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Playing with math 
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Post Playing with math
Here is where I/you can play with math.
A
{1,2,4,5}
B
{1,4,6,7}
A=12
B=18
A+B\ne47
A+B=30

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Mon Nov 18, 2013 10:37 am
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Post Re: Playing with math
:?: :?: :?:
Um... Yes? :|

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Post Re: Playing with math
I'm not sure what you're trying to get across. What's goin' on here?

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Mon Nov 18, 2013 4:49 pm
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Post Re: Playing with math
... confused...

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Mon Nov 18, 2013 9:00 pm
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Post Re: Playing with math
09
18
27
36
45
54
63
72
81
90

The x9 multiplication table are basically palindromes!

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Mon Nov 18, 2013 10:42 pm
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Post Re: Playing with math
kurowolfe wrote:
The x9 multiplication table are basically palindromes!
That's funny. Never thought about it. :D

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Tue Nov 19, 2013 7:11 am
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Post Re: Playing with math
Well, it's only a palindrome if you arrange the numbers in a straight line.

x + y = 35
x + z = 45

y - z = ?

Also, in Malay, subtract is usually translated as 'tolak'. But 'tolak' also means push. So a common Malaysian joke is the statement "a number x - a number y equals the number y falling over and lying on the floor".

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Tue Nov 19, 2013 11:04 am
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Post Re: Playing with math
That'd be -10, Kuro!

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Tue Nov 19, 2013 12:12 pm
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Post Re: Playing with math
Render wrote:
kurowolfe wrote:
The x9 multiplication table are basically palindromes!
That's funny. Never thought about it. :D

it goes beyond that.
for every 11 multiples of 9, the digits in the first n multiples (where n is a number less than 11) add up to 9, and the digits of the next (11)-n multiples add up to 18
(first 10 multiples of 9)0+9=9, 1+8=9, 2+7=9, 3+6=9, 4+5=9, 5+4=9, 6+3=9, 7+2=9, 8+1=9, 9+0=9
(next 1 multiple of 9)9+9=18
(next 9 multiples of 9)1+0+8=9, 1+1+7=9, 1+2+6=9, 1+3+5=9, 1+4+4=9, 1+5+3=9, 1+6+2=9, 1+7+1=9, 1+8+0=9
(next 2 multiples of 9)1+8+9=18, 1+9+8=18
(pattern continues)
(next 2 multiple of 9)8+0+1=9, 8+1+0=9
(next 9 multiples of 9)8+1+9=18, 8+2+8=18, 8+3+7=18, 8+4+6=18, 8+5+5=18, 8+6+4=18, 8+7+3=18, 8+8+2=18, 8+9+1=18
(next 1 multiple of 9)9+0+0=9
the pattern kind of changes there, and I can;t be bothered to try to figure out a pattern that takes the change into account, but how often will someone need to know a pattern for finding more than 100 multiples of 9?

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Post Re: Playing with math
OK, fun with math:
Let a = b

Then a^2 = ab (multiplying both sides by a)

a^2 + a^2 = a^2 + ab (adding a^2 to both sides)

Which is: 2a^2 = a^2 + ab

2a^2 - 2ab = a^2 + ab - 2ab (adding -2ab to both sides)

Which is: 2a^2 - 2ab = a^2 - ab

Which can be written as: 2(a^2 - ab) = 1(a^2 - ab)

Cancelling the (a^2 - ab) out leaves us with:
2 = 1

Isn't math fun :D

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Post Re: Playing with math
That's not math because you divided by zero and then multiplied by zero again.

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Thu Nov 21, 2013 12:14 pm
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Post Re: Playing with math
It's a study on the reason you can't divide by zero :P
OK, how about this one:

-1/1 = 1/-1

√(-1/1) = √(1/-1) (Square root both sides)

√(-1)/√(1) = √(1)/√(-1) (Simplify)

Which is: i/1 = 1/i

i/2 = 1/2i (Multiple both sides by 1/2)

i/2 + 3/2i = 1/2i + 3/2i (Add 3/2i to both sides)

i(i/2 + 3/2i) = i(1/2i + 3/2i) (Multiply both sides by i)

Which is: i^2/2 + 3i/2i = i/2i + 3i/2i

Which is: -1/2 + 3/2 = 1/2 + 3/2

Which is: 1 = 2

And, yes, there is an error, but the point is that it is not an obvious error. It is disguised so as to pass by unnoticed to make you do brain work. :D

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Post Re: Playing with math
Obbl wrote:
√(-1/1) = √(1/-1) (Square root both sides)

√(-1)/√(1) = √(1)/√(-1) (Simplify)

If a and b aren't positive then √(a / b) = √a / √b doesn't always hold. No amount of disguises can hide that! ^_^

The fun part of math is figuring out what happens when your "ordinary" algebraic rules don't hold. Like, what happens if A BB A? (I happen to run into a lot of these lately.)


Thu Nov 21, 2013 5:48 pm
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Post Re: Playing with math
Gah, I thought I was done with math!

fylwind wrote:
what happens if A B ≠ B A?

... *Mind explodes*



I would probably be a little better off if my last year of High School math (Caculus no less) hadn't been with three different teachers. First a teacher who was experienced in teaching, but a little rusty in her calc skills. Second a teacher who was pretty fresh with his calc skill, but mostly because he was just out of college and didn't have a whole lot of real-world teaching skills yet. My third teacher was when the second one was fired and he was just out of college too, except he didn't really know calc. He knew his algebra really well and that was good because 4 out of the 7 classes he had were algebra, but he was kinda learning the material MY class had at the same time we were.

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Thu Nov 21, 2013 6:58 pm
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Post Re: Playing with math
JeffCvt wrote:
Gah, I thought I was done with math!

One does not simply become done with math, for the journey never ends! :]

JeffCvt wrote:
I would probably be a little better off if my last year of High School math (Caculus no less) hadn't been with three different teachers. First a teacher who was experienced in teaching, but a little rusty in her calc skills. Second a teacher who was pretty fresh with his calc skill, but mostly because he was just out of college and didn't have a whole lot of real-world teaching skills yet. My third teacher was when the second one was fired and he was just out of college too, except he didn't really know calc. He knew his algebra really well and that was good because 4 out of the 7 classes he had were algebra, but he was kinda learning the material MY class had at the same time we were.

That's quite unfortunate. Oftentimes I think if we had better math instructors (and similar subjects) perhaps more people would be interested in it! *sigh*

On a personal note, little of my math (and most nonhumanities subjects) was actually learned from my instructors. (Prior to the internet,) I learned most of the materials by reading textbooks. (Corollary: I tend to suck more in classes that had poor or nonexistent textbooks). For new material, the lecturing format often doesn't work for me since I often like to pause and think about things before continuing. The classes where I actually enjoyed the lectures were (ironically) the classes where I either knew the material or the material was already easy to grok.


Thu Nov 21, 2013 8:19 pm
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Post Re: Playing with math
JeffCvt wrote:
Gah, I thought I was done with math!

fylwind wrote:
what happens if A B ≠ B A?

... *Mind explodes*



I would probably be a little better off if my last year of High School math (Caculus no less) hadn't been with three different teachers. First a teacher who was experienced in teaching, but a little rusty in her calc skills. Second a teacher who was pretty fresh with his calc skill, but mostly because he was just out of college and didn't have a whole lot of real-world teaching skills yet. My third teacher was when the second one was fired and he was just out of college too, except he didn't really know calc. He knew his algebra really well and that was good because 4 out of the 7 classes he had were algebra, but he was kinda learning the material MY class had at the same time we were.


That really stinks man. Sorry to hear about that. Mathematics is a difficult subject and can be very difficult to understand without a good teacher. Especially without any stable instruction.

fylwind wrote:
That's quite unfortunate. Oftentimes I think if we had better math instructors (and similar subjects) perhaps more people would be interested in it! *sigh*

On a personal note, little of my math (and most nonhumanities subjects) was actually learned from my instructors. (Prior to the internet,) I learned most of the materials by reading textbooks. (Corollary: I tend to suck more in classes that had poor or nonexistent textbooks). For new material, the lecturing format often doesn't work for me since I often like to pause and think about things before continuing. The classes where I actually enjoyed the lectures were (ironically) the classes where I either knew the material or the material was already easy to grok.


I have to agree about math instruction, it's an important thing to get right and do well. I think sometimes math education gets too broad in the early levels. But then again, it is a very in depth subject. It would also be nice to have better science education programs as well.

I find it admirable that you learned your mathematics from textbooks. I personally have never found a textbook on mathematics that I've thought was good, although some that I have had to use were particularly awful. It's a difficult subject to write about, I'll admit.



A*B≠ B*A
The only cases where I've seen that not hold true is when multiplying matrices when A or B or both have determinants of zero. I'm not sure if there are others or not.


Let's play with matrices then.
3x+6y=10
2x+9y=5

What are x and y?



And a fun sciency question too.

Na+H2O->H2+NaOH
How many grams of sodium do I need to generate 100g of hydrogen gas through the reaction with water?


Sat Nov 23, 2013 1:53 am
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Post Re: Playing with math
Squirrelboy265 wrote:
A*B≠ B*A
The only cases where I've seen that not hold true is when multiplying matrices when A or B or both have determinants of zero. I'm not sure if there are others or not.

There are actually whole other number systems that play by different rules than our own, and in some, things like A*B or even A+B are not commutative. ;) (Though that may not be what fylwind is talking about)
I haven't looked into them because I like my mind intact. :D

Squirrelboy265 wrote:
Let's play with matrices then.
3x+6y=10
2x+9y=5

What are x and y?

I don't need a matrix to solve this though...
It's just a system of equations.
Is there a way to do it with matrices? :|

Squirrelboy265 wrote:
And a fun sciency question too.

Na+H2O->H2+NaOH
How many grams of sodium do I need to generate 100g of hydrogen gas through the reaction with water?

Well, according to your equation, I need 2 parts sodium and 2 parts water for every 1 part hydrogen gas and 2 parts sodium...chemistry-type name ending
So it's however many parts hydrogen gas equals 100 grams times 2 times the weight of sodium, which I am not calculating at this hour @_@ (I've been programming, and I'm tired... so quit intriguing me with your intriguing questions...zzzzzzzzzzzzzzzzzzzzz)

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Post Re: Playing with math
Obbl wrote:
Squirrelboy265 wrote:
Let's play with matrices then.
3x+6y=10
2x+9y=5

What are x and y?

I don't need a matrix to solve this though...
It's just a system of equations.
Is there a way to do it with matrices? :|

yes there is, but that's usually only used when there are more than two or three variables, since the faster way gets kind of hard at that point

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Post Re: Playing with math
Squirrelboy265 wrote:
Let's play with matrices then.
3x+6y=10
2x+9y=5

What are x and y?
You don't even need to use Gaussian elimination for that one. Make it bigger. :P

x = 4
y = -1/3
Squirrelboy265 wrote:
And a fun sciency question too.

Na+H2O->H2+NaOH
How many grams of sodium do I need to generate 100g of hydrogen gas through the reaction with water?
Stoichiometry is not exactly what I would call fun. It's the boring part of chemistry!

1.15 kg assuming perfect efficiency.

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Sat Nov 23, 2013 11:19 am
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Post Re: Playing with math
NOOOOOOO NOT THE MATH. *flings pillows at Sleet*


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Post Re: Playing with math
Sleet wrote:
Squirrelboy265 wrote:
Na+H2O->H2+NaOH
How many grams of sodium do I need to generate 100g of hydrogen gas through the reaction with water?

Stoichiometry is not exactly what I would call fun. It's the boring part of chemistry!
1.15 kg assuming perfect efficiency.


Close, but the reaction I gave you was unbalanced.
The balanced reaction is 2Na+2H2O->H2+2NaOH. Using this reaction gives you a value of 2.28 ish Kg of sodium needed. A rather terrifying amount of a highly reactive metal.

Stoichiometry isn't the fun part? I mean, sure, the reactions are more fun, but you need to know how much to add. Otherwise it gets a little messy.

Obbl wrote:
There are actually whole other number systems that play by different rules than our own, and in some, things like A*B or even A+B are not commutative. ;) (Though that may not be what fylwind is talking about)
I haven't looked into them because I like my mind intact. :D

Squirrelboy265 wrote:
And a fun sciency question too.
Na+H2O->H2+NaOH
How many grams of sodium do I need to generate 100g of hydrogen gas through the reaction with water?

Well, according to your equation, I need 2 parts sodium and 2 parts water for every 1 part hydrogen gas and 2 parts sodium...chemistry-type name ending
So it's however many parts hydrogen gas equals 100 grams times 2 times the weight of sodium, which I am not calculating at this hour @_@ (I've been programming, and I'm tired... so quit intriguing me with your intriguing questions...zzzzzzzzzzzzzzzzzzzzz)


It was rather late wasn't it?
Sounds like a good area to not play in. The regular number system we use is tough enough, but entirely different number systems? That seems very very frightening.


RandomGeekNamedBrent wrote:
Obbl wrote:
I don't need a matrix to solve this though...
It's just a system of equations.
Is there a way to do it with matrices? :|

yes there is, but that's usually only used when there are more than two or three variables, since the faster way gets kind of hard at that point


Attached is a quick rough example of how you would use a matrix to solve it.
It is a rather small one to do matrices on.

sleet wrote:
You don't even need to use Gaussian elimination for that one. Make it bigger. :P

Here's a larger one.

1.5x+2.20y+3.5z+4.0t=34
2.0x+7.0y+5.0z+8.0t=22
2.0x+2.0y+2.0z+2.0t=18
1.0x+3.0y+5.0z+9.0t=5

Also, forgive me for any quoting mistakes or whatnot. I haven't been on a forum like this for a long time.


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Solve Matrix.png
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Sat Nov 23, 2013 2:43 pm
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Post Re: Playing with math
It never occurred to me that the equation wasn't balanced. Once you get past that part of elementary chemistry no one bothers leaving reactions for you to balance. :P

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Sun Nov 24, 2013 2:09 am
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Post Re: Playing with math
You're right. I didn't realize it was unbalanced when I posted it though. I was just doodling with chemistry.
The fun part is when you get to organic chemistry and it starts to stop mattering if the reaction is balanced, because it won't happen like that anyways. Or many things happen at once that ruin the simple reaction schematic.


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Post Re: Playing with math
Usually when you get that far out of the theoretical realm, people will take percent yield into effect.

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Post Re: Playing with math
:shock: so much maths... This is giving me bad flashbacks... From algebra class... *thousand yard stare*

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Post Re: Playing with math
Obbl wrote:
There are actually whole other number systems that play by different rules than our own, and in some, things like A*B or even A+B are not commutative. ;) (Though that may not be what fylwind is talking about)

Indeed. In quantum mechanics (QM) there are these things called "operators" that represent quantities (like energy, position, momentum, etc) and the fact that they don't commute is what makes QM interesting.

Some other examples of noncommutative operations are:
  • matrices (as everyone has said),
  • quaternions (used for calculating rotations in 3D space in e.g. rendering),
  • function composition,
  • string concatenation,
etc. In fact, noncommutative operations occur quite frequently in the real world, just that most people don't notice them!


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Post Re: Playing with math
I miss learning pure maths as it is. I used to know how to do matrices and stuff. Now I don't.

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Post Re: Playing with math
I'm really rusty with matrices, myself. I could pick them back up if I have a use for them pretty easily though.

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Wed Nov 27, 2013 6:51 pm
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Post Re: Playing with math
Hey, there's always Wolfram Alpha if you want to do some simple matrix stuff! :)


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Post Re: Playing with math
Wolfram Alpha is an incredible thing. Mathematics, some Chemistry, it does an incredible amount of data lookup,solving, and tabulation.

Every time I look at quantum mechanics I become rather befuddled. They call it physics, but sometimes a more appropriate term seems to be "Magic". I'm just glad that those microscopic properties of matter tend to be of a low level of importance in calculating macroscopic changes. While the quantum interactions may drive the reactions and other chemically things I deal with, there are equations and mechanisms in place that I can understand and use that are much easier to do.

It's rather intimidating to think about all the things that we don't notice. The mountains of work that went into the very computers we use to communicate. From the design of the screen and the chemical process that made them, the discovery and application of the transistors, the refining of the silicon, the complicated programming, and the list just goes on and on. It makes you think that our society may reach a point where we're all just technicians maintaining things that we don't truly understand the inner workings of. Perhaps we're at that stage now. I'm reasonably sure that this topic has been often explored in science fiction, but I'm failing to remember the particular novels. It also tends to crash on the situation where we can't know everything.


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Post Re: Playing with math
I always thought it's math playing with you.

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Post Re: Playing with math
Squirrelboy265 wrote:
Wolfram Alpha is an incredible thing. Mathematics, some Chemistry, it does an incredible amount of data lookup,solving, and tabulation.

I just used WA yesterday to calculate a rather messy expression containing the fine structure constant, Bohr radius, Planck's constant, electron mass, etc. I was honestly surprised that it worked, and certainly saved me a lot of time looking up the values of those constants. (FYI: it was to find the lifetime of the 2p state in hydrogen) Also used it to calculate an integral that I was too lazy to do by hand. Sometimes I use Google to do simple numerical calculations with quantities, but alas Google doesn't have the fine structure constant nor the Bohr radius. :(

Squirrelboy265 wrote:
Every time I look at quantum mechanics I become rather befuddled. They call it physics, but sometimes a more appropriate term seems to be "Magic". I'm just glad that those microscopic properties of matter tend to be of a low level of importance in calculating macroscopic changes. While the quantum interactions may drive the reactions and other chemically things I deal with, there are equations and mechanisms in place that I can understand and use that are much easier to do.

I've read a few popular science books and I have some mixed feelings about them when it comes to topics like quantum. Sure they introduce physics to laypeople, but they tend to give off the wrong impression about things. Quantum phenomena is really cool, mind you, but the real meat of quantum is the mathematics, which is not only beautiful but also surprisingly simple. The various effects such as "entanglement" and "collapse" merely features that emerge from this simple theory. (And maybe I'm just jaded but these "weird phenomena" don't seem all that weird to me. :) )


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Post Re: Playing with math
I'm curious as to the reason why you would need to calculate the lifetime of the 2p state in hydrogen. I think that might illuminate the difference in our perspectives. My basis for viewing things is more on the terms of "What can be done with ___ chemical/thing?" which is why i'm studying to be a chemical engineer. When it comes to microscopic interactions and the details of why things occur, I'm more concerned with the effects of the phenomena. I.E. how much heat will burning x amount of natural gas produce, rather than why that heat is made. It also helps that I can touch a kilogram of water 8-) .

I will definitely concede, however, that engineering and pure sciences go hand in hand and that both are necessary.

I haven't had a reason to look into the mathematics of quantum mechanics, but from what you said, such an endeavor might help with my lack of understanding the various phenomena associated with this. It most probably won't be easy, but could be worth it.

Now to tack on a math/physics problem, since this is the "playing with math" thread.
Or perhaps it is where "Math Plays With You".

The cork in a bottle is a cylinder 5 cm long with a 1 cm diameter with the top of the bottle flush with the top of the cork. If the pressure inside the bottle is constant at 5 atm throughout the popping off of the cork, how high will the vertically launched cork go neglecting air resistance but not gravity?

Cork density=.24g/cubic centimeter


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Post Re: Playing with math
Squirrelboy265 wrote:
I'm curious as to the reason why you would need to calculate the lifetime of the 2p state in hydrogen. I think that might illuminate the difference in our perspectives.

It was a homework problem! :) But from a practical perspective knowing the lifetime (equivalently, the transition rate) of a state is useful if you're, say, building a laser, or trying to understand how the substance interacts with light. The 2p state in hydrogen is very well studied and makes a good homework problem since it's (relatively) easy to calculate using the dipole approximation.

Squirrelboy265 wrote:
My basis for viewing things is more on the terms of "What can be done with ___ chemical/thing?" which is why i'm studying to be a chemical engineer. When it comes to microscopic interactions and the details of why things occur, I'm more concerned with the effects of the phenomena. I.E. how much heat will burning x amount of natural gas produce, rather than why that heat is made. It also helps that I can touch a kilogram of water 8-) .

One of the goals of the more theoretical parts of physics and chemistry is to understand how to derive the properties of things from first principles (the fancy terminology used in literature is ab initio), so that no phenomenological assumptions need to be made. In the case of molecules, for example, it would be how do you go from our quantum model of the atom to reproduce the various properties of the substance. In this area, quantum chemistry and condensed matter theory (from physics) overlaps greatly and is a very active area of research.

Squirrelboy265 wrote:
The cork in a bottle is a cylinder 5 cm long with a 1 cm diameter with the top of the bottle flush with the top of the cork. If the pressure inside the bottle is constant at 5 atm throughout the popping off of the cork, how high will the vertically launched cork go neglecting air resistance but not gravity?

Cork density=.24g/cubic centimeter

A little too early for champagne, isn't it? :)
ΔP / (ρ g) = 4atm / (.24g/cm^3 * 9.81m/s^2) [click to calculate!]
The interesting thing is that for a cylindrical geometry it's actually independent of the volume. In reality it's probably a lot less given air resistance and also falling pressure as it expands.


Mon Dec 02, 2013 10:43 am
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Post Re: Playing with math
Any of y'all ever hear of Mathematica?

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Tue Dec 03, 2013 11:10 am
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Post Re: Playing with math
Mmhmm!

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Wed Dec 04, 2013 1:21 am
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Post Re: Playing with math
That makes sense then, knowing how long the 2p state lasts would help with the laser building. It would have to do with the rate of light generation right, since the transition of an electron from a high energy level to a low one releases a photon, yes?


While I had to look up "phenomenological", I can see the usefulness of such research. Although, I'm curious as to how far we'll be able to progress in that respect. So many equations I've been using to predict material properties, such as the antoine equation for vapor pressures, in chemical processes are just equations fitted to empirical data. While accurate enough, they're incomplete and subject to error. A quantum model would be far superior and accurate, but only possible I should think because of computer calculations.
In regards to the problem I posed

A little too early for champagne, isn't it? :)
Quite so, Finals aren't over yet.

The first time I solved this I neglected to take into account the fact that the atmosphere is exerting a pressure on the cork as well, such that the gauge pressure is 4 atmospheres.
That's also an incredible height, 172 meters. No wonder it hurts so much in movies to get hit by the cork.
I solved this a bit differently than you did though.

ΔEsys=W=ΔK+ΔU (change in energy of the system=work done on system= change in kinetic energy+change in potential gravitational energy)
W=Kf-Ki+Uf-Ui
Kf=Ki=Ui=0
W=Uf=0
W=F*dl (Force*displacement=mass*gravitational acceleration*height) (Force=area of bottom of the cork*pressure) (displacement=5 cm, the length of the cork since the top is flush with the top of the bottle.)
Uf=m*g*hf (Mass*gravitation acceleration*final height) (mass of cork cylinder=pi*radius of cork^2*height of cork cylinder*density of cork)
W/(mg)=hf
Calculate!



I've heard of mathematica, but I've never used it. I typically use Matlab for numerical computations, and I'm not often doing symbolic things.


Fri Dec 06, 2013 1:21 am
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Post Re: Playing with math
Wolfram Alpha is like a gentler version of the more expensive Mathematica. It has a much more lenient syntax compared to Mathematica (although the leniency gets reduced as the complexity of the expression increases). On the other hand, Wolfram Alpha runs from the "cloud" so the length of the computation is limited by what the server gives you (and is very short), whereas you can leave Mathematica running overnight on your own computer.

Squirrelboy265 wrote:
That makes sense then, knowing how long the 2p state lasts would help with the laser building. It would have to do with the rate of light generation right, since the transition of an electron from a high energy level to a low one releases a photon, yes?
Yes, the 2p state decays spontaneously to the ground state (1s) by emitting a photon. This decay occurs with a mean lifetime of 1.6 nanoseconds (equivalently, a half-life of 1.1 nanoseconds). The mechanism of decay is called the electric dipole transition.

Yesterday, our professor outlined the steps to calculate the 2s to 1s transition via a single-photon emission. It turns out that because the electric dipole transition is forbidden here, the calculation of the mean lifetime is a lot harder as you have to consider effects from special relativity (Mathematica helps here). Still doable though, and the result you get is a whopping 4.6 days. Does this mean that if you excite a hydrogen atom to the 2s state, then it won't decay for days!?

Well, no. There is a much easier way to decay: if you calculate the lifetime of the 2s to 1s transition by emitting two photons instead of just one, then you get a lifetime of 0.14 seconds, which is what is experimentally observed! So the 2s to 1s transition occurs primarily by emitting two photons simultaneously, rather than emitting a single photon (which is much more rare event).

Well, so what's the moral of the story? Uh ... none really! :)


Fri Dec 06, 2013 2:33 pm
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Post Re: Playing with math
It's a really awesome tool. The class for it is more than undesirable, but whatever I guess. My favorite thing about it is that it can write Taylor Polynomials for you. I don't have a problem with writing them myself, they're just really really meticulous. The university gives it to all math students for free which is phenomenal. TrigExpand[ * * expression * * ] is really helpful as well.

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Mon Dec 09, 2013 5:48 pm
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Post Re: Playing with math
You usually don't need Taylor polynomials past the first few terms, though. 'Cept in pure math where you need their explicit formula in the first place. :P

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Tue Dec 10, 2013 1:08 am
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