Favorite Equation: Quadratic Formula - Solving for X

[rock4christ]

list your favorite equation and why

mine is -b + or - the square root of b² -4ac all over 2a

the quadratic formula. I found it easier than factoring for finding my x's

 

[FunkyDwarf]

e^(i.pi) = -1

just because its so Goddamn freaky

 

[murshid_islam]

without a doubt: e^{i\pi} = -1

i think it will be the favourite equation of many ppl around here.

 

[rdt2]

More suggestions (some of the symbols may not reproduce correctly):

http://www.mech.gla.ac.uk/~rthomson/teaching/equcharts.htm

 

[Gib Z]

Please, if it is your favourite, give Euler's Identity the respect it deserves:

e^{i\pi} + 1 = 0

I probably wouldn't be able to state my favorite, but here's one I found VERY useful: \int^b_a f(x) dx = F(b) - F(a) where F'(x)= f(x)

EDIT: Favorite equation of mine, here it is. I am a mathematician, but this is really beautiful.
F^{\rightarrow} = \frac{dp^{\rightarrow}}{dt}. Simple, effective, and has withstood the test of time. It's still correct to this date.

 

[neutrino]

The Euler's identity, of course, because I didn't realize a thing about it when I first saw it; it didn't hit me a like a rock in the stomach, not like a lightning out of the blue, and so on. And I still don't understand what the hype is all about! People see it like some sort of Hollywood movie featuring all the top stars, that's all.

 

[murshid_islam]

Please, if it is your favourite, give Euler's Identity the respect it deserves:

e^{i\pi} + 1 = 0

is that the way Euler originally wrote it? i think i read somewhere that Euler origially wrote it as e^{i\pi} = -1 and not in the more beautiful way you or other ppl nowadays writes it.

Favorite equation of mine, here it is. I am a mathematician, but this is really beautiful.
F^{\rightarrow} = \frac{dp^{\rightarrow}}{dt}. Simple, effective, and has withstood the test of time. It's still correct to this date.

isn't it \overrightarrow{F} \propto \frac{d\overrightarrow{p}}{dt}
and is it true for velocities close to the velocity of light?

 

[uart]

Although I think it's definitely a bit too geeky to have a "favorite" equation I have to say that the first time I saw the following it equation it really impressed me.

\prod_{n=1}^\infty \frac{1}{1-(1/p_n)^a} = \sum_{n=1}^\infty 1/n^a

where p_n is the n_th prime and a>1.

 

[Hootenanny]

isn't it \overrightarrow{F} \propto \frac{d\overrightarrow{p}}{dt}
and is it true for velocities close to the velocity of light?

No, its as Gib Z written it;

\vec{F} = \frac{d\vec{p}}{dt}

And yes, if applied correctly, is valid in Special Relativity. However, note that F=ma is not valid when v>0.01c

 

[Curious3141]

Definitely Euler's identity for me. When I first saw it (I think when I was around 15 or so), my mind was blown because the equation immediately suggested a way to define the logarithms of negative reals. And that's really, really cool.

 

[ranger]

I'll go simple. My favs are Kirchoff's voltage and current law. I use 'em practically everyday.

 

[3trQN]

Hmm, i have a few favourites, in order of their cognitive bias from greatest to least:

a^2 + b^2 = c^2

e^{ix} = cos(x) + i sin(x)

F(t) = \frac{1}{\sqrt{(2\pi)}}\int^{\infty}_{-\infty} e^{itx}f(x)dx

and finally Maxwell's equations, which i don't fully remember or understand but when i do they will be next in the list...

 

[Gib Z]

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

BTW uart, its not too geeky for have a favorite equation :) and that's a nice relation you've chosen, perhaps you could prove the related Riemann hypothesis for me? :PEDIT: Forgot to address this. Euler Originally wrote it as Murshid_islam states it, but on later realisation of the equations profound consequences, changed it to the new form. He was inclined to do so by many collegues and espically number theorists, who found beauty in equations that were equated to zero.

 

[quasar987]

F(t) = \frac{1}{\sqrt{(2\pi)}}\int^{\infty}_{-\infty} e^{itx}f(x)dx

Written as such, this is nothing. The cool-looking thing is that

f(t)=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}f(\chi)e^{-i\omega\chi}d\chi \right)e^{i\omega t}d\omega

 

[rock4christ]

I also kinda like aⁿ + bⁿ =/= cⁿ

Fermat's theorem. was a pain to prove

 

[MeJennifer]

Forgive my ignorance but why do some here consider Euler's identity so special?

To me it seems that Euler's identity is a trivial instance of Euler's formula.
Furthermore the presence of \pi is not significant, it is only there if you decide to measure angles in radians.

 

[Curious3141]

Forgive my ignorance but why do some here consider Euler's identity so special?

I don't know about others, but speaking for myself :

1) It's a breathtakingly simple looking result that beautifully ties up four mathematical constants (e, pi, 0 and 1) in a single equation (at least when you write it with the RHS equal to zero).

2) It allows one to define the logarithm of a negative number as a complex number, as I've already alluded to.

3) It is an important result that allows a quick proof of pi's transcendence via the Lindemann-Weierstrass theorem (actually the exp(2*pi*i) = 1 variant is the one used most often here).

To me it seems that Euler's identity is a trivial instance of Euler's formula.

Maybe trivial to derive (from Euler's formula, which in itself is a beautiful tie-up between exponentiation and trig and leads to the shortest possible proof of De Moivre's theorem), but hardly trivial in its significance.

Furthermore the presence of is not significant, it is only there if you decide to measure angles in radians.

Pi is a mathematical constant, it need have nothing to do with measuring any angle as far as (the "four constant") Euler's identity goes. As for Euler's original formula, well, it's understood that the trig ratios have to be evaluated with arguments in radians. There is nothing arbitrary in this, radian measure is also considered by most to be far more fundamental than any other commonly used unit of angle measure (degrees, grad, etc.) It's the same sort of "natural bias" (pun not intended) when one compares a natural log with a common one.

And if you want to treat trig functions as abstractions without any immediate reference to angles or triangles (as is often the case in analysis), then you should leave everything in radians, far more elegant that way.

 

[DieCommie]

Well, it contains unique numbers... The square root of negative one, pi, e, 1 and 0. All these are very special numbers and they can be equated. Who would have thought that raising e (crazy number with elegant properties) to the power of i (an imaginary number that doesn't quite make sense) times pi (the ratio of circumference/diameter) and add one (an obvious important number) and it all equals zero (a number humanity struggled to come to grips with)? wtf? <--thats what I think. Like it was said before, kind of creepy.

 

[d_leet]

Well, it contains unique numbers... The square root of negative one, pi, e, 1 and 0. All these are very special numbers and they can be equated. Who would have thought that raising e (crazy number with elegant properties) to the power of i (an imaginary number that doesn't quite make sense) times pi (the ratio of circumference/diameter) and subtract one (an obvious important number) and it all equals zero (a number humanity struggled to come to grips with)? wtf? <--thats what I think. Like it was said before, kind of creepy.

That should be add.

 

[Gib Z]

Fermats LAST Theorem, n has to be an integer larger than 2, in the case of 2 its just Pythagoras...a b and c have to be postive integers as well. Very painful to prove.

 

[MeJennifer]

There is nothing arbitrary in this, radian measure is also considered by most to be far more fundamental than any other commonly used unit of angle measure (degrees, grad, etc.) It's the same sort of "natural bias" (pun not intended) when one compares a natural log with a common one.

Well, then I respectfully disagee with those.

In my opinion, more fundamental would be to use for instance the simple range [0, 1] or even better [-1,1]. To me to use of the term \pi is just getting fancy, it is really completely insignificant to me.

 

[Gib Z]

LOL Using the range [-1,1] would completely destroy a huge chunk of calculus. No disrespect Jennifer, I can tell you know a lot more than myself, but radian measure is the most natural. Heres one example, the derivative of sin. In radians, its a nice cos function. In degrees, its not so nice. Also, it can be shown that any number that is not a rational multiple of PI, other than zero, the sin, cos, or tan of that number will be transcendental. No beautiful relation like that arises from any other angle measure.

Not to mention, when using radian measure with certain taylor series, than integrating them, it gives series for pi. That does not happen for any other angle measure, and it wouldn't give a series for what the measure is based on either, incase you were thinking its cause pi is the radian measures base.

Just like previous posts, you just wuoldnt expect it! It leads to many beautilful results.

 

[theperthvan]

x^x(1+ln(x))

 

[Curious3141]

x^x(1+ln(x))

That's not an equation unless you prepend 'd/dx(x^x) ='.

 

[rock4christ]

Im also curious:
most useful equation and why(this will definitely depend on what you do)
least useful and why

most:grams/molar mass=mol

least: quadratic formula. I love it but its useless

 

[arunma]

If this were one of the physics forums, I'd probably cite Maxwell's Equations. But since this is the world of math, I'm definitely going to have to go with the Fundamental Theorem of Calculs. But Uart's example was also interesting...

Although I think it's definitely a bit too geeky to have a "favorite" equation I have to say that the first time I saw the following it equation it really impressed me.

\prod_{n=1}^\infty \frac{1}{1-(1/p_n)^a} = \sum_{n=1}^\infty 1/n^a

where p_n is the n_th prime and a>1.

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.

 

[Gib Z]

Most useful - \int^b_a f(x) dx = F(b) - F(a) where F'(x)=f(x).
Why: Helps alot.

Least Useful - \sum^{\inf}_{n=1} n = \frac{-1}{12}

Why: Its cool, but I've never found a use for it. Maybe when I do string theory >.<

 

[Gib Z]

arunma- That equation is derived here: http://en.wikipedia.org/wiki/Riemann_zeta_function

 

[AsianSensationK]

for r > 0,

\Sigma{(\frac{1}{1+r})^n} = \frac{1-(\frac{1}{1+r})^n}{r}

This is the multiplying factor for present value of an annuity with level payments. Simply marvelous in it's uses in the field of finance.

 

[murshid_islam]

I also kinda like aⁿ + bⁿ =/= cⁿ

Fermat's theorem. was a pain to prove

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.

Least Useful - \sum^{\inf}_{n=1} n = \frac{-1}{12}

is that a Ramanujan summation? or am i confusing it with something else?

 

[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.