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which is your favourite mathematical prof/teorem?
Posted: Fri Feb 08, 2013 12:12 pm
by marvin
Which is your favourite mathematical prof/teorem?
Mine:
Newton–Leibniz
http://en.wikipedia.org/wiki/Fundamenta ... f_calculusIdea of linkind definite and indefinite integral
or (Lagrange's) Mean Value Theorem:
http://upload.wikimedia.org/wikipedia/c ... heorem.svghttp://en.wikibooks.org/wiki/Calculus/S ... t_TheoremsWe use it all the time:P
Re: which is your favourite mathematical prof/teorem?
Posted: Fri Feb 08, 2013 12:26 pm
by Solomon
Re: which is your favourite mathematical prof/teorem?
Posted: Fri Feb 08, 2013 1:56 pm
by speedchase
I don't know if it is technically a theorem, but I love l'hopitals rule.
if lim x-> a (f(x)) is zero or infinity, and if lim x->a (g(x)) is zero or infinity
lim x->a (f(x)/g(x)) = lim x ->a (f'(x)/g'(x))
edit: Just to be clear, there are some other conditions, but what I wrote above is the jist of it
http://en.wikipedia.org/wiki/Lhopitals_rule
Re: which is your favourite mathematical prof/teorem?
Posted: Fri Feb 08, 2013 4:07 pm
by TheBigH
I've always like Euclid's proof of the infinitude of primes, and Cantor's diagonal argument that there are more real numbers than there are integers.
But my favourite would have to be:
Posted: Fri Feb 08, 2013 4:39 pm
by EdLee
In college, we encountered this particular wording by our professor
and it has remained my classmates' and my all-time favorite:
Picard's theorem --
In any neighborhood of an essential singularity,
a function takes on every possible value,
except perhaps one, an infinite number of times.
(my emphasis in red)
Re: which is your favourite mathematical prof/teorem?
Posted: Fri Feb 08, 2013 6:37 pm
by GoRo
Ptolemy's theorem:
Select any four points on a circle.
Call one of them A, the next one B, and following this
direction call the next one C and the last one D.
Measure the sides of that quadrilateral
a = AB
b = BC
c = CD
d = DA
and measure the two diagonals
e = AC
f = BD
Magically it turns out that a*c + b*d = e*f
Cheers,
Rainer
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 11:53 am
by drmwc
I like this proof, beacuse it uses a nuclear weapon to crack a nut:
Theorem
2^(1/n) is irrational for n>=3.
Proof
Suppose otherwise. Then there exists integers n (which is 3 or greater), p and q such that:
(2^(1/n))^n=(p/q)^n
Hence q^n+q^n=p^n.
However, this contradicts Fermat's Last Theorem.
QED
A separate argument is needed for n=2 - Fermat's Last Theorem is not strong enough!
(Unfortunately, the proof above actually turns out to be a circular argument if one follows the details through, and so it's not actually valid.)
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 12:42 pm
by GoRo
drmwc wrote:(1)... I like this proof
(2)... Fermat's Last Theorem is not strong enough!
(3)... circular argument ... not actually valid.
@(1): me too, thanks for sharing!
@(2): that's a fancy way to put it, but "not actually valid"
@(3): I can't detect a circulus vitiosus here.
Let me repeat - slowly and for the sake of simplicity for n=3.
Proposition: There are no natural numbers p and q
such that (p/q)^3 = 2.
Proof: Assuming the opposite we would have natural numbers
p and q such that (p/q)^3 = 2.
Then p^3 / q^3 = 2,
i.e. p^3 = 2 * q^3 = q^3 + q^3.
Let a = q, b = q, c = p.
Then a^3 + b^3 = c^3 for some natural numbers a, b and c.
That contradicts Fermat.
Thus the assumption led to a contradiction.
q.e.d.
Cheers,
Rainer
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 1:13 pm
by Bill Spight
I like the proof that the base angles of an isosceles triangle are equal.
The triangle is congruent to its own reflection. QED.
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 1:38 pm
by drmwc
Rainer
The circularity is non-trivial.
Assume we use Wiles' proof of FLT - as far as I know, it's the only known proof.
The proof shows equivalance of integer a^n+b^n=c^n with n>=3 to a "Frey curve" of the form y^2=x(x-a^n)(x+b^n). At this step of the proof, certain restrictions are placed on (a,b,c) such as them being pairwise coprime. Consider the equation we had p^n=q^n+q^n. The pairwise co-prime restiction applied to (p,q,q)is equivalent to a straightforward Euler proof that 2^(1/n) is irrational.
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 1:53 pm
by GoRo
drmwc wrote:The circularity is non-trivial.
Seems so
But the circularity is not only non-trivial, but non-existent, too.
Pardon me: I don't have to go into the details of the proof
of any theorem I use for my proof. If the proposition, I am going
to prove, is already a trivial subcase of the theorem, then so
what? That is no problem and there is no circulus vitiosus.
If you want to convince me of having used a circular argument
you should point to a certain line in my proof and say:
"Ha, here you are - that is something you use as if already
proved!"
For your convenience and easier pointing-out here are the
lines of my proof, numbered:
1. Assume natural numbers p and q such that (p/q)^3 = 2.
2. Then p^3 / q^3 = 2,
3. i.e. p^3 = 2 * q^3 = q^3 + q^3.
4. Let a = q, b = q, c = p.
5. Then a^3 + b^3 = c^3 for some natural numbers a, b and c.
6. That contradicts Fermat.
7. Thus the assumption led to a contradiction.
8. q.e.d.
The first error is in line ....?
Cheers,
Rainer
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 2:14 pm
by Fedya
TheBigH wrote:I've always like Euclid's proof of the infinitude of primes, and Cantor's diagonal argument that there are more real numbers than there are integers.
But my favourite would have to be:
http://xkcd.com/179/
Re: which is your favourite mathematical prof/teorem?
Posted: Sat Feb 09, 2013 2:15 pm
by jts
GoRo wrote:drmwc wrote:The circularity is non-trivial.
Seems so
But the circularity is not only non-trivial, but non-existent, too.
Pardon me: I don't have to go into the details of the proof
of any theorem I use for my proof. If the proposition, I am going
to prove, is already a trivial subcase of the theorem, then so
what? That is no problem and there is no circulus vitiosus.
If you want to convince me of having used a circular argument
you should point to a certain line in my proof and say:
"Ha, here you are - that is something you use as if already
proved!"
For your convenience and easier pointing-out here are the
lines of my proof, numbered:
1. Assume natural numbers p and q such that (p/q)^3 = 2.
2. Then p^3 / q^3 = 2,
3. i.e. p^3 = 2 * q^3 = q^3 + q^3.
4. Let a = q, b = q, c = p.
5. Then a^3 + b^3 = c^3 for some natural numbers a, b and c.
6. That contradicts Fermat.
7. Thus the assumption led to a contradiction.
8. q.e.d.
The first error is in line ....?
Cheers,
Rainer
6. (Or are you being pedantic about the fact that no step in the proof assumes Fermat, so 6 -> 7 is non sequitur?)
Re: which is your favourite mathematical prof/teorem?
Posted: Mon Feb 11, 2013 1:34 am
by perceval
a similar thread on this forum:
http://www.lifein19x19.com/forum/viewtopic.php?f=8&t=4885hCantors proof is also mentionned
Re: which is your favourite mathematical prof/teorem?
Posted: Mon Feb 11, 2013 2:07 am
by shapenaji
I always liked the integral of a gaussian over the reals.
Integral(e^(-x^2)) from -Inf to Inf
1) Square the integral,
2) Then change from cartesian to polar,
3) Solve integral
4) Take the square root
5) Square again
6) Multiply by 2 EDIT: BAD, ignore my silly half-infinites
7) Have dessert
