Today I Learned

[Stephanus]

63! is 88 digits
1,982,608,315,404,440,064,116,146,708,361,898,137,544,773,690,227,268,628,106,279,599,612,729,753,600,000,000,000,000

64! is 89 digits
126,886,932,185,884,164,103,433,389,335,161,480,802,865,516,174,545,192,198,801,894,375,214,704,230,400,000,000,000,000

I think X! = 1.2E89 is somewhere between 63! and 64!

 

[OmCheeto]

63! is 88 digits
1,982,608,315,404,440,064,116,146,708,361,898,137,544,773,690,227,268,628,106,279,599,612,729,753,600,000,000,000,000

64! is 89 digits
126,886,932,185,884,164,103,433,389,335,161,480,802,865,516,174,545,192,198,801,894,375,214,704,230,400,000,000,000,000

I think X! = 1.2E89 is somewhere between 63! and 64!

According to my calculator, 64! ≈ 1.27E89

hmmm... I wonder if factorials would be a more convenient way to memorize large numbers.

Oh, now this is weird:

Stirling's approximation
n! \approx \sqrt[]{2\pi n} \big(\frac{n}{e}\big) ^{n}

e = \sum_{n=0}^\infty \frac{1}{n!}

I have no idea what that means. But I always find it weird when e & π get mixed up. I guess I don't know enough maths to know why that's not weird.

ps. TIL that "Numbers", which is the Mac spreadsheet, can calculate up to 170! ≈ 7.25E306

For n ≥ 171, I get the following message; "The formula contains a number outside the valid range."
So I guess I know the approximate valid range limit of "Numbers".

 

[Stephanus]

Stirling's approximation
n! \approx \sqrt[]{2\pi n} \big(\frac{n}{e}\big) ^{n}

e = \sum_{n=0}^\infty \frac{1}{n!}.

Wow, pi and e, how did this guy derive factorial from pi and e?

 

[Stephanus]

e = \sum_{n=0}^\infty \frac{1}{n!}

Ahh, the last formula, I forgot. It's how our teachers taught us how to find e, has nothing todo with finding factorial

 

[OmCheeto]

Ahh, the last formula, I forgot. It's how our teachers taught us how to find e, has nothing todo with finding factorial

Sorry about that. I included it, because I thought it was weird, that a constant, based on a factorial, was used to estimate, really big factorials.

​

One day, in the far distant future, I'll figure out, why maths, is so weird.
I'm sure there's a very reasonable explanation.

 

[mfb]

WolframAlpha can calculate factorials of very large numbers. An exact value of 200! ? No problem. An approximate value for (10^10)! ? No problem.

And it can link you to a proof of the stirling formula. And much more.

 

[OmCheeto]

WolframAlpha can calculate factorials of very large numbers. An exact value of 200! ? No problem. An approximate value for (10^10)! ? No problem.

And it can link you to a proof of the stirling formula. And much more.

As I said, it will be in the far distant future, when I can again comprehend recreational maths...

Speaking of "Wolves"

TIL, that Wolf 359, was not only the name of an epic battle in the Star Trek saga, and the name one of our nearest stars, but was also the title of a 1964 episode on The Outer Limits.

Closing narration
There is a theory that Earth and sun and galaxy and all the known universes are only a dust mote on some policeman's uniform in some gigantic super-world. Couldn't we be under some super-microscope, right now?​

My brother, who is an absolute movie geek, is always telling me, that "such and such" movie, is filled with cliches, and therefore, sucks.
After watching these retro-TV shows, I've discovered, that he is correct.
Not in that new movies suck, but, that history, is, really, the motto of my submarine.

The way I remember it; "What is past, is prologue."

Wiki leaves out the comma. A faux pax(-1sp for me ), imho.

"What's past is prologue" is a quotation by William Shakespeare from his play The Tempest. The phrase was originally used in The Tempest, Act 2, Scene I. Antonio uses it to suggest that all that has happened before that time, the "past," has led Sebastian and himself to this opportunity to do what they are about to do: commit murder.
In contemporary use, the phrase stands for the idea that history sets the context for the present. The quotation is engraved on the National Archives Building in Washington, D.C. and is commonly used by the military when discussing the similarities between war throughout history.

 

[nuuskur]

I learned that $\sum\limits_{k=0}^\infty \frac{1}{k^2} = \frac{\pi ^2}{6}$. Amazing what you can find out about through a trigonometric series.

 

[Stephanus]

Sorry about that. I included it, because I thought it was weird, that a constant, based on a factorial, was used to estimate, really big factorials.

​

One day, in the far distant future, I'll figure out, why maths, is so weird.
I'm sure there's a very reasonable explanation.

OmCheeto, would you please write that formula without graph? I can't grasp your graph.
Thanks.

 

[Stephanus]

WolframAlpha can calculate factorials of very large numbers. An exact value of 200! ? No problem. An approximate value for (10^10)! ? No problem.

Oh my God!. And in 1 second, too. What kind of method that they use??
If I run my method, it will take 1000 years! (or more?) And takes my entire RAM 4 x 10^9, but I think windows will switch to hard drive, but still 1000 years.

 

[mfb]

For (10^10)! ? The Stirling formula, of course.
For 200! I guess they have the result stored somewhere.

 

[SteamKing]

Oh my God!. And in 1 second, too. What kind of method that they use??
If I run my method, it will take 1000 years! (or more?) And takes my entire RAM 4 x 10^9, but I think windows will switch to hard drive, but still 1000 years.

It's not clear what method you are using to calculate 200!, nor why this computation would swallow up 4GB of RAM. But that's the secret surprise in studying numerical analysis: you learn to do more with less.

Clearly, using floating-point arithmetic would be a non-starter. The result WA gives for 200! seems to be expressed in a reasonably finite number of decimal digits.

This article:
http://en.wikipedia.org/wiki/Arbitrary-precision_arithmetic

presents the code of an algorithm for doing arbitrary-precision integer arithmetic, specifically geared to calculating factorials.

I suspect that WA has a similar routine built into its programming somewhere. There are other examples of arbitrary-precision arithmetic on the web.

 

[Stephanus]

Hi SteamKing, hi Mfb, hi everybody

It's not clear what method you are using to calculate 200!, nor why this computation would SWALLOW 4 GB

Hi SteamKing, glad to see you again after you answer my thread https://www.physicsforums.com/threads/neutron-star-temperature.814116/
My method? Actually it's a simple one.
The most significant and accurate floating point variable in Delphi (and I guess many programming languages) is 10 bytes real. Extended, in Delphi. long double in C.
So I have to make my own variable type: (I convert it to C, because many familiar in C more than Pascal)

[FONT=Courier New]char[2000] Number;
/* for 2000 digits base ten number,
   should use BCD style, but more complex. You can guess for (10^10)!
   which takes billions of digits, the array should be bigger */

int Carry;   // to store for example x = 6 * 4, then Carry = 2
int Modulus; // to store x = 6 * 4, then Modulus = 4
int NumLen;  // for 100 NumLen = 2, for 1000 NumLen = 3, etc...

void MultiplyNumberByX(int ByX); // you can guess
                                 // the rest of the function.

For (10^10)! ? The Stirling formula, of course.

≈[PLAIN]http://mathworld.wolfram.com/images/equations/StirlingsApproximation/Inline11.gif[PLAIN]http://mathworld.wolfram.com/images/equations/StirlingsApproximation/Inline13.gif
Well, glancing for a while from http://mathworld.wolfram.com/StirlingsApproximation.html, using that integral sign (and of course approximation symbol),
no mystery here that wolfram is very quick. But it's still an impressive software.
And natural number is impressive, too.
ln n! = ln 1 + ln 2 + ln 3.
Like OmCheeto said, the number that's derived from a sequence of factorial, can be used to find factorial itself.

Talking about natural number, do anybody know what other constant like Pi, Ln(1) sorry, my poor latex, and Golden ratio?
I mean constants that are not bound by physical law.
Supposed other aliens using a length unit. They wouldn't measure in Earth metric, right. They don't have to divide their length unit from the circumference of their planet by 40,000 (and why should 40,000 at all).
All their constant would be different from us. Gravity, Planck (if they do have someone named Planck, there), Avogadro, etc...
I think an advanced civilization will eventually suspect about Golden Ratio.

So, do anybody know what other constants other than Pi, Ln(1), and Golden Ratio, that are not bound by physical law? Their Pi, Golden Ratio, would still match ours even if they use different base number.
Is Planck constant theoretical or not?

Thanks.

 

[mfb]

There are many mathematical constants.
The planck units would be known, and all dimensionless physical constants (like the proton to electron mass ratio, for example).

But this is getting off-topic.

 

[zoobyshoe]

Back on topic here:

I made a list. The first column is just the uncalculated factorials from 1 to 69, which is the limit of my calculator. The second column is the corresponding number of digits for each factorial, and the third column is the difference between the number and the number of digits in its factorial. Like this:

12! (9) -3
13! (10) -3
14! (11) -3

The results were interesting and show the 22! 23! 24! triplet is inevitable. On the whole it looks like two different frequencies accelerating at different rates but which happen to "beat" right there. Generally it looks like the factorials of any two successive numbers will jump one order of magnitude from one to the next, but this is is increasingly interrupted by larger jumps in order of magnitude. The frequency of those jumps, and the amount of them, seemed to be accelerating in it's own right in a way that might be predictable, but my calculator started returning "error" results at 70!.

 

[mfb]

Remember that 20!=20*19!, for example.
If you multiply 1... to 4... by 20=10*2 you get one digit more (factor 10) and the first digit increases to 2 to 9 (factor 2). If you multiply 5... to 9... by 20, you get two digits more and the first digit decreases to 1.

That way, factors of 11 to 99 are always adding one or two digits, with larger numbers adding two digits more frequently. 100 adds two digits, following factors add at least two digits (occasionally three), and so on.

Challenge: find the smallest base b such that there is no n! with n digits, or proof there is no such base. I would expect there to be some base, which means there is a smallest one.Today I learned that I don't want to live on exoplanet Kepler-10b. The weather forecast is horrible.

 

[Stephanus]

Challenge: find the smallest base b such that there is no n! with n digits, or proof there is no such base. I would expect there to be some base, which means there is a smallest one.

I've found some
9, 13, 31.
But why?

The number of digits in an f! for n digits is
int(ⁿ Log f!)
So what we have to find is f = int(ⁿ log f!)
stuck here.

 

[zoobyshoe]

Remember that 20!=20*19!, for example.
If you multiply 1... to 4... by 20=10*2 you get one digit more (factor 10) and the first digit increases to 2 to 9 (factor 2). If you multiply 5... to 9... by 20, you get two digits more and the first digit decreases to 1.

That way, factors of 11 to 99 are always adding one or two digits, with larger numbers adding two digits more frequently. 100 adds two digits, following factors add at least two digits (occasionally three), and so on.

This explains it! Thanks! It took me the longest time to fathom what you were saying, but I see now that, given any factorial, it's simple to predict how many digits will be in the next one; how many digits it will jump, and also explains why the breaks are more frequent as the factorials get higher. That's the "acceleration" I saw but couldn't explain.

 

[edward]

Today I learned that if I drop a warm can of diet coke on a ceramic tile floor the can instantly becomes a self opening container. This is not an absolute but I am not willing to drop enough cans to find out. Now back to factorials.

 

[Stephanus]

Today I learned that if you shake a closed cup glass tupperware filled with hot water, it will pop (kinda explode) the top.
It's just that I didn't bring spoon to brew some coffee, so instead of stirred it with spoon, I shook it. The top tumbled, the coffe spilled. I brew some coffee again, now I USE SPOON!

 

[chloeangelique]

Today I learned that studying abroad is a great way to learn new things, new culture, ideas, lifestyle as well as education system. We all know that we are living in a world with full of diverse people, and by studying abroad, we can try to experience dealing with those people and learn something from them that would help us be a better individuals and be prepared for next generation's global leaders.

 

[thankz]

today I learned that if you learn socket programming in vb.net the world is your oyster

 

[Jonathan Scott]

Today I learned why bees (and other insects) are all over our cherry laurel, even though it's much too early for flowers. It turns out that young leaves of that plant ooze a bit of nectar (from "extrafloral nectaries"). I cut it back a lot last year, so there are lots of new leaves coming out all over it.

 

[Silicon Waffle]

And I learned that a worker bee is not necessarily a male bee but it also doesn't have a reproductive system. A real queen bee is well covered and over protected!

 

[aquitaine]

Today I learned that PF looks way different from what I remember and I suddenly got a bunch of rewards just for showing up. Awesome.

 

[Entanglement]

Today I learned how transistors work. [emoji4]

 

[Lisa!]

YILT I try so hard to look helpful and good! Maybe that's because I've made a bad image of myself in the past and now I'm trying to change it.

 

[jim hardy]

Being kind makes us better. Where the conscious goes, the subconscious must follow.

 

[Stephanus]

Today I learned that the speed of light is much faster than the speed of sound.
Is that why that someone looks bright until you hear him/her speaking?

 

[Astronuc]

Origin-of-Life Story May Have Found Its Missing Link
http://news.yahoo.com/origin-life-story-may-found-missing-123319318.html

But exactly how that creature arose has long puzzled scientists. For instance, how did the chemistry of simple carbon-based molecules lead to the information storage of ribonucleic acid, or RNA? The RNA molecule must store information to code for proteins. (Proteins in biology do more than build muscle — they also regulate a host of processes in the body.)

The new research — which involves two studies, one led by Charles Carter and one led by Richard Wolfenden, both of the University of North Carolina — suggests a way for RNA to control the production of proteins by working with simple amino acids that does not require the more complex enzymes that exist today. [7 Theories on the Origin of Life on Earth]