What's the most beautiful definition you've ever seen? For me, it has to be the definition of a free basis in group theory.
Perpendicular distance from a point to a line in coordinate geometry I was the only one in my class to appreciate the formula though.
Well call me premature if you will, but I reckon it is: e is a number such that: d/dx (e^x) = e^x I mean so much can be drawn from this...
YES IT IS! Give me one "Definition" that boils to this one... Using this definition one can derive the Maclaurin Series for e... Using this definition one can use l'Hopital's Rule to derive: e = lim (1+1/n)^n x->inf And by defining ln(x) to be the inverse function of e^x (i.e. Logarithm base e), one can go further and get Integral of ln(x) is 1/x - which some claim to be the first definition...
you could also define e to be e = lim (1+1/n)^n, x->inf and then derive the other results, which I think is a more common definition. Ontopic: I dont get how a definition can be beautiful? Sure, a proof or a theorem can be elegant, but what is a "beautiful" definition?
The number 0 also satisfies the condition above, so e is not unique in that case. (well if you consider [tex]x\neq 0[/tex].)
True... I still view that as the standard definition. To make it better how about: xER; obviously 0 can no longer work. Also on topic: I believe a "beautiful" definition in simple refers to one that is simple but a lot can be done with it/derived from it...
It still doesn't define e^x uniquely, because any c.e^x with c in R is good too. You can define f(x) = e^x as the function satisfying f(x)' = f(x) and f(0) = 1.
I think the most beautiful definition for me (simple though it is!) would be Gauss' definition of congruence classes mod m. I think the intent was to define e, not e^x. e is the unique positive solution of a^x = d/dx a^x.
Just like a beautiful proof. When it has a lot in it (i.e get something out of it every time you think about it) and gets to your heart.
I think a beautiful proof is totally different. Imo a beautiful proof is one that has some elegant and creative "trick" in it, that usually makes the proof short without a lot of messy computation, and usually makes you think "how did he think of that?"... Altough, when thinking about it, the construction of the real numbers with dedekind cuts is imo very cool and elegant, so that would maybe qualify as a beautiful definition for me..
To me a beautiful proof shows that two a priori unrelated things can be used together to show something interesting. A proof that comes out of nowhere (but gives no insight on why the unusual step is chosen) is less beautiful than one that is straightforward, to me at least.
I appreciate most definitions because they are what expresses the true intuition of the mathematician, like the spark of motivation that starts the tedious process of deduction. (S)he starts with what seems sensible and can't be unarguably justified. We sure could invent a lot of mathematics which has absolutely no interpretation, but the only mathematics that survives is the one that makes sense. I'm aware of the fact that most of mathematics is very far from reality, but it still makes sense to somebody, even if it's in a "fantasy" of the minds of a small group of mathematicians.
A good and beautiful proof is a constructive one, though general, which gives insight and suggests a direction of attack to any problem related to it. The "trick" is usually the discovery of this method of attack.
Here ya go: By a "set" we mean any collection M into a whole of definite, distinct objects m (which are called the "elements" of M) of our perception or of our thought.
I like Euler's identity: [tex] e^{i \pi} + 1 = 0 [/tex] allthough I am not sure if it strictly qualifies as a definition, the implicit relationship between the important mathematical constants e, pi and i is beautiful to me anyway. Ref http://en.wikipedia.org/wiki/Euler%27s_identity Euler's identity also implies: [tex] i = \frac{\ln(-1)}{\pi} [/tex] and [tex] \pi = \frac{\ln(-1)}{i} [/tex] but there may be restrictions on the above "definitions".