.
V.
 

I got 2/35 from an (emulated) HP 50g.
(07-23-2019 05:07 PM)Dave Shaffer Wrote: [ -> ]
(07-22-2019 07:48 PM)Claudio L. Wrote: [ -> ]I got the simplest formula:

3

Sorry, I'm an engineer, that's enough precision most of the time...

As long as you construct square buildings only!

Or a molten sea (1 Kings 7:23).
(07-23-2019 03:15 PM)Gerson W. Barbosa Wrote: [ -> ]
(07-22-2019 10:49 PM)Valentin Albillo Wrote: [ -> ] 
...your Pi Approximation present. Smile
 

Thank you, Valentin!

Now, let’s unwrap it:

On the HP-49G, in exact mode,

'∫(0,1,X^4*(1-X)^4/(1+X^2),X)'

EVAL

—> '-((7*π-22)/7)'

→NUM

—> 1.26448927143E-3

= 22/7 - π

On the 50G: (noting Gerson's slight modification)

[attachment=7502]

EVAL

[attachment=7503]

FDIST

[attachment=7504]
(07-23-2019 05:31 PM)BartDB Wrote: [ -> ]FDIST

FDISTRIB... That’s what I was looking for. It’s available on the 49G as well. Thanks!

Alternatively,

'22/7-∫(0,1,X^4*(1-X)^4/(1+X^2),X)'

EVAL

—> 'π'


That is,

π = 22/7 - ∫(0,1,X^4*(1-X)^4/(1+X^2),X)
(07-23-2019 05:09 PM)ijabbott Wrote: [ -> ]
(07-22-2019 10:49 PM)Valentin Albillo Wrote: [ -> ] 
Hi, all:

Happy Pi Approximation Day (i.e.: 22/7)

Use your CAS (or your math ingenuity) to compute the exact symbolic value (not numeric) of:

          Integral between 0 and 1 of:     x4(1-x4)/(1+x2) . dx

and there you are, your Pi Approximation present. Smile

Regards to all.
V.
 

I got 2/35 from an (emulated) HP 50g.

Hi Valentin,

wondering that you didn't put the "old" HP71b code in your post Smile So I'll do that.

Code:
PI = 22/7-INTEGRAL(0,1,1E-11,(IVAR^4*(1-IVAR)^4/(1+IVAR^2)))

best regards
Erwin
(07-22-2019 07:48 PM)Claudio L. Wrote: [ -> ]I got the simplest formula:

3

Sorry, I'm an engineer, that's enough precision most of the time...

“Pi is about 22/7”, says the engineer in this math jokes page. Roman engineers would say it was close to 25/8. The builders of King Solomon’s Temple knew it was a bit great than 3, I think, but apparently the reporters believed it was exactly 3.

I’ve read 3.1416 is good enough in Mechanical Engineering. Other types of engineering might use other aproximate values for pi, though. I remember 377 was commonly used as an approximation for 2\(\pi\times\)60 in EE texbooks, for convenience.
(07-24-2019 12:35 AM)Gerson W. Barbosa Wrote: [ -> ]I remember 377 was commonly used as an approximation for 2\(\pi\times\)60 in EE texbooks, for convenience.

Indeed... The good old 2πf we used in EE formulas... In Mozambique we used 2*3,14*50 though.
(07-23-2019 06:18 PM)Gerson W. Barbosa Wrote: [ -> ]That is,

π = 22/7 - ∫(0,1,X^4*(1-X)^4/(1+X^2),X)


I entered this into my Sharp Writeview EL-W506T and got the result of 'π'
Admittedly I expected the numerical value of π
[attachment=7522]

EDIT: when I enter the numerical value of pi to more than 10 digits (correctly rounded) it will convert to the symbol 'π'
(07-24-2019 10:30 AM)BartDB Wrote: [ -> ]
(07-23-2019 06:18 PM)Gerson W. Barbosa Wrote: [ -> ]That is,

π = 22/7 - ∫(0,1,X^4*(1-X)^4/(1+X^2),X)


I entered this into my Sharp Writeview EL-W506T and got the result of 'π'
Admittedly I expected the numerical value of π

Same on the CASIO fx-991 LA X CLASSWIZ.
There’s a setting for numerical values:

SHIFT SETUP
1: Input/Output
2: MathI/DecimalO

Regardless the setting, the S<=>D key cycles between both formats.

On the EL-W506T you can try the ->CHANGE<- key.
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