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tgt
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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.
prasannaworld said: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...
tgt said:Is that a 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.prasannaworld said:Give me one "Definition" that boils to this one...
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].)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...
fluidistic said: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].)
TD said: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.
Kurret said:but what is a "beautiful" definition?
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?"...tgt said: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.
Kurret said: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?"...
kev said: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".
Diffy said:Here you 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.
quasar987 said:And CRGreathouse, what is this definition of congruence class mod m you are referring to?
HallsofIvy said:I have no idea how you think that shows "the connection between mathematics and physics". That's from Euclid and surely is not the way the word "point" is used in physics.
HallsofIvy said:I have no idea how you think that shows "the connection between mathematics and physics". That's from Euclid and surely is not the way the word "point" is used in physics.
mathwonk said:i agree that the free basis definition is typical of the most beautiful definitions, namely definitions by universal mapping properties.