Favorite Equation

Originally posted by amolv06
Do you guys have a favorite equation?

Mine is e^(i*pi)+1=0.

It's also my favourite equation/identity. 5 of the most important numbers in mathematics elegantly combined.

Originally posted by joe shmo
A mathematical identity is one that remains constant regardless of the values of the variables in it.

ex.

(sin(A))^2 + (cos(A))^2 = 1

no matter what value you substitue for "A" the equation is true.

although now I'm slightly confused because there are no variables in e^(pi*i) = -1

?

Edit: ok actually its just a special case (x=pi) of

e^(ix) = cos(x) + i*sin(x)

Identities are equations, just a particular kind of equation.

Originally posted by Palynka
Identities are equations, just a particular kind of equation.

Depends. In general, it's true. But in many contexts "equation" is used to mean specifically those expressions that aren't identities.

From Wikipedia:

"An equation is a mathematical statement that asserts the equality of two expressions."

but then

"Many authors reserve the term equation exclusively for the second type, to signify an equality which is not an identity."

Originally posted by mtthw
Contrasting with an equation, which is true for particular values. A simple example:

2x = x + x, is an identity
2x = x + 1, is an equation with the solution x = 1.

So an identity is an equation whose solution is R for each variable in the equation.

I suspect that it is cheating to say
e^(i*pi)+1=0

is just a special case of
e^(ix) = cos(x) + i*sin(x)

Surely there are other possible identities it could be a 'special case' of?
eg: e^(ix) + x = cos(x) + i*sin(x) + x

And we still haven't really settled the question of whether an equation or identity must have variables. Presumably and equation with only constants has to be an identity.

I think we all know the difference between an identity and an equation.
Now we're just splitting hairs.
Is this really interesting, regarding the title of the thread?

Originally posted by mtthw
"Many authors reserve the term equation exclusively for the second type, to signify an equality which is not an identity."

I agree that in general identities are called identities (and not equations) but that doesn't mean they are not considered to be equations. It's just more precise to specify them as identities. But I agree with Fabian, maybe we're just splitting hairs. 🙂

Originally posted by Palynka
I agree that in general identities are called identities (and not equations) but that doesn't mean they are not considered to be equations. It's just more precise to specify them as identities. But I agree with Fabian, maybe we're just splitting hairs. 🙂

I know, but he seemed to suggest that being pedantic isn't interesting! 🙂

In the interests of actually contributing to the original intent of the thread...as an ex-fluid dynamicist, I'd probably have to go for the Navier-Stokes equation. Lots of interesting physical behaviour all tied up in a single equation. And although it's only an approximation (the continuum approximation) it gives an accurate representation of reality over a huge range of scales.

Originally posted by twhitehead
Don't all equations have that property?

No.

For instance in F=ma only works for certain values of F, m and a.

While (a+b)^2=a^2+b^2+2ab works for all values of a and b.

Originally posted by mtthw
That is a nice one, though you could argue it's an identity rather than an equation. If you wanted to be picky. 🙂

My bad. I was under the impression that an equation was just a statement which asserted the equality of two expressions i.e. anything with an equal sign.

Originally posted by FabianFnas
One of the most interesting equations, and one of the most famous ones too, I believe, is this:
a^n + b^n = c^n, where a,b,c is an integer >0 and n is an integer >2.
Fermat got interested in it, and myriads of mathematicians thereafter.

One of the matematicians working with this equation is Sophie Germain (1776 - 1831) who has one of the most interes re.

Simon Sing brought the attention to this equation in his book "Femat's Last Theroem".

but the recent proof, while correct, doesn't have the qualities that led Fermat to state that:

"I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

perhaps someone will eventually recreate Fermat's actual "truly marvelous" proof - it would have to use only techniques and theories that Fermat himself could possibly have developed or been aware of during his lifetime.

Originally posted by Melanerpes
but the recent proof, while correct, doesn't have the qualities that led Fermat to state that:

"I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

perhaps someone will eventually recreate Fermat's actual "truly marvelous" proof - it would have to use only techniques and theories that Fermat himself could possibly have developed or been aware of during his lifetime.

unfortunatley the mother of invention is necessity, so it may be a while before anyone descover the elegent proof. Or maybe fermat just made a mistake? What was he working on prooving while he came to this side problem that he didn't feel the need to outline a proof? I won't understand specifics, just trying to get the big picture.

Originally posted by joe shmo
Or maybe fermat just made a mistake?

That would be my guess - that he came up with something that looked right initially, but was missing something. It's entirely possible someone has since come up with the same "proof", but discarded it after realising the problem.

We'll never be sure, though.

Originally posted by Melanerpes
but the recent proof, while correct, doesn't have the qualities that led Fermat to state that:

"I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain."

perhaps someone will eventually recreate Fermat's actual "truly marvelous" proof - it would have to use only techniques and theories that Fermat himself could possibly have developed or been aware of during his lifetime.

I don't think that Fermat really had a proof. However, I think that Wiles' proof is unnecessary complicated. I can't wait to see a proof that is simpler and more beautiful.

Originally posted by amolv06
Do you guys have a favorite equation?

Mine is e^(i*pi)+1=0.

3 x 4 = 12