Favorite Equation: Quadratic Formula - Solving for X

[Gib Z]

Ahh but both sides are TRUE, shown by sign to be not equal :) Its still an equation.

and yes, that's Ramanujans Summation, or zeta(-1).

 

[Gib Z]

O and not to mention, they can be equal, he didn't say that a, b and c had to be positive integers.

 

[Hootenanny]

In fact, F=ma is incorrect for any velocity larger than zero :P as small as the error is

So small in fact that we can't measure it ; so one must ask, is F=ma incorrect if we can't detect any discrepancy between it and F=dp/dt?

 

[Gib Z]

Ahh I am sure theoretically, we could be able to measure it, if the velocity in itself was measurable.

 

[uart]

But Uart's example was also interesting...
$$\prod_{n=1}^\infty \frac{1}{1-(1/p_n)^a} = \sum_{n=1}^\infty 1/n^a$$

This, I must admit, is pretty awesome. I wonder how it's derived, especially since there's no obvious formula for calculating the nth prime.

Yes that's the thing that impressed me about this equation. It looks so unlikely and yet the proof is actually very simple, requiring nothing more than a binomial expansion and the fundamental theorem of arthimetic.

Start with the application of the binomial expansion, $$(1 - x)^{-1} = 1 + x + x^2 + x^3 + x^4 +\ldots$$ to each of the product terms on the LHS of the original equation.

This gives the LHS of the orgiinal equation as,

$$\prod_{i=1}^\infty (1+(1/p_i^a)+(1/p_i^a)^2+(1/p_i^a)^3+ ...)$$

Or if you prefer to put in some numbers its,

$$(1 + \frac{1}{2^a} + \frac{1}{4^a} + \frac{1}{8^a} + \ldots)\, (1 + \frac{1}{3^a} + \frac{1}{9^a} + \frac{1}{27^a} + \ldots) \, (1 + \frac{1}{5^a} + \frac{1}{25^a} + \frac{1}{125^a} + \ldots) \ldots (1 + \frac{1}{p_k^a} +\frac{1}{p_k^{2a}} + \frac{1}{p_k^{3a}} + \ldots) \ldots$$

Now for the interesting part. If you understand the fundamental theorem of arithmetic (uniqueness of prime factorization) and you stare at the above expansion for long enough you'll suddenly realize why it is equal to the infinite sum on the RHS of the original equation. Try it and see, it's quite a revalation if you haven't seen it before and a startling demonstration of the uniqness of prime factorization.

 

[murshid_islam]

Ahh but both sides are TRUE, shown by sign to be not equal :) Its still an equation.

no it isn't. how can it be an "equation" if two sides of it are NOT equal?

O and not to mention, they can be equal, he didn't say that a, b and c had to be positive integers.

yeah, they can be equal. but then he would be wrong because he wrote it to be NOT equal.

 

[Gib Z]

aww fine be that way :P Its his favorite INEQUALITY :P

2nd bit, ill be a turd and say maybe he had cartain values of a,b and c in mind and forgot to tell us :P

 

[Alkatran]

isn't this thread about favourite "equations"? but what rock4christ mentioned is not technically an "equation". the two sides are not equal for $$n \geq 3$$. is that a Ramanujan summation? or am i confusing it with something else?

I typed "Ramanujan summation" into my address bar, forgetting to prefix it with "google" and a PDF download immediately started. Apparently Firefox uses I'm feeling lucky, scary.

I've always liked the infinite series equalities learned in first year calculus (power series, e^x, sin/cos, etc). Those blew my mind a lot more than Euler's identity (mainly because they implied it)

 

[Gib Z]

Yes those amazed me a lot as well. Its interesting to observe the properties on the series, and see how they match the original function. I have countless hours of fun finding the derivative of the sine series to get cosines and using the series for e^x to derive eulers formula :P. Eulers formula is cool because its simple, and its not implyed but true :)

 

[murshid_islam]

another fun thing i did with the maclaurin series is finding out different series to calculate the value of pi, e.g., the series for arctan(x).

 

[Gib Z]

Ahh yes that was fun, but usually converge slowly >.< The simplest case, and also the slowest converging is $$\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{-1^n}{2n+1}$$. I read somewhere it takes 10000 terms just to converge to 3 decimal places >.<

 

[complexPHILOSOPHY]

$$z_{0}=C$$;

$$z_{n+1}=z^2_n+C$$

Not that I pretend to understand it but I certainly appreciate the elegance involved in the mathematics and the visuals it produces are trippy.

 

[tehno]

You like fractals complexPHILOSOPHY,don't you?
I like your choice.

 

[disregardthat]

(cosx)^2+(sinx)^2 = 1

 

[complexPHILOSOPHY]

You like fractals complexPHILOSOPHY,don't you?
I like your choice.

I think all psychedelic heads do, my friend! I was enthralled by the visuals it produced before I even had an interest in mathematics so after discovering the maths behind it, it quickly became something that I truly appreciated.

 

[murshid_islam]

Ahh yes that was fun, but usually converge slowly >.< The simplest case, and also the slowest converging is $$\frac{\pi}{4} = \sum_{n=0}^{\infty} \frac{-1^n}{2n+1}$$. I read somewhere it takes 10000 terms just to converge to 3 decimal places >.<

yeah its the slowest i think. and as far as i remember, the fastest series for $$\pi$$ is the series given by ramanujan.

 

[Gib Z]

Just to re iterate my love for Eulers Formula to Jarle-

$$e^{ix}=\cos x + i \sin x$$
$$e^{-ix}=\cos x - i\sin x$$
Multiplying these two together:
$$\sin^2 x + \cos^2 x=1$$

Euler's formula can lead to that great Pythagorean Identity :D

 

[Gib Z]

Response to murshid_islams comment, there are actually faster series now, but they are all based on the work by ramanujan anyway :)

 

[ssd]

I am the first to mention,
E=mc^2.

 

[radou]

Blah, all of this is too fancy. I'll take 0 + 1 = 1.

 

[ranger]

I am the first to mention,
E=mc^2.

Thats my least favorite equation. I don't have use for it. I dislike it because it gave rise to atomic bombs and so on.

 

[robphy]

I am the first to mention,
E=mc^2.

Thats my least favorite equation. I don't have use for it. I dislike it because it gave rise to atomic bombs and so on.

This page "From E=mc² to the atomic bomb" (from Einstein Online, Max Planck Institute for Gravitational Physics)
http://www.einstein-online.info/en/spotlights/atombombe/index.html
has an enlightening discussion.
(I have no association with that site.)

 

[theperthvan]

If 1+1 doesn't equal 2, then the whole of maths EVER has been in vain.

 

[Gib Z]

Thank God Hilbert lobbied for the axiomization of mathematics and we have DEFINED 1+1 to equal 2 :)

 

[ssd]

Thats my least favorite equation. I don't have use for it. I dislike it because it gave rise to atomic bombs and so on.

For that matter, the person who invented 0, the person(s) invented calculus, laws of physics, quantum theory, Bose (for his statistic) ...every body and ennumerable men of pure math... and big number of elementory results of Physics, Math, Chemistry ... all are to be blamed for the atom bomb.

I like E=mc^2 not because of the fact that it gives rise to atom bombs... but because of the enormous talent, imagination and brain work behind the derivation of it and the scope of human knowledge to get extended (to have a true picture of universe) standing on shoulder of it (I mean relativity theory).

 

[theperthvan]

Thank God Hilbert lobbied for the axiomization of mathematics and we have DEFINED 1+1 to equal 2 :)

Yeah, it obviously does. But WHAT if it didn't?

 

[Gib Z]

I think you wanted to bold text with the IF...but well yea, if it didn't, we're screwed :)

 

[spacemonkey]

$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

 

[robert Ihnot]

fedora: Thats my least favorite equation. I don't have use for it. I dislike it because it gave rise to atomic bombs and so on.

E=MC^2, I am not sure it had much to do with the atomic bomb. Einstein at first called it only a "theoretical" value, but later wrote a letter to Pres. Roosevelt because the Germans had split the uranium atom in 1938. Under Hitler the Nazis were very skeptical of anything Einstein did, but were seeking to build the bomb.

It certainly was known that some things were radioactive. Then we have the energy of the sun. A really important discovery was the possibility of a chain reaction on Dec 2, 1942 by Enrico Fermi at University of Chicago. For the bomb, we need a chain reaction.

Einstein, you know, was no engineer and did not build things. Some have said his greatest contribution to the atomic bomb was, along with Szilard, his letter to President Roosevelt. As for a chain reaction, Einstein is quoted by Szilard as saying, "It never occurred to me." http://www.doug-long.com/einstein.htm

 

[d_leet]

If 1+1 doesn't equal 2, then the whole of maths EVER has been in vain.

Can I say, consider Z₂ (Z is the set of integers).