Today I Learned

[NascentOxygen]

Today ( actually yesterday ) I learned coffe is just as good with molasses.
Ran out of sugar and I was too lazy to go out and buy some.

Recipes calling for molasses require caution. You may not be aware that "molasses" means different things in different countries.

 

[1oldman2]

TIL, a

is 6.022 × 10²³ units of anything.

 

[Astronuc]

TIL Paul (Louis-Toussaint) Héroult (10 April 1863 – 9 May 1914), a French scientist and the inventor of the aluminium electrolysis process that bears his name, developed the first successful commercial electric arc furnace. EAF melting and refining became important for clean steels.

 

[Stephanus]

TIL, a View attachment 103334is 6.022 × 10²³ units of anything.

Hahahahahaha. English is not my first language. Guess what when I tried:
https://www.google.co.id/search?q=mole&biw=1366&bih=643&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjt2vmn5fnNAhVFNI8KHRu6CKIQ_AUIBigB
I would expect pictures of laborartory apparatus, but... Well try it yourself.

Of course this will give the chemisty link.
Below is Google search
https://www.google.co.id/search?q=mole&biw=1366&bih=643&source=lnms&sa=X&ved=0ahUKEwj9zuCr5fnNAhXIKo8KHfHRCYUQ_AUIBygA&dpr=1
[Add: ahhh "mole hunt" is the term in spy novel about searching for a mole inside a secret service agency]

 

[1oldman2]

Hahahahahaha. English is not my first language. Guess what when I tried:
https://www.google.co.id/search?q=mole&biw=1366&bih=643&source=lnms&tbm=isch&sa=X&ved=0ahUKEwjt2vmn5fnNAhVFNI8KHRu6CKIQ_AUIBigB
I would expect pictures of laborartory apparatus, but... Well try it yourself.

Of course this will give the chemisty link.
Below is Google search
https://www.google.co.id/search?q=mole&biw=1366&bih=643&source=lnms&sa=X&ved=0ahUKEwj9zuCr5fnNAhXIKo8KHfHRCYUQ_AUIBygA&dpr=1
[Add: ahhh "mole hunt" is the term in spy novel about searching for a mole inside a secret service agency]

https://www.khanacademy.org/science...oichiome/stoichiometry-ideal/v/stoichiometry#!

 

[Stephanus]

https://www.khanacademy.org/science...oichiome/stoichiometry-ideal/v/stoichiometry#!

Yep, 1 mole = 6 * 10²³ is NOT (added) a Today I learned thing
But a mole =

is really a Today I learned thing. Yeah , today I learned that that is a mole.

 

[Stephanus]

TIL: This rule is true in general. Proof, with an interesting approach I think:

Well this is not as straight forward for me as for everyone else. I have to break it down.

Spoiler: I'm going to need a proof for n*a+m*b mod ab will produce different results.

Let's consider the set of n*a+m*b mod (ab) for n=0 to b-1 and m=0 to a-1. We get (ab) different results ..

Fact 1. for n=0 to b-1 and m=0 to a-1 we get ab diffrent results. Will it give us different results?
Fact 2: For every n mod b where n<b there will be unique/different results.
Fact 3: for every n*a mod b*a where n<b then there will be unique results and some gaps. This will also work for n = b-1, so
Fact 4: for every (b-1)*a mod b*a there will be unique results and some gaps.
Fact 5: (b-1)*a + (a-1) * b might be more than ab, seems contradict fact 1. Will (a-1)*b fil the gaps?
5.1 Let's add (b-1)*a with sequences of m*b where m=0..a-1
Before that. Sequence of (b-1)*a mod a is 0.
((b-1)*a + a*b) will give an overlap result.
because ((b-1)*a + a *b) mod a = 0.
and
((b-1)*a + a*b) mod ab = -a; ((b-1)*a) mod ab = -a

But (b-1)*a + (a-1)*b will not give an overlap result.
((b-1)*a + (a-1)*b) mod a = -b
((b-1)*a + (a-1)*b) mod ab = -a-b.
So, for ((b-1)*a + m*b) mod a where m = 1..a-1 there wouldn't be overlap result.
So the (a-1)*b gaps will be reduced by a-1. So there will be (a-1)*b-(a-1) = (a-1)*(b-1) gaps.

5.2 Do 5.1 again with (b-2) * a
Again ((b-2) * a + m*b) mod a wil produce the exact result as in 5.1
But there wouldn't be any conflict with ((b-2) * a + m*b) mod ab, because it's like shifting/decreasing all the filled number by a
So there will be (a-1)*(b-1)-(a-1) = (a-1)*(b-2) gaps.

5.3 Do 5.1 again until (b-b) * a
So there will be (a-1)*(b-b) gaps. = 0 gaps!

So here is the proof for

Let's consider the set of n*a+m*b mod (ab) for n=0 to b-1 and m=0 to a-1. We get (ab) different results ..

There are ab different results.

Let's call the numbers a and b.
They key number here is a*b, which can be expressed as "a times b" or "b times a". It is the first number that has this ambiguity, as it is the least common multiple.

Let's consider the set of n*a+m*b mod (ab) for n=0 to b-1 and m=0 to a-1. We get (ab) different results (due to (ab) being the least common multiple - getting the same result twice would lead to a smaller common multiple) - but there are just ab possible results, so we get every result exactly once.

Okay...

In absolute terms, the values n*a+m*b go from 0 to 2ab-a-b.

Okay...

Let's consider (b-1)*a+(a-1)*b = 2ab-a-b. We know it is in the set above.

Okay..

But that means ab-a-b cannot be in this set, otherwise we would have a contradiction to the unique results mod (ab).

Okay..

Spoiler: Here is the proof

Sequence of ((b-1)*a+(a-1)*b) mod ab will give ab different results.

void Fill(int a, int b)
{
  int Numbers1[a*b]; // I know this is error :smile:
  int Numbers2[2*a*b];
  int c=0;
  int n,m;
  for (n=0;n<b;n++) for (m=0;m<a;m++) Numbers1[(a*n+b*m) % (a*b)]++;
  for (n=0;n<b;n++) for (m=0;m<a;m++) Numbers2[(a*n+b*m)]++;
  for (n=0;n<b;n++) for (m=0;m<a;m++) c++; // capturing the last element of the counter
}

All elements in Numbers1 will be filed by 1. But there elements in Numbers2 that are zero and the others are of course 1.
So there's no overlap here. I'm sorry. I'm not a mathematician, I don't know math language. But that's what I mean.
Because all Numbers1 element already filled with 1 this instruction
for(n=c;n<2*a*b;n++) if(!Numbers2[n]) {Numbers1[n%(a*b)]++; break;} will set one of the Numbers1 element by 2.
I swear I just write this code in the browser not in the compiler. But here's is the proof that ab-a-b cannot be in the set. I really can't express myself in math language!

We also cannot reach ab-a-b (edited by me) with larger n or m - the sum would be too large. There is no way to get to ab-a-b with positive n,m.

Yes. (ab-a-b) mod ab = -a-b. And we already have Numbers1[(-a-b)% (a*b)]==1 from (b-1)a + (a-1)b. I don't have the proof that we can reach ab-a-b from na+mb, but this proof is sufficient if we combine the previous proofs.

What about all the values between 2ab-a-b and 2ab? They are not in the set above,

Yes, sure.

But that means the values between ab-a-b and ab have to be in the set, because we know those values appear as remainder mod (ab).

Yes, all of them.

=> ab-a-b = (a-1)(b-1)-1 cannot be reached, while (a-1)(b-1) to (ab) can be reached. It is a well-known result that all numbers above (ab) can be reached as well. Therefore, (a-1)(b-1)-1 is the last number that cannot be reached.

Oh my. I've never thought of that. Although, about the "gaps" my software has already shown that. They are all before my eyes. I just don't realize them.

Reds row are the gap. But my software doesn't continue the proof until 2ab-a-b. It just counter from ab-a-b+1 to the least of a or b. Supposed if a is the smaller
then if there's a sequence from ab-a-b+1 to ab-a-b+1 + (a-1) or sequence from ab-a-b+1 to ab-b then it just cuts. And continue the next sequence in another window as I upload yesterday.
And today I learned that I do believe that all the mentors/staffs/science advisors inf PF Forum are really geniuses! What's been shown before my eyes a long time can be solved just in 5 minutes by them. I do hope that this community either direct/indirectly can contribute many things to human wealth and continuity.

 

[Stephanus]

Spoiler: Today I leaned

That we can use spoiler to shorten the details in our post although it's not intended as a spoiler.

Spoiler: Details

Because the details tends to divert our attention to the main message.
They are best kept in a spoiler so it does not crowd our message.
Only for those who really need to read the proof/detail can expand the spoiler.
Perhaps in the future, the PF Forum allow the word SPOILER to be replaced by user?

Spoiler: And another one

We can use spoiler inside spoiler

We can also insert graph in a spoiler

 

[mfb]

A mole of moles?

2ab-a-b is the largest number that has a unique way to being expressed as n*a+m*b with non-negative n,m, by the way, all larger numbers have at least two options (because you can replace "a times b" by " b times a" if n>=b or m>=a).

 

[jtbell]

When I was in high school many years ago, my chemistry teacher once posed this question on a test:

If a mole could dig a mole of holes in a day, how many holes could a mole of moles dig in a mole of days?

 

[Stephanus]

When I was in high school many years ago, my chemistry teacher once posed this question on a test:

If a mole could dig a mole of holes in a day, how many holes could a mole of moles dig in a mole of days?

Wow, around 216 * 10⁶⁹?

 

[DrGreg]

 

[Pepper Mint]

Wow, around 216 * 10⁶⁹?

Are "Colorless green ideas sleep furiously"'s spread either by default or by design ? I guess it is the latter that is to hook up something systematically.

 

[256bits]

Recipes calling for molasses require caution. You may not be aware that "molasses" means different things in different countries.

Never knew the caution.
So that is something new I learned today, but not sure what it is.

A Molasse is a sedementary rock formation - stones in coffee, might be gritty.

Definitions doesn't show any other than the refining process towards sugar of beets and sugarcane, and some other plants.

 

[Pepper Mint]

Wow I think I learn what molasses is today.
By the way, stevia can make sugar too.

 

[1oldman2]

A mole of moles?

Excellent! I love xkcd.

 

[fresh_42]

And I always thought moles had to do with avocados ...

 

[Jonathan Scott]

And I always thought moles had to do with avocados ...

Isn't that just the wacky ones?

 

[1oldman2]

And I always thought moles had to do with avocados ...

Good one !

 

[1oldman2]

Isn't that just the wacky ones?

TIL, that would be a lot of "whack a mole"

 

[Larry Gopnik]

Today I learned that I should never dye my hair by myself.

I got it dyed black a few months back and decided today to return it to my natural colour - sort of gingery brown. I went wrong. It's now highlighter orange, the shops are shut and I have work tomorrow...

 

[fresh_42]

Today I learned that I should never dye my hair by myself.

I got it dyed black a few months back and decided today to return it to my natural colour - sort of gingery brown. I went wrong. It's now highlighter orange, the shops are shut and I have work tomorrow...

Take the positive part of it. I have an Irish friend. No matter how large a crowd might be, she cannot get lost.

 

[jtbell]

Today I learned about "corn sweat" and how it contributes to summer heat waves in the midwest US.

http://www.cnn.com/2016/07/17/weather/extreme-weather-heat-dome/index.html

(I actually grew up in the midwest, but it was in a steel-mill town, not among the cornfields.)

 

[OCR]

Today I learned about "corn sweat" and how it contributes to summer heat waves in the midwest US.

Lol... I just learned that myself, about two hours ago... 'cause my wife told me...So, to borrow from your signature...

I don't need no stinkin' signature...

Internet??... Internet?

I don't need no stinkin' internet...

 

[Stephanus]

Today I learned,
that if you pack 250 moles of moles in a mole hole, it will become a black hole.

Spoiler: Schwarzschild Radius

G = 6.674 * 10^-11
Mole = 6.022 * 10²³
Weight of a, ehm, mole ≈ 500 gr
$R_{schwarzschild} = \frac{2GM}{c^2}$
$R_s = \frac{2 * 250 * 6.022 * 10^{23}\text{ } * 500gr * 6.674 * 10^{-11}}{9 * 10^16}$
$R_s = 1.12 * 10^{-1}m = 0.112 m = 10 cm$ About the size of a mole hole.

Even if my calculation is wrong, the hole still black.

 

[Stephanus]

And I always thought moles had to do with avocados ...

Ahhh, I didn't get the joke. Avogadro?

 

[Jonathan Scott]

Ahhh, I didn't get the joke. Avogadro?

Double joke (not sure whether intentional). Firstly, Avocado could be confused with Avogadro. Secondly, Avocado is used to make Guacamole (sounds something like "whack a moley").

 

[Kaura]

Today I learned,
that if you pack 250 moles of moles in a mole hole, it will become a black hole.

Spoiler: Schwarzschild Radius

G = 6.674 * 10^-11
Mole = 6.022 * 10²³
Weight of a, ehm, mole ≈ 500 gr
$R_{schwarzschild} = \frac{2GM}{c^2}$
$R_s = \frac{2 * 250 * 6.022 * 10^{23}\text{ } * 500gr * 6.674 * 10^{-11}}{9 * 10^16}$
$R_s = 1.12 * 10^{-1}m = 0.112 m = 10 cm$ About the size of a mole hole.

Even if my calculation is wrong, the hole still black.

Today I learned through using that equation that one would have to shrink a mole weighing 100 grams to a size of about 1.48x10^-28 which is smaller than a Neutrino in order to make the mole into a black hole

 

[mfb]

which is smaller than a Neutrino

It is expected that neutrinos are point-like, and there are no indications of any size. Experiments set upper limits on a possible size.

 

[1oldman2]

neutrinos are point-like

TIL, https://docushare.icecube.wisc.edu/dsweb/Get/Document-58012/dumm_thesis.pdf
Very soon into reading this I was wondering why the southern sky registered so many more events than the north.
("The data sample contains 36,900 events: 14,121 from the northern sky, mostly
muons induced by atmospheric neutrinos and 22,779 from the southern sky, mostly
high energy atmospheric muons.")
I quickly came upon,
( "in the southern hemisphere these neutrinos are swamped by the downgoing cosmic-ray muon background.") Which of course brought up...http://icecube.wisc.edu/
This led to wondering about tau... http://www.scientificamerican.com/article/let-s-use-tau-it-s-easier-than-pi/,
Which in turn brought me to...

Soon I'm watching...

(Ain't the internet great!)
Thanks mfb, this has been a good "TIL day" (I'm hoping for some feedback/opinions on pi v.s. tau)