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7 Favorite Equations

Fkkize
Posts: 2,149
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8/29/2016 11:45:25 PM
Posted: 3 months ago
Recently I was reading Sean Carroll's blog where he wrote about a hashtag he started on twitter, #fav7equations. Which is a beautiful idea, so I thought DDO needs that, too.

(no particular order)

1. S = k log(w) - statistical entropy
2. dG = dH -TdS - Gibbs free energy
3. F = 1/(4 Pi epsilon0) * Q1 * Q2/r^2 - Coulomb force
4. H F = E F - Schrodinger equation
5. lambda = h/p - De-Broglie wavelength
6. Position/ momentum uncertainty
7. E = 3/2 * R * T - Kinetic energy/ temperature relation

What are yours?
: At 7/2/2016 3:05:07 PM, Rational_Thinker9119 wrote:
:
: space contradicts logic
Stronn
Posts: 318
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8/30/2016 6:28:56 AM
Posted: 3 months ago
Euler's identity, since it elegantly relates five fundamental mathematical constants.

e^{i * pi} + 1 = 0
Nottingham
Posts: 19
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8/30/2016 6:49:17 AM
Posted: 3 months ago
At 8/30/2016 2:45:43 AM, dee-em wrote:
From the holey book:

DELUSION = WHAT I THINK - THE FACTS : :

Another derailment by a hypocrite.

http://www.debate.org...
AlyceTheElectrician
Posts: 233
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8/30/2016 2:29:50 PM
Posted: 3 months ago
V=IxR

BMR= 66.5 +(9.6 x weight) +(1.8 x height) -(4.68 x age)

UKISS Circle of trust = r2pi
Be who you are, Say what you feel, Because those who mind don"t matter, And those who matter don't mind.

BANGTAN! Blood, Sweat, & Tears> Check it out yes! https://www.youtube.com...
Diqiucun_Cunmin
Posts: 2,710
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9/1/2016 1:38:45 PM
Posted: 3 months ago
-Pythagoras' theorem: a^2 + b^2 = c^2
-sin^2 theta + cos^2 theta = 1
-Fundamental Theorem of Calculus: int(a to b) f(x) dx = F(b) - F(a)
-Euler's formula: e^(i theta) = cos theta + i sin theta
-Bayes' theorem: P(A|B) = P(A)P(B|A)/P(B)
-Central Limit Theorem: lim(n -> infinity) P(sum(i=1 to n) (X_i - mu_i)/sqrt(sum(i=1 to n) sigma^2_i <= a) = Phi(a)
-Moment generating function theorem: d^k/dt^k E(e^(tX)) |(t = 0) = mu'_k

If inequalities were allowed (I know, I'm already stretching the definition of 'equation' a loooooot by including the CLT), then I would add the Triangle Inequality and the Cauchy-Schwartz Inequality, and take away the trig and MGF. If the CLT isn't allowed, then I'd add:
-Taylor's theorem: f(b) = f(a) + f'(a)(b-a) + f'(a)/2! (b-a)^2 + ... + f^(n)(a)/n! (b-a)^n + f^(n+1)(c)/(n+1)! (b-a)^(n+1)
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
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Response to conservative views on deforestation:
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Diqiucun_Cunmin
Posts: 2,710
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9/1/2016 1:43:03 PM
Posted: 3 months ago
gamma(1/2) = sqrt pi might be an honourable mention :P
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
http://www.debate.org...

Response to conservative views on deforestation:
http://www.debate.org...

Topics I'd like to debate (not debating ATM): http://tinyurl.com...
Diqiucun_Cunmin
Posts: 2,710
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9/3/2016 4:38:05 AM
Posted: 3 months ago
To make things more interesting, I suggest that we all justify our choices of equations and compare each others' criteria :P

There are several general criteria that I used when selecting the equations:
-I know the equation. This is obviously a necessary condition, because if I weren't acquainted with the equation, I'm not qualified to include it. So, while I'm aware of the existence of Green's theorem and its generalisations, I cannot include them.
-Beauty. Mathematical beauty is part of what justifies mathematics, outside of practical value (cf. Hardy.) Some beauty cannot be expressed in words but can only be felt after one has attained a certain level of mathematical maturity; other kinds of beauty are accessible to everyone (e.g. the Koch snowflake).
-Utility and influence. Equations whose influence extend well beyond the field where they originated should be included in the list. Unfortunately, this is why I did not include the Fisher equation, the equation of exchange or C + I + G + X - M in my list, as important as they are for economics.
-Non-obviousness. Some theorems merely reflect our intuition about a certain interpretation of mathematics, and are not surprising at all. This is why I excluded, for example, the theorem of total probability, Fubini's theorem, Rolle's theorem, the Fundamental Theorem of Arithmetic, and others.

Now I'll briefly write up why I included each one:
1. Pythagoras' theorem
I think this should be a no-brainer. It is neat, it is not obvious to anyone who is yet to learn it, yet a^2 + b^2 = c^2 and its generalisation to n-space are crucial in our understanding of space, mensuration and dimension. It is not surprising that the theorem is a cornerstone of mathematical education, and its entry in the Elements is one of the most celebrated.
2. Trigonometric identity sin^2 theta cos^2 theta = 1
Granted, you can prove this in one minute if you know Pythagoras' theorem, but this doesn't mean it's obvious: if nobody ever told you about it, you may never be able to derive it in the first place. The reason I included it is because of the way it often simplifies complicated calculations, such as allowing us to derive reduction formulae for integration and solve trigonometric equations, and because, like Pythagoras' theorem, it is one of any mathematics student's early excursions into mathematical beauty.
3. Fundamental Theorem of Calculus
Calculus is basically the mathematical foundation of much of modern science. It is essential to an understanding of classical mechanics, probability theory, area and volume, and more. The Fundamental Theorem of Calculus unites the two main branches of basic calculus - differentiation and integration - and its proof, involving the rate of change of area, is ingenious.
4. Euler's formula
Euler's formula is a generalisation of Euler's identity, which is beautiful for reasons Stronn has explained above. However, unlike Euler's identity, Euler's formula is very useful formula at the foundation of complex analysis, which is why I chose to include it.
5. Bayes' theorem
Whether you're a Bayesian or a frequentist, it's hard not to appreciate the beauty and utility of Bayes' theorem. You may think that Bayes' theorem fails the 'obviousness' criterion. While it's true that Bayes' theorem is close to the intuitive workings of our brains, we often ironically forget it when we try to think rationally. As an analogy, our neurons add up weights infallibly, but our brains often make mistakes when adding numbers up. Similarly, though our brains are inclined to intuit in a Bayesian fashion, our higher-level thought processes often violate Bayes' theorem. What I have in mind is, of course, the confusion of the inverse, which has led humanity to false conclusions and even false convictions. Only a good understanding of Bayes' theorem can fix such problems.
6. The moment-generating function theorem
Using the Taylor series expansion of e^x, the moment-generating function is an ingenious method of deriving (literally - it involves derivatives) the moments of a probability distribution. It is not infallible, as it either doesn't work (Cauchy) or complicates matters (Hypergeometric) for some distributions. However, it is one of the major methods of simplifying functions of random variables - thanks to the fact that the set of all moments uniquely identifies a distribution - and is also of great theoretical importance, as it features prominently in the proofs of fundamental results in probability theory, not least the Central Limit Theorem.
7. The Central Limit Theorem
Some people have put the normal CDF or PDF in their list of equations, but I went a step further (and cheated) by including the CLT (which isn't technically an equation). The CLT is among the most important results in modern probability theory and theoretical statistics and, I dare say, much of contemporary science. Its most important application is in the field of statistical inference, where it proves that the sample mean from a large enough sample follows the normal distribution. This has huge implications for the construction of confidence intervals and hypothesis tests, which in turn are of great importance in the experimental sciences, particularly the social and life sciences.
The thing is, I hate relativism. I hate relativism more than I hate everything else, excepting, maybe, fibreglass powerboats... What it overlooks, to put it briefly and crudely, is the fixed structure of human nature. - Jerry Fodor

Don't be a stat cynic:
http://www.debate.org...

Response to conservative views on deforestation:
http://www.debate.org...

Topics I'd like to debate (not debating ATM): http://tinyurl.com...
Axonly
Posts: 1,802
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9/9/2016 7:17:40 AM
Posted: 2 months ago
At 8/29/2016 11:45:25 PM, Fkkize wrote:
Recently I was reading Sean Carroll's blog where he wrote about a hashtag he started on twitter, #fav7equations. Which is a beautiful idea, so I thought DDO needs that, too.

(no particular order)

1. S = k log(w) - statistical entropy
2. dG = dH -TdS - Gibbs free energy
3. F = 1/(4 Pi epsilon0) * Q1 * Q2/r^2 - Coulomb force
4. H F = E F - Schrodinger equation
5. lambda = h/p - De-Broglie wavelength
6. Position/ momentum uncertainty
7. E = 3/2 * R * T - Kinetic energy/ temperature relation


What are yours?

http://www.lukewallin.co.uk...
Meh!
Dirty.Harry
Posts: 1,584
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9/11/2016 5:05:31 PM
Posted: 2 months ago
At 8/30/2016 6:28:56 AM, Stronn wrote:
Euler's identity, since it elegantly relates five fundamental mathematical constants.

e^{i * pi} + 1 = 0

Yep this is definitely up there.
Dirty.Harry
Posts: 1,584
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9/11/2016 5:08:39 PM
Posted: 2 months ago
At 8/29/2016 11:45:25 PM, Fkkize wrote:
Recently I was reading Sean Carroll's blog where he wrote about a hashtag he started on twitter, #fav7equations. Which is a beautiful idea, so I thought DDO needs that, too.

(no particular order)

1. S = k log(w) - statistical entropy
2. dG = dH -TdS - Gibbs free energy
3. F = 1/(4 Pi epsilon0) * Q1 * Q2/r^2 - Coulomb force
4. H F = E F - Schrodinger equation
5. lambda = h/p - De-Broglie wavelength
6. Position/ momentum uncertainty
7. E = 3/2 * R * T - Kinetic energy/ temperature relation


What are yours?

I'm quite fond of Stirling's formula a very fast way to approximate n! for larger n.

This lame website's tools prevent me from stating it clearly so look it up, pretty neat.
zaarbuc
Posts: 43
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9/12/2016 8:02:42 AM
Posted: 2 months ago
At 8/29/2016 11:45:25 PM, Fkkize wrote:
Recently I was reading Sean Carroll's blog where he wrote about a hashtag he started on twitter, #fav7equations. Which is a beautiful idea, so I thought DDO needs that, too.

(no particular order)

1. S = k log(w) - statistical entropy
2. dG = dH -TdS - Gibbs free energy
3. F = 1/(4 Pi epsilon0) * Q1 * Q2/r^2 - Coulomb force
4. H F = E F - Schrodinger equation
5. lambda = h/p - De-Broglie wavelength
6. Position/ momentum uncertainty
7. E = 3/2 * R * T - Kinetic energy/ temperature relation


What are yours?

The quadratic formula
Simply because I have a humorous story about my wife getting drunk at her former math teacher's house and trying to remember it on the back of a paper plate.