Theorems every mathematician should know

[Char. Limit]

This is getting off topic. Therefore, I'm going to submit Fubini's Theorem.

 

[Mathitalian]

Banach fixed point theorem. I love this theorem :)

 

[Curl]

Fundamental theorem of line integrals.

Clairaut's theorem

 

[discrete*]

How about the always overshadowed Fermat's Little Theorem.

 

[discrete*]

Banach fixed point theorem. I love this theorem :)

Nice one. And because you mentioned Banach, how about the Banach-Tarski Paradox. I have been fascinated by this theorem for years.

 

[fourier jr]

How about the always overshadowed Fermat's Little Theorem.

or euler's theorem! one of the first things I thought of were the isomorphism theorems, especially the first one.

 

[jasonRF]

How about Cauchy's theorem from complex analysis, as proven by Goursat.

 

[LCKurtz]

Stone-Weierstrass theorem:

If X is any compact space, let A be a subalgebra of the algebra C(X) over the reals R with binary operations + and ×. Then, if A contains the constant functions and separates the points of X, A is dense in C(X) equipped with the uniform norm.

 

[nonequilibrium]

$$\forall$$ metric spaces $$\exists$$ metric space such that he can say "you complete me"

 

[Landau]

I feel like people are responding to either 'what is your favourite theorem?' or 'name a random theorem'. Of course it's quite subjective which theorems every mathematician "should" know, but perhaps we could try to give arguments?

For example, I can imagine a number theorist or logician never have to use or know Fubini's theorem. So why should he know it?

 

[micromass]

I feel like people are responding to either 'what is your favourite theorem?' or 'name a random theorem'. Of course it's quite subjective which theorems every mathematician "should" know, but perhaps we could try to give arguments?

For example, I can imagine a number theorist or logician never have to use or know Fubini's theorem. So why should he know it?

Well perhaps the question isn't well-frazed. The way I interpret the question is "What theorem do you want every math student to know". And certainly Fubini's theorem is something that every math student should have heard about. Maybe they will never use it later on, but I think they should still know it as a form of general culture.

Another theorem I would like to nominate is Taylor's theorem. It's importance is well-established. I have used it in analysis, probability theory, number theory,... Moreover, you can use the theorem to give approximations to a variety of functions. And a lot of useful inequalities are coming from the theorem. I don't think I could call anybody a mathematician if they have never heard of Taylor's theorem...

 

[discrete*]

I feel like people are responding to either 'what is your favourite theorem?' or 'name a random theorem'. Of course it's quite subjective which theorems every mathematician "should" know, but perhaps we could try to give arguments?

For example, I can imagine a number theorist or logician never have to use or know Fubini's theorem. So why should he know it?

Perhaps the best way to avoid the differences of opinions between different types of mathematicians is to state what theorems everyone should know based on their historical significance and general impact.

That's the basis of my argument for Godel's Incompleteness Theorems. They may not have particular utility for many (most) mathematicians, however the impact that Godel made was meteoric.

Of course, theorems that come into use a lot, like the aforementioned Taylor's Theorem have a particular utility that automatically earns them a spot on such a list.

 

[micromass]

Perhaps the best way to avoid the differences of opinions between different types of mathematicians is to state what theorems everyone should know based on their historical significance and general impact.

That's the basis of my argument for Godel's Incompleteness Theorems. They may not have particular utility for many (most) mathematicians, however the impact that Godel made was meteoric.

Of course, theorems that come into use a lot, like the aforementioned Taylor's Theorem have a particular utility that automatically earns them a spot on such a list.

Yes, I completely agree with Godel's Incompleteness Theorems in that respect. Another theorem that is important for the same reason is Cohen's result that the continuum hypothesis and the axiom of choice is independent of ZF. I have never used this result, but I think the significance of the theorem is huge!

 

[Landau]

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.

 

[micromass]

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!

 

[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!