# Favorite equation

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.

Homework Helper
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.

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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. :tongue2:

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.

Gib Z said:
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?

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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.

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Staff Emeritus
Gold Member
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

Homework Helper
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.

Gold Member
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....

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Homework Helper
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? :P

EDIT: 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.

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

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.

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Homework Helper
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.

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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.

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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.

Homework Helper
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.

Homework Helper
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
$$x^x(1+ln(x))$$

Homework Helper
$$x^x(1+ln(x))$$

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.

Homework Helper
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 >.<

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.

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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?

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Homework Helper
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).

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

Staff Emeritus
Gold Member
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 :tongue2: ; so one must ask, is F=ma incorrect if we can't detect any discrepancy between it and F=dp/dt?

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

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.

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