A Few Math Jocks Might Appreciate This...

[Code Doc]

If interested, please do the following:

(1) Open the Windows Calculator. Set View to Scientific. Got it? If so, proceed.

(2) Click on the Pi key.

(3) Click on the M+ key to store it in memory.

(4) Enter the following integer: 5419351

(5) Press the / key.

(6) Enter the following integer: 1725033

(7) Press the = Key.

(8) Press the - key.

(9) Press the MR key.

(10) Press the = key.

Do you see a rather small number on the screen? How small is it and what does this mean?

-------------------
Originally posted in a different fashion by Anhn on another VbForums thread.

[wy125]

it's about 2e-14. That's the error in the estimate of pi you got by dividing those two numbers. It's interesting because it's correct to 13 places but of course the number of digits in those numbers is 14 so you're not gaining anything.

[NickThissen]

Indeed. 5419351 / 1725033 is an approximation of pi, just like 22/7 (only much more precise).

If 5419351 / 1725033 is almost pi, then that number minus pi is almost zero.

[Code Doc]

Indeed. 5419351 / 1725033 is an approximation of pi, just like 22/7 (only much more precise).

If 5419351 / 1725033 is almost pi, then that number minus pi is almost zero.

Then somebody went to work and did this for us:

0.000000000000022144779300394027945726930849601

How on earth did he crank out that many digits to the right of the decimal?

[NickThissen]

You can theoretically crank out as many digits as you want as long as you know both pi and 5419351/1725033 to at least that many digits, and assuming you have to computational power to calculate it.

[Logophobic]

How on earth did he crank out that many digits to the right of the decimal?

My Windows Calculator gives exactly that many digits:

5419351 / 1725033 - pi = 2.2144779300394027945726930849601e-14

I'm guessing your version is only displaying 16 digits?

[dbasnett]

when using VB .Net and Math.Pi I get
5419351 / 1725033 - pi = 0.000000000000030198066
or
5419351 / 1725033 - pi = 3.01980662698043E-14

[NickThissen]

Be aware though that you can't be sure that that many digits are correct. Computers all have a lower limit on the smallest number they can calculate with, often called e (actually epsilon but I cannot type that ).

This means that, for a computer, 1 + e = 1.
In other words, adding e (or smaller) to any number is 'not recognized'; the computer cannot calculate that anymore.

In most computers, e is around 3E-16 so the large number of digits windows calculator chucks out may be incorrect (noise) after 16 digits orso.


You can get a rough approximation for e using something like this:

Dim eps As Double = 1
        Dim x As Double = 1 + eps
 
        Do While x > 1
            x = 1 + eps
            eps = eps / 10
        Loop
 
        MessageBox.Show(CStr(eps))

My computer tells me eps = 1E-17.

[Logophobic]

Windows Calculator uses arbitrary precision, which is very much unlike the intrinsic variable types of Visual Basic. Its value for pi is accurate to 46 digits*, and it can calculate 5419351 / 1725033 far beyond that. The 32-digit precision it displays is correct.

*Windows Calculator gives this result:
pi - 3.141592653589793238462643383279 = 5.0288419716939931148196659300057e-31

The actual value, with as many digits, is:
pi - 3.141592653589793238462643383279 = 5.0288419716939937510582097494459e-31

[jemidiah]

Then somebody went to work and did this for us:

0.000000000000022144779300394027945726930849601

How on earth did he crank out that many digits to the right of the decimal?

If you google either the numerator or denominator in that fraction you can find some programs that generated this approximation. Since the numbers are so large, the probability of finding an unrelated page to the pi approximation is low enough that it doesn't really happen.

[Code Doc]

A few more "magic Pi" integer pairs with even more accuracy:

80143857 / 25510582 = 3.14159265358979267191
(Error = 0.00000000000000044409)

245850922 / 78256779 = 3.14159265358979311600
(Error = 0.00000000000000000000) Gasp!

[Code Doc]

And now for the grand finale:

2646693125139304345 / 842468587426513207 = 3.14159265358979323846264338327950288418

error = 0.00000000000000000000000000000000000001

[NickThissen]

I just come up with this one myself:
5293386250278608690 / 1684937174853026414

Or
326750146457197957216320 / 104007801929327614483392

(They are just multiples of your last one )

[Code Doc]

I just come up with this one myself:
5293386250278608690 / 1684937174853026414

Or
326750146457197957216320 / 104007801929327614483392

(They are just multiples of your last one )

Nick, did you use the Windows calculator or did you program your computer to obtain the results in VB6 or any other language? And, what do you mean by "Multiples". Multiples of what?

You see, Nick, Excel and VB6 cannot reach more than 15 to the right of the decimal. Therefore, how does anyone know that your "magic Pi" integers or the pair that I just posted are valid? Just curious. I think Logophobic already hinted at the answer.

[Logophobic]

Nick's fractions are equivalent to your final fraction. The numerator and denominator are each multiplied by 2 in the first and 123456 in the second.

As for your ratios, I have taken this a bit further using an iterative process that produces the same results you have posted.

Iteration 5: 5419351 / 1725033
Iteration 6: 80143857 / 25510582
Iteration 7: 245850922 / 78256779

Iteration 22: 2646693125139304345 / 842468587426513207

Iteration 35: 18644210947563865148979297792 / 5934636664705637943635533097

I₃₅ = 3.14159265358979323846264338327950288419716939937510582096659008

pi - I₃₅ = 0.00000000000000000000000000000000000000000000000000000000835451 = 8.35451E-57

[jemidiah]

You see, Nick, Excel and VB6 cannot reach more than 15 to the right of the decimal. Therefore, how does anyone know that your "magic Pi" integers or the pair that I just posted are valid? Just curious. I think Logophobic already hinted at the answer.

There are various infinite- or high-precision calculators in existence. The PowerToy Calc (XP) offers up to 512 bits of precision under Advanced Options, or around 154 decimal digits:

3.1415926535897932384626433
832795028841971693993751058
209749445923078164062862089
986280348253421170679821480
865132823066470938446095505
822317253594081284811174502
841027019385211055596446229
489549303819644288109756659
334461284756482337867831652
712019091456485669234603486
104543266482133936072602491
412737245870066063155881748
815209209628292540917153643
678925903600113305305488204
665213841469519415116094330
572703657595919530921861173
819326117931051185480744623
799627495673518857527248912
279381830119491298336733624

[Kekeb]

It's funny to me that the PowerToy Calculator would bother to go to that many digits. It's essentially over-kill at that point

Elementary applications, such as calculating the circumference of a circle, will rarely require more than a dozen decimal places. For example, a value truncated to 11 decimal places is accurate enough to calculate the circumference of a circle the size of the earth with a precision of a millimeter, and one truncated to 39 decimal places is sufficient to compute the circumference of any circle that fits in the observable universe to a precision comparable to the size of a hydrogen atom.

[jemidiah]

Well, the PowerToy Calculator just has an extended-extended precision mode, and since it has a few pre-defined constants the creators must not have wanted to give a false value for, they included the extra-extra precision on pi in that mode. I agree it's completely unnecessary in almost all real-world applications. Highly iterative algorithms might need all of that precision--maybe.

To be honest a huge part of math ends up being edge cases nobody in their right mind would ever encounter, except a mathematician looking for a counterexample to some theorem. The real numbers themselves are an example of this--realistically, you only need a finite amount of precision to calculate anything in the real world to an accuracy where quantum uncertainty takes over. Regardless, the irrationals are still very important theoretically.

See this and this for examples of math's silliness. The hyperreals are awesome but nuts; unmeasurable sets are just nuts.

[ForumAccount]

Pi: To a million decimal places

[Code Doc]

Pi: To a million decimal places

BUT, how could anyone ever prove that he was wrong (or right)?

Finally, that picture is so ugly that I have a feeling this is a complete hoax.

[jemidiah]

BUT, how could anyone ever prove that he was wrong (or right)?

There are infinite series expansions for pi that should converge quickly enough to do 1,000,000 digits. The error term after you take some number of terms is provably small, so you can be sure you have the first _x_ digits correct.