02 Mar 10
Originally posted by joe shmoIdentities are equations, just a particular kind of equation.
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)
02 Mar 10
Originally posted by PalynkaDepends. In general, it's true. But in many contexts "equation" is used to mean specifically those expressions that aren't identities.
Identities are equations, just a particular kind of equation.
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."
02 Mar 10
Originally posted by mtthwSo an identity is an equation whose solution is R for each variable in the equation.
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.
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.
02 Mar 10
Originally posted by mtthwI 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. 🙂
"Many authors reserve the term equation exclusively for the second type, to signify an equality which is not an identity."
02 Mar 10
1 edit
Originally posted by PalynkaI know, but he seemed to suggest that being pedantic isn't interesting! 🙂
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. 🙂
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.
02 Mar 10
Originally posted by mtthwMy 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.
That is a nice one, though you could argue it's an identity rather than an equation. If you wanted to be picky. 🙂
12 Mar 10
2 edits
Originally posted by FabianFnasbut the recent proof, while correct, doesn't have the qualities that led Fermat to state that:
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".
"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.
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14 Mar 10
Originally posted by Melanerpesunfortunatley 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.
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.
14 Mar 10
Originally posted by joe shmoThat 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.
Or maybe fermat just made a mistake?
We'll never be sure, though.
14 Mar 10
Originally posted by MelanerpesI 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.
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.