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Most beautiful definition? 
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#1
Oct1108, 06:15 AM

P: 468

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.



#2
Oct1108, 08:44 PM

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Perpendicular distance from a point to a line in coordinate geometry
I was the only one in my class to appreciate the formula though. 


#3
Oct1108, 11:55 PM

P: 21

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... 


#4
Oct1208, 05:01 AM

P: 468

Most beautiful definition?



#5
Oct1208, 05:33 AM

P: 21

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... 


#6
Oct1208, 09:17 AM

P: 124

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? :o 


#7
Oct1208, 09:24 AM

PF Gold
P: 3,189




#8
Oct1208, 01:29 PM

P: 21

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... 


#9
Oct1208, 02:16 PM

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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. 


#10
Oct1208, 02:41 PM

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I think the most beautiful definition for me (simple though it is!) would be Gauss' definition of congruence classes mod m.



#11
Oct1208, 02:45 PM

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Oh of course, I misread!



#12
Oct1208, 03:47 PM

P: 308

Beautiful is the way that you write down the definition in SYMBOLS and not the definition itself



#13
Oct1208, 04:27 PM

PF Gold
P: 3,189

[tex]\vec{L}=\vec{r}\times \vec{p}[/tex] definition of angular momentum.



#14
Oct1308, 05:33 AM

P: 468




#15
Oct1308, 12:25 PM

P: 124

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.. 


#16
Oct1308, 12:57 PM

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#17
Oct1808, 07:28 AM

P: 44

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.



#18
Oct2208, 12:44 PM

Sci Advisor
P: 1,797

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.



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