Theorems every mathematician should know

[discrete*]

Well, a theorem is important not only because it is useful. A theorem can also be important because it has been an open question for a long time, or because it has made an impact on the mathematical world.

I think the continuum hypothesis satisfies that. When studying countability and uncountability, students of mathematics naturally come up with the continuum hypothesis. In fact, every first-year student of mathematics is confronted with the continuum hypothesis in some way. And it's the first statement that is shown to be independent of ZFC, something which is often hard to grasp for students. I don't think there is any mathematician out there which has never heard of the continuum hypothesis, and regardless of it's usefulness, that implies that the theorem is important.

And then there's the fact that Cohen won the Fields medal for his work, which means that the question must have had some importance. The continuum hypothesis was also one of Hilbert's millenium problems, which further adds to it's significance.

There are another set of results that satisfy thesame criteria: the insolvability of the quintic, the parallel postulate, the transcendence of pi and e,...
While these may not seem to be important, their historic significance is overwhelming!

Well said. I agree with every bit.

 

[discrete*]

That the axiom of choice is independent of ZF: agreed, significant! But the continuum hypothesis? Nice to know, but I don't see the significance... The difference is that Choice leads to many important and useful results (hahn-banach, existence of maximal ideals, etc.), while I don't know any 'use' of the continuum hypothesis.

Why does the Continuum Hypothesis have to have a "use" for it to be of importance?

You say that the AoC is important -- which we can all agree upon -- because it leads to important results. How can you deny that of the Continuum Hypothesis? It may not lead to important results outside of the Foundations (which is probably arguable), but it has weight none-the-less.

 

[adoado]

The prime number theorem!

 

[adoado]

But the continuum hypothesis? Nice to know, but I don't see the significance...

Surely even the new insight itself brought by the theorem is enough? 'Usefulness' in terms of practicality and extension is great, but sometimes the theorem itself is just beautiful for it's intricacies and logical outcome. Maybe not every person will agree, but any theorem that has these qualities I personally feel I need to know..

 

[disregardthat]

My opinion is that a prerequisite for a theorem that "every mathematician should know" should be a broad scope of application throughout different fields in mathematics. The independence of the Continuum hypothesis from the axioms of set theory is hardly relevant to any field outside the study of formal set theory, and independence results came before that. If you want to go down that road, I believe that Gödels incompleteness theorems are way more important.

I can't think of a theorem more satisfactory to this criterion than Zorn's lemma. It is essential to great many vitally important theorems throughout mathematics, something which can be said for few other theorems that still are non-trivial.

And how are first-year students confronted with the continuum hypothesis?

 

[micromass]

And how are first-year students confronted with the continuum hypothesis?

When I was a first-years student, I was confronted with the continuum hypothesis while I was learning basic set theory. I was immediately intrigued by the theorem and shifted my entire world-view of mathematics. Before, I thought everything can be proven by math, but thanks to (CH) I realized that this is not true, and that it's the choice of axioms that matter.

It's not only the mathematics consequences that matter to me, it's also the philosphical consequences. And that's why I think (CH) is quite important. In fact, I don't think any mathematician has never heard of the continuum hypothesis. Not because it is important, but because it has a lot of consequences about how you think of math.

 

[fourier jr]

definitely the 1st isomorphism theorem. It almost always comes in handy in two fairly common situations
1. showing a subset is a normal subgroup (or ideal or submodule, etc) by showing that it's the kernel of a homomorphism
2. showing two things are isomorphic

 

[fourier jr]

the chinese remainder theorem is another good one imho

 

[Robert1986]

So, I think that the title of this thread is kind of silly. IMO, there are very few theorems that EVERY mathematician should know. However, I think that these are very important in combinatorics:

Dilworth's Theorem and its dual

In all likelihood, only a Combinatorialist would REALLY need to know this; nonetheless, it is vitally important.

 

[disregardthat]

So, I think that the title of this thread is kind of silly. IMO, there are very few theorems that EVERY mathematician should know.

There are many theorems every mathematician must know. Basic knowledge of analysis is for example always necessary.

 

[Calrid]

1+1=2

After all strictly speaking its the basis of all maths.

I kid probably go for

e^{i \pi} + 1 = 0\,\!

Eulers proofs and the application of this to all trigonometric functions.

 

[Calrid]

That the axiom of choice is independent of ZF: agreed, significant! But the continuum hypothesis? Nice to know, but I don't see the significance... The difference is that Choice leads to many important and useful results (hahn-banach, existence of maximal ideals, etc.), while I don't know any 'use' of the continuum hypothesis.

No I started a thread about how it is basically not even philosophically consistent.

I like it as an aesthetic idea but it cannot ever have any use IMO so its kind of like claiming fairies exist, great I can draw nice pictures of them or how I think they might look if I actually saw one but what does that prove?

Infinity is not equal to aleph 0 it is symbolically approximate and larger than anything I can conceive, so in fact infinity is in fact larger than the continuum or the same size making it trivial and useless, bigger than or smaller than the limit is useless and the same as it is again trivial.

It's a quaint little idea that probably anamours people to epistemologically dubious axioms without actually saying anything about anything. In other words pure mathematicians will probably love it and applied mathematicians will think it is pointless.

 

[Landau]

Infinity is not equal to aleph 0 it is symbolically approximate and larger than anything I can conceive, so in fact infinity is in fact larger than the continuum or the same size making it trivial and useless, bigger than or smaller than the limit is useless and the same as it is again trivial.

I have no idea what you are saying. But let's stay on-topic.

 

[myth_kill]

i request you people to go through this:

http://mathoverflow.net/questions/19356/how-has-what-every-mathematician-should-know-changed

 

[Robert1986]

There are many theorems every mathematician must know. Basic knowledge of analysis is for example always necessary.

I'm only an under-grad so I cannot profess to speak from experience as a mathematician. When you say basic knowledge of analysis, what exactly do you mean? It seems to me that someone who is concerned with say, number theory and algebra, should never "need" to know that a set is compact if and only if it is closed, but this is used quite a lot in introductory analysis classes.


But, perhaps I am wrong as the title of the thread is "Theorems every mathematician should know", not "Theorems every mathematician needs to know to be able to barely function in his field of research." So, yes, on second thought, you are probably correct, mathematicians should probably understand some analysis.

 

[disregardthat]

It seems to me that someone who is concerned with say, number theory and algebra, should never "need" to know that a set is compact if and only if it is closed, but this is used quite a lot in introductory analysis classes.

In algebra it is vital to have solid knowledge of topology which basically requires or incorporates elementary analysis. Modern number theory makes heavy use of complex analysis which also requires analysis.

A set is not compact if and only if it is closed, by the way. Neither of the implications are true.

 

[Pythagorean]

v-e+f=2

 

[Alan1000]

There are 10 kinds of people in the world: those who understand binary notation, and those who don't.

Seriously though, I second the nomination of Pythagoras' Theorem. This was the earliest example of a genuinely profound mathematical insight, built upon axiomatic foundations which were probably barely as old as Pythagoras himself. An awe-inspiring achievement, the mathematical equivalent of the Parthenon.

 

[Robert1986]

In algebra it is vital to have solid knowledge of topology which basically requires or incorporates elementary analysis. Modern number theory makes heavy use of complex analysis which also requires analysis.

A set is not compact if and only if it is closed, by the way. Neither of the implications are true.

Ahh, yes, closed and bounded.


Anyway, I am only in my second semester of algebra, and I haven't come across topology, as of yet. Does the knowledge of topology come in more advanced algebra?

 

[micromass]

Ahh, yes, closed and bounded.

I don't mean to be annoying, but not even closed and bounded is equivalent with compact You'll need complete and totally bounded, and even that is only true is metric spaces... Indeed, compactness is quite a sensitive property...

Anyway, I am only in my second semester of algebra, and I haven't come across topology, as of yet. Does the knowledge of topology come in more advanced algebra?

Topology is very important in advanced algebra. In particular, you can't do algebraic geometry without knowing your topology!

 

[Robert1986]

I don't mean to be annoying, but not even closed and bounded is equivalent with compact You'll need complete and totally bounded, and even that is only true is metric spaces... Indeed, compactness is quite a sensitive property...



Topology is very important in advanced algebra. In particular, you can't do algebraic geometry without knowing your topology!

Ok, so a subset of R^n is compact if and only if it is closed and bounded. This is the Heine Borel Theorem, isn't it (at least restricted to R^n)?

At any rate, and I admit I was wrong in my first post in this thread, if you are doing just general ring theory, for example, do you really need to know topology? Or do you only need to know topology if you are doing something like algebraic geometry, for example?

 

[micromass]

Ok, so a subset of R^n is compact if and only if it is closed and bounded. This is the Heine Borel Theorem, isn't it (at least restricted to R^n)?

At any rate, and I admit I was wrong in my first post in this thread, if you are doing just general ring theory, for example, do you really need to know topology? Or do you only need to know topology if you are doing something like algebraic geometry, for example?

I've still seen topology when doing rings. Specifically, when discussing completions of rings. But I know where your coming from and I think you're probably right. There are many fields in mathematics where one can do research and not use topology or analysis at all. So in that respect, there is no single theorem that a mathematician should know. All the theorems one needs to know can differ from field to field.

But I still think that there are an amount of theorems that a mathematician should know as "culture", not because it is useful. And Heine-Borel certainly is one of them. Even if some mathematicians see no use for it...

 

[mathwonk]

rank-nullity, riemann roch, fundamental theorem of calculus, poincare - hopf theorem, gauss bonnet, big-little picard thorems, mittag leffler, Fourier transform, taylor theorem, cauchy theorem, green's - stokes theorem, hurewicz theorem on homotopy/homology groups, archimedes formulas on areas and volumes of spheres, pappus' theorems, pythagoras' theorems including law of cosines, riemann's theorem and riemann's singularities theorem, mordell's theorem and faltings' theorems, unique factorization theorems, zariski's main theorem,...

 

[Pythagorean]

There are 10 kinds of people in the world: those who understand binary notation, and those who don't.

Seriously though, I second the nomination of Pythagoras' Theorem. This was the earliest example of a genuinely profound mathematical insight, built upon axiomatic foundations which were probably barely as old as Pythagoras himself. An awe-inspiring achievement, the mathematical equivalent of the Parthenon.

Awww, thanks guy

 

[Robert1986]

I've still seen topology when doing rings. Specifically, when discussing completions of rings. But I know where your coming from and I think you're probably right. There are many fields in mathematics where one can do research and not use topology or analysis at all. So in that respect, there is no single theorem that a mathematician should know. All the theorems one needs to know can differ from field to field.

But I still think that there are an amount of theorems that a mathematician should know as "culture", not because it is useful. And Heine-Borel certainly is one of them. Even if some mathematicians see no use for it...

I agree completely. What I wrote in my first post of this thread was wrong. There are certain things that every mathematician should know; not for any particular reason other than the fact that the person is a mathematician.

 

[myth_kill]

v-e+f=2

the Euler's formula

but i recently read an article which states that archimedes proved this one quite before euler although he stated it differently

 

[Jmf]

I study control engineering rather than mathematics, but:

Taylor's theorem, in all it's forms (one variable, multiple variables, complex variables, using vectors/matrices, with and without the bounds on the error. etc) since I think it's useful in so many different contexts.

Lots of people seem to learn the basic form of the series but not the error term when they learn basic calculus, which is a shame.

Incidentally I think all engineers, physicists, mathematicians etc. should learn the basics of vector and inner product spaces, and understand why function spaces can be viewed a vector space - the number of electrical engineering students I've seen that know how to find a Fourier series, but don't really understand them, is somewhat frightening.

 

[Rap]

Theorem: 5 out of 4 people have problems with rational numbers.

I thought it was 10 out of 8 - I stand corrected.. Did you know that 1 in \pi people have trouble with real numbers?

 

[Char. Limit]

And 1 in 3-i people have trouble with complex numbers.

 

[micromass]

And 0/0 persons have trouble with "indeterminate" terms...

 

[jhae2.718]

And \geqslant\!1 people have trouble with division by zero...

 

[wisvuze]

rank - nullity

 

[micromass]

The Stone-Weierstrass theorem! Which is, I think, one of the most beautiful results there is...

 

[LCKurtz]

The Stone-Weierstrass theorem! Which is, I think, one of the most beautiful results there is...

Hey! No fair! I already named that one.