What is the most beautiful definition you've encountered?

[tgt]

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.

 

[Mentallic]

Perpendicular distance from a point to a line in coordinate geometry

I was the only one in my class to appreciate the formula though.

 

[prasannaworld]

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]

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...

Is that a definition?

 

[prasannaworld]

Is that a definition?

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...

 

[Kurret]

Give me one "Definition" that boils to this one...

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 don't get how a definition can be beautiful? Sure, a proof or a theorem can be elegant, but what is a "beautiful" definition?

 

[fluidistic]

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...

The number 0 also satisfies the condition above, so e is not unique in that case. (well if you consider $$x\neq 0$$.)

 

[prasannaworld]

The number 0 also satisfies the condition above, so e is not unique in that case. (well if you consider $$x\neq 0$$.)

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...

 

[TD]

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.

 

[CRGreathouse]

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

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 intent was to define e, not e^x. e is the unique positive solution of a^x = d/dx a^x.

 

[TD]

Oh of course, I misread!

 

[evagelos]

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

 

[fluidistic]

$$\vec{L}=\vec{r}\times \vec{p}$$ definition of angular momentum.

 

[tgt]

but what is a "beautiful" definition?

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]

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..

 

[CRGreathouse]

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?"...

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.

 

[Jame]

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.

 

[disregardthat]

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.

 

[Diffy]

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.

 

[yuiop]

I like Euler's identity:

$$e^{i \pi} + 1 = 0$$

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:

$$i = \frac{\ln(-1)}{\pi}$$

and

$$\pi = \frac{\ln(-1)}{i}$$

but there may be restrictions on the above "definitions".

 

[yuiop]

I like Euler's identity:

$$e^{i \pi} + 1 = 0$$

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:

$$i = \frac{\ln(-1)}{\pi}$$

and

$$\pi = \frac{\ln(-1)}{i}$$

but there may be restrictions on the above "definitions".

and

$$e = (-1)^{(\frac{1}{i \pi})}$$

?

 

[quasar987]

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.

This is cool sounding. Have you seen this definition in any particular book?

And CRGreathouse, what is this definition of congruence class mod m you are referring to?

 

[Diffy]

At the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, gave the following definition of a set:[5]
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 [Anschauung] or of our thought.

 

[CRGreathouse]

And CRGreathouse, what is this definition of congruence class mod m you are referring to?

Just the ordinary one -- {mk + n, k in Z}. It's essentially the first step toward p-adics.

 

[JasonRox]

The definition of a topological space.

 

[atyy]

I know this is actually meaningless for mathematicians, but I like it because it shows the connection between mathematics and physics: "A point is that which has no part."

 

[HallsofIvy]

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.

 

[tgt]

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.

Point particles have no parts, no? Although electrons have spin. How can something that have no parts have spin?

 

[atyy]

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.

My understanding is that "has no part" is not formally used to deduce geometrical statements. But if I want to draw a picture representing the geometry, then I use the idea that a point "has no parts". Drawing a triangle or a circle on a flat piece of paper is the physical representation of the geometry.

 

[quasar987]

Hey, I just stumbled upon this on the wiki page about relations:

"When two objects, qualities, classes, or attributes, viewed together by the mind, are seen under some connexion, that connexion is called a relation." (Augustus De Morgan[1])

 

[quasar987]

I don't know if I find the definition of a topological space beautiful in itself, but the whole abstractization of the "theory of the neighborhood" (i.e. freeing it from epsilons!) is certainly a beautiful feat of the mind. Although yes, since the definition of a topological space captures the essence of the idea of a neighborhood and all its power in 3 simple yet mysterious axioms, then I can definitely see how it has some beauty in itself! I guess it's my favorite definitions too then.

Not a mathematical dfn but it appears in an old absolutely beautiful philosophy book attempting to explain the human mind by starting from a handful of definition and axioms about the most elementary but fundamental concepts about the universe and working upwards by means of the "Lemma, Theorem, Corolary" formula. This is one of the first definitions of the book I think:

"Per aeternitatem, intelligo ipsam existensiam" (By eternity, I mean existence itself)

:!)

 

[Psycopathak]

I don't know if this is ok, but It's just amazing

Euler's Identity:

e^i*pi = -1

It relates the exponential base which is found with calculus, the imaginary unit which literally has to be made up to solve functions where there are no real solutions, and pi, which links all of geometry together. And it all equals -1!

It relates algebra, geometry and calculus to the most basic number.

 

[mathwonk]

i agree that the free basis definition is typical of the most beautiful definitions, namely definitions by universal mapping properties.

 

[tgt]

i agree that the free basis definition is typical of the most beautiful definitions, namely definitions by universal mapping properties.

It took ages to understand them. Do you know who invented them?

 

[mathwonk]

i believe peter freyd, in his book on categories and functors, credits maclane with this type of definition in a paper on groups.