Today I Learned

[JorisL]

TIL there is an inequality named after one of my professors.
In fact I need it for an assignment which requires a continuity argument.

I don't like the fact that nobody hinted us to use that though. Luckily I found it before I set out to do just that.

 

[OmCheeto]

TIL that I started mentally cataloguing TV show scenes when I was about 5 years old.

I've been watching retro TV from 1956 through 1964, over the last few months.
It's quite strange, halfway through a 50 year old TV show, that you remember the gills in someones chest, and feathers flying around the room.
And a chicken man, in an amusement park flying saucer, that turned out to be real.
And some sissy guy, who kept calling for his mommy, when attacked by the serrated boomerang wielding alien creature.

Brains, are amazing.

 

[zoobyshoe]

Today I learned about the Banana Equivalent Dose (BED) of radioactivity:
http://en.wikipedia.org/wiki/Banana_equivalent_dose

A banana equivalent dose (abbreviated BED) is an informal expression of ionizing radiation exposure, intended as a general educational example to indicate the potential dose due to naturally occurring radioactive isotopes by eating one average-sized banana. One BED is often taken as 0.1 µSv, however, in practice this dose is not cumulative, as the principal radioactive component is excreted to maintain metabolic equilibrium. The BED is only an indicative concept meant to show the existence of very low levels of natural radioactivity within a natural food and is not a formally adopted dose quantity...

This part was interesting:

Although the amount in a single banana is small in environmental and medical terms, the radioactivity from a truckload of bananas is capable of causing a false alarm when passed through a Radiation Portal Monitorused to detect possible smuggling of nuclear material at U.S. ports.[6]

 

[OCR]

Today I learned that Johnny Carson graduated with a Bachelor of Arts Degree in radio and speech with a minor in physics in 1949.

 

[Astronuc]

Death toll in Amtrak derailment increased from 5 to 7. It appears the train was speeding (102 mph) where it should have been going slower. NTSB and Amtrak investigating.

Amtrak derailment victims: CEO, Naval Academy student and software architect dead
http://news.yahoo.com/passengers-st...ain-derailment-in-philadelphia-153358441.html

Among the deceased are:

Rachel Jacobs, 39, the CEO of Philadelphia-based technology education company ApprenNet. Jacobs, a wife and mother of a 2-year-old, commuted to Philadelphia twice a week from New York, according to the CW affiliate.

Justin Zemser, 20, student at U.S. Naval Academy. The young man’s mother, https://gma.yahoo.com/amtrak-crash-victims-ceo-still-missing-navy-midshipman-152252891--abc-news-topstories.html , said her only son was heading home to Rockaway Beach, N.Y., after finishing his second year at the academy in Annapolis, Md.

Jim Gaines, a 48-year-old father of two who worked as a video software architect for the Associated Press, also died in the crash. He was returning home to Plainsboro, N.J., after attending meetings in Washington, D.C.

Peace be upon them and the families, friends and colleagues.

Profiles of Philadelphia Amtrak train derailment victims
http://news.yahoo.com/profiles-philadelphia-amtrak-derailment-victims-163042443.html

Navy Midshipman, AP Employee Among Dead In Amtrak Derailment
http://www.npr.org/2015/05/13/406505182/navy-midshipman-ap-employee-among-dead-in-amtrak-derailment

http://www.npr.org/sections/thetwo-...s-cars-roll-in-philadelphia-injuries-reported

"Amtrak Train 188 was traveling at 106 mph moments before it derailed Tuesday night. Investigators said the engineer applied the emergency brakes, but could only slow the train to 102 before it crashed seconds later. "You are supposed to enter the curve at 50 miles per hour," said Robert Sumwalt, NTSB board member."
Ref: http://news.yahoo.com/live-updates--amtrak-train-derailment-near-philadelphia-030241888.html

Engineer Applied Emergency Brake Before Fatal Amtrak Derailment
http://www.npr.org/sections/thetwo-...m-on-its-way-to-investigate-amtrak-derailment

 

[Enigman]

http://www.slate.com/content/dam/slate/blogs/future_tense/2014/08/WaterGallonsUsed.png.CROP.original-original.png

 

[mfb]

Today I learned: a vomitorium in ancient Rome was a gate crowds used to enter and exit a stadium.

 

[thankz]

that my first attempt at crab cakes are going to be delicious, in the oven right now.

here's the recipe:

lump crap
plain breadcrumbs
old bay
chives
egg whites
butter

 

[Stephanus]

Today I learned that 23! is 25,852,016,738,884,976,640,000 which has, coincidentally, 23 digits. The same coincidence does not occur in any other factorial except 1!.

Yep, if you use base 10 number

 

[zoobyshoe]

Yep, if you use base 10 number

And note I had to correct that a few posts later. There are actually three in a row: 22! has 22 digits, and 24! has 24 digits.

 

[Stephanus]

And note I had to correct that a few posts later. There are actually three in a row: 22! has 22 digits, and 24! has 24 digits.

Yes, yes, it's very cunning that you find it. It's interesting, beyond 24! say 25! I think it's 26 digits, and for 21! it's 20 digits. But it can only be manifested in base ten number.
Do you have any idea what is n?
n! is n digits in base n number?
Should program the computer to find out.

 

[Lisa!]

Now I'm 10 years old in PF years!

 

[zoobyshoe]

Yes, yes, it's very cunning that you find it. It's interesting, beyond 24! say 25! I think it's 26 digits, and for 21! it's 20 digits.

Yes, it's a strange and interesting little island.

But it can only be manifested in base ten number.
Do you have any idea what is n?
n! is n digits in base n number?
Should program the computer to find out.

That's beyond me. You go ahead and work on that.

 

[Stephanus]

Yes, it's a strange and interesting little island.
That's beyond me. You go ahead and work on that.

So far, no. Only base 2 and 2! is two digits 10
This is the list that I make. For each base. Remember, A -> 10, B -> 11, C -> 12. Maximum is 62 base number
base; len; result
2: 2; 10
3: 2; 20
4: 3; 120
5: 3; 440
6: 4; 3200
7: 5; 20460
8: 6; 116600
9: 6; 612700
10: 7; 3628800
11: 8; 205940A0
60: 47; 1K34lTABjkLQij9TkGanwv0XwJVAfZ89`00000000000000
61: 47; b2tipGOC8axb22OlaFskBeQEgCbeEuo6`vixIIk7DhScsx0
62: 48; I0YAwAmEHVRb1wt5AfbjF1UTOi1IvOT64r2X5ZK2cbErxG00
But with higher base number, the digits seems left behind. I think there's no such n! is n digits for n base number. But can it be proven otherwise?

 

[zoobyshoe]

Today I learned the Stephanus theorem:

there's no such n! is n digits for n base number

Will it be proven?

 

[Stephanus]

Today I learned the Stephanus theorem:

there's no such n! is n digits for n base number

Will it be proven?

Naah, it's not a theorem. It's not mine either.
It's this':

[FONT=Courier New]  for n:=2 to 62 do begin
    FBase:=n;
    for m:=1 to n do FNumber[m]:=0;
    FNumber[0]:=1;
    FNumber[1]:=1;
    for f:=1 to n do Multiply(f);       
  end;[/FONT]

Part of the code.

 

[Stephanus]

Congratulations! You are correct. Also, 24! is a 24 digit number. It's a hat trick: 22!, 23!, 24! .

Yes zoobyshoe, I remember saw your post earlier.

 

[OmCheeto]

Now I'm 10 years old in PF years!

Congratulations!

Yes, yes, it's very cunning that you find it. It's interesting, beyond 24! say 25! I think it's 26 digits, and for 21! it's 20 digits. But it can only be manifested in base ten number.
Do you have any idea what is n?
n! is n digits in base n number?
Should program the computer to find out.

Yes!
And please find out what Lisa! is in base 36.
Thanks!

Oh, never mind. It's her birthday. I'll do it.

Lisa₃₆ = 1,004,122₁₀
1,004,122! = ∞?
Stupid google calculator...

hmmmm...

per wiki:

1,000,000! ≈ 8.263931688×10^5,565,708
1,723,508! ≈ 5.290070307×10^10,000,001
​

hmmm...

In his book The Emperor's New Mind, Penrose estimates the number of baryons in the observable universe to be of the order of 10⁸⁰​

So Lisa₃₆!, is a lot.

ps. TIL that sometimes, wiki gets things right; "The calculator seen in Mac OS X handles up to 92!"
My calculator claims 93! is; "Not a number"

 

[mfb]

2 is the only number n such that n! has n digits in base n.
There is an intuitive way to see this: n^(n-1) is the smallest number with n digits in base n, but we only have n! = n(n-1)(n-2)*...*2*1 < n*n*...*n*1 = n^(n-1) with n=2 as exception.

You can also prove it via the stirling formula.

 

[Stephanus]

All in base ten
Has to make the software by hand. Delphi has no variable that can handle that much.
You can check my software with calculator for low number, but for higher number, I don't know if there's bug or not. Beside, who know?

10!: 3,628,800
20!: 2,432,902,008,176,640,000
30!: 265,252,859,812,191,058,636,308,480,000,000
100!: 93,326,215,443,944,152,681,699,238,856,266,700,490,715,968,264,381,621,468,592,963,895,217,599,993,229,915,608,941,463,976,156,518,286,253,697,920,827,223,758,251,185,210,916,864,000,000,000,000,000,000,000,000
200!: 788,657,867,364,790,503,552,363,213,932,185,062,295,135,977,687,173,263,294,742,533,244,359,449,963,403,342,920,304,284,011,984,623,904,177,212,138,919,638,830,257,642,790,242,637,105,061,926,624,952,829,931,113,462,857,270,763,317,237,396,988,943,922,445,621,451,664,240,254,033,291,864,131,227,428,294,853,277,524,242,407,573,903,240,321,257,405,579,568,660,226,031,904,170,324,062,351,700,858,796,178,922,222,789,623,703,897,374,720,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
300!: 306,057,512,216,440,636,035,370,461,297,268,629,388,588,804,173,576,999,416,776,741,259,476,533,176,716,867,465,515,291,422,477,573,349,939,147,888,701,726,368,864,263,907,759,003,154,226,842,927,906,974,559,841,225,476,930,271,954,604,008,012,215,776,252,176,854,255,965,356,903,506,788,725,264,321,896,264,299,365,204,576,448,830,388,909,753,943,489,625,436,053,225,980,776,521,270,822,437,639,449,120,128,678,675,368,305,712,293,681,943,649,956,460,498,166,450,227,716,500,185,176,546,469,340,112,226,034,729,724,066,333,258,583,506,870,150,169,794,168,850,353,752,137,554,910,289,126,407,157,154,830,282,284,937,952,636,580,145,235,233,156,936,482,233,436,799,254,594,095,276,820,608,062,232,812,387,383,880,817,049,600,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

 

[Stephanus]

2 is the only number n such that n! has n digits in base n.
There is an intuitive way to see this: n^(n-1) is the smallest number with n digits in base n, but we only have n!=n(n-1)(n-2)*...*2*1 < n*n*...*n*1 with n=2 as exception.

You can also prove it via the stirling formula.

You're right mfb!. I tought 1 is also the answer beside 2. But I just realized that there's no base 1

 

[OmCheeto]

All in base ten
Has to make the software by hand. Delphi has no variable that can handle that much.
You can check my software with calculator for low number, but for higher number, I don't know if there's bug or not. Beside, who know?

10!: 3,628,800
20!: 2,432,902,008,176,640,000
30!: 265,252,859,812,191,058,636,308,480,000,000
...

um... given that the numbers >92! are not numbers...

What is the highest factorial you can calculate, if all the baryons and neutrino's were converted into binary bits?
Given that it is predicted that neutrinos outnumber baryons by a billion to 1.

"So the total number of neutrinos in the observable universe is about 1.2 x 10⁸⁹ !"
​

 

[Stephanus]

What is the highest factorial you can calculate, if all the baryons and neutrino's were converted into binary bits?
Given that it is predicted that neutrinos outnumber baryons by a billion to 1.

"So the total number of neutrinos in the observable universe is about 1.2 x 10⁸⁹ !"​

What is this? Are you reading my mind?
A week ago I created a thread about neutrino.
Yes, you're right OmCheeto, neutrinos outnumber baryons, but baryon are still heavier then neutrino.

https://www.physicsforums.com/threads/are-neutrinos-much-more-abundant-than-atoms.813792/

But to calculate the factorial of 1.2E89 is a very difficult trick. Try if I can solve this. But I think, I might take 1 week. I'm still searching the algorithm of 4 pegs Hanoi Tower.
​

 

[OmCheeto]

What is this? Are you reading my mind?
A week ago I created a thread about neutrino.
Yes, you're right OmCheeto, neutrinos outnumber baryons, but baryon are still heavier then neutrino.

https://www.physicsforums.com/threads/are-neutrinos-much-more-abundant-than-atoms.813792/

But to calculate the factorial of 1.2E89 is a very difficult trick. Try if I can solve this. But I think, I might take 1 week. I'm still searching the algorithm of 4 pegs Hanoi Tower.
​

Actually, I meant to say, the inverse of the factorial. In other words, find x when x! = 1e89.

But that should probably be asked in the maths section of PF.

TIL, that in the 1964 TV version of "I Robot", they pronounced "robot" as "row butt".
I kind of giggled, like a minion.
But then I thought about the origin of the word, and decided it should be pronounced "row boat".
That made me giggle too.

I also learned today, that Leonard Nimoy played different rolls in both the 1964 and 1995 "The Outer Limits" versions. (I thought I was losing my mind for a moment. I'd seen the 1995 version 3 months ago, and didn't know there was a previous one.)

I also learned that "I Robot", was a story originally written by someone named Eando Binder, and not Isaac Asimov.
Then I learned that Eando was not really a person, but two people: Earl and Otto Binder.
That made me giggle again.

It's been a funny day.

 

[Stephanus]

2 is the only number n such that n! has n digits in base n.
There is an intuitive way to see this: n^(n-1) is the smallest number with n digits in base n...

How foolish I am. Of course. Should have take a momen to think rather than find the answer through coding.

Actually, I meant to say, the inverse of the factorial. In other words, find x when x! = 1e89

" So the total number of neutrinos in the observable universe is about 1.2 x 10⁸⁹!"

So the "!" is only an exclamation mark
I tought you want me to search 1.2E89!, instead what is X! = 1.2E89, I think, x is somewhere around 100 or 110, but I have to use software to find that.
Give me time.

 

[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.