Theorems every mathematician should know

[jgm340]

Let's compile a list of theorems we think every mathematician ought to know!

I'll start:

Stoke's Theorem: If M is a smooth n-dimensional manifold, and \omega is a compactly supported (n-1) form on M, then \int_{M} d\omega = \int_{\partial M} \omega

 

[gb7nash]

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

Besides that theorem, I would go with Pythagorean Theorem. If you don't know this, then you're really screwed.

 

[Char. Limit]

Let ƒ be a continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by

F(x) = \int_a^x f(t) dt

Then, F is continuous on [a, b], differentiable on the open interval (a, b), and

F'(x) = f(x)

for all x in (a, b).

 

[mjpam]

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

How meta is this joke?

 

[gb7nash]

How meta is this joke?

Yes.

 

[micromass]

Nice topic!

Something every mathematician should know is Zorn's lemma: If X is a nonempty partially ordered set such that every chain in X has an upper bound, then X has maximal elements.

If you don't consider Zorn's lemma a theorem, then I suggest this alternative:
Lagrange's theorem: Let H be a subgroup of a finite group G, then |H||G/H|=|G|. In particular, |H| divides |G|.

A bit of a generalization is of course the
orbit-stabilizer theorem: Let G be a group that acts on the set X. Let G_x denote the stabilizer of x and let G(x) denote the action of x. Then |G|=|G_x||G(x)|.

I think these three theorems should be known to every mathematician!

 

[camilus]

I say Euclid's theorem of the infinitude of primes.

 

[micromass]

I say Euclid's theorem of the infinitude of primes.

And add to that the fundamental theorem of arithmetic: every natural number greater than 1 can be written as the product of primes. And this (up to order) the unique way of writing that number...

 

[sin2beta]

The mean value theorem.

A technique that I have found useful in many surprising instances is Gauss's trick. (See Knuth's Concrete Mathematics) It is not a theorem of course, but something that has been very useful in my bag of tricks and favored approaches.

 

[Char. Limit]

The proof of Fermat's Last Theorem is something every mathematician should at least TRY to understand.

 

[gb7nash]

The proof of Fermat's Last Theorem is something every mathematician should at least TRY to understand.

Conjecture, maybe. The actual proof by Wiles though? I'm not sure about that. It's at least 100 pages long and very complex (not to mention took many many years to perfect). Maybe if someones forte is algebra, but if not I wouldn't expect someone to understand the proof.

 

[Char. Limit]

Conjecture, maybe. The actual proof by Wiles though? I'm not sure about that. It's at least 100 pages long and very complex (not to mention took many many years to perfect). Maybe if someones forte is algebra, but if not I wouldn't expect someone to understand the proof.

I didn't say understand it. I said try.

 

[gb7nash]

I didn't say understand it. I said try.

I would try it and probably get lost at page 1. Algebra isn't really my subject though.

 

[camilus]

And add to that the fundamental theorem of arithmetic: every natural number greater than 1 can be written as the product of primes. And this (up to order) the unique way of writing that number...

aww man, how could I forget that one, that one is even more important ;P

 

[camilus]

and Godel's incompleteness theorems! they are extremely important to the foundations of math.

 

[Curl]

Let A be an n x n matrix. If the matrix is singular, det(A)=0 and there exists a nontrivial solution for Ax=b.

Let B be an n x n matrix. If det(B)=0, B is singular.

Any elementary row operation can be written as the original matrix multiplied by another matrix.

These are pretty fundamental to linear algebra, I guess mathematicians should know them.

 

[disregardthat]

and Godel's incompleteness theorems! they are extremely important to the foundations of math.

Not very useful however to the average mathematician.

My vote goes for Zorn's lemma as previously mentioned here. It's basic and incredibly useful at many levels, but still non-trivial in more than one sense.

 

[usljoo]

every mathematician should know that math is maybe the easies discipline around cos all u have to do is sit in your chair :D and this can easily be stated as a theorem hehe

 

[discrete*]

This is a great topic!

De Morgan's Laws.
\overline{A\cup B}=\overline{A}\cap\overline{B}

\overline{A\cap B}=\overline{A}\cup\overline{B}

Chosen for their brevity and clarity, and the uncanny ability to have applications in many fields of mathematics. The way I stated them is using Set Theoretic notation, however we can just as easily state them in other forms. Also, if I'm not mistaken, these are some fundamental theorems that one learns when doing proofs; that alone makes them worth knowing.

 

[discrete*]

Not very useful however to the average mathematician.

My vote goes for Zorn's lemma as previously mentioned here. It's basic and incredibly useful at many levels, but still non-trivial in more than one sense.

I disagree. I'd like to give the nod to Godel's theorems as well.

As for their usefulness, well I won't say that logic is the most studied of mathematical specialties, but one cannot deny that the field -- and in particular Godel's Incompleteness Theorems -- underscores all of mathematics, and for some it is the foundations on which all of mathematics is built. And Godel changed that field in such a profound way that it cannot, and ought not, be ignored. Those two theorems profoundly changed the way (pure) mathematicians operate for better or for worse. One also should consider the far-reaching philosophical ramifications of Godel's work.

 

[micromass]

Some other theorems I think every mathematician should know:

Tychonoff's theorem: The product of compact spaces is a compact space.

Heine-Borel theorem: A subspace of \mathbb{R}^n is compact iff it is closed and bounded. In general, a subspace of a metric space is compact iff it is complete and totally bounded.

Strong law of large numbers: If X_n are iid random variable such that the first moments exist, then \overline{X_n}\rightarrow E[X_1] a.e.

 

[micromass]

I agree with Godels incompleteness theorem. It's not that it is very useful to the working mathematician these days, but I do think it is certainly something every mathematician should have heard of. It's a key result in mathematics and philosophy!

 

[discrete*]

While not a theorem, I'd like to put the Peano Axioms on the table for discussion. In my opinion, they are of eminent importance.

 

[disregardthat]

I disagree. I'd like to give the nod to Godel's theorems as well.

As for their usefulness, well I won't say that logic is the most studied of mathematical specialties, but one cannot deny that the field -- and in particular Godel's Incompleteness Theorems -- underscores all of mathematics, and for some it is the foundations on which all of mathematics is built. And Godel changed that field in such a profound way that it cannot, and ought not, be ignored. Those two theorems profoundly changed the way (pure) mathematicians operate for better or for worse. One also should consider the far-reaching philosophical ramifications of Godel's work.

I don't see how you disagree, I only said it's not very useful to the average mathematician. And I didn't imply that it should be ignored nor denied! What I mean is that the theorem is seldom used outside the study of logic and axiomatic theory. Sure it is a profound theorem which has changed the view of the power of axiomatic systems, but that doesn't make it essential in ordinary discourse. Therefore, as much as it is celebrated it isn't my first choice as a theorem.

The problem with many theorems essential to ordinary mathematical discourse is that the more you get used to them, the more they seem like trivialities; special cases of a broader theory. I don't think that's the case with Zorn's lemma, which is one reason for why I pick it.

 

[discrete*]

I don't see how you disagree, I only said it's not very useful to the average mathematician. And I didn't imply that it should be ignored nor denied! What I mean is that the theorem is seldom used outside the study of logic and axiomatic theory. Sure it is a profound theorem which has changed the view of the power of axiomatic systems, but that doesn't make it essential in ordinary discourse. Therefore, as much as it is celebrated it isn't my first choice as a theorem.

Well, I guess it depends on how you define ordinary discourse and average mathematician. But I see now where you're coming from.

The problem with many theorems essential to ordinary mathematical discourse is that the more you get used to them, the more they seem like trivialities; special cases of a broader theory. I don't think that's the case with Zorn's lemma, which is one reason for why I pick it.

This is a very good point. I'm sure that we could easily say that every mathematician ought to know the Division Algorithm, however it a trivial piece of mathematics and we don't really consider it in a list such as this one.

Zorn's Lemma is a good counter example to this. It has far reaching implications and is mathematically interesting in many ways, whereas some other theorems are seemingly 1-dimensional in comparison. Good choice, I think.

 

[Curl]

every mathematician should know that math is maybe the easies discipline around cos all u have to do is sit in your chair :D and this can easily be stated as a theorem hehe

If all you're doing is sitting in your chair then you ain't doing math, are you?

 

[usljoo]

If all you're doing is sitting in your chair then you ain't doing math, are you?

well i guess youre doing some thinking too, but its not nearly as hard and pleasing as physics

 

[micromass]

If you think mathematics is not all that hard, then I guess you don't really know a lot of mathematics...

 

[gb7nash]

well i guess youre doing some thinking too, but its not nearly as hard and pleasing as physics

You'll think differently when somebody asks you to solve a nonhomogeneous partial differential equation.

 

[discrete*]

well i guess youre doing some thinking too, but its not nearly as hard and pleasing as physics

Surely, you're joking. I'm going to try and say this without insulting anyone.

First of all, if you want a hard job, become an iron worker or something. Secondly, what is pleasing is a very relativistic topic. I would argue that physics gives me nearly zero pleasure, while (some branches of) mathematics provide me with an immense amount of satisfaction. You obviously would argue the contrary. It's all a matter of opinion.

Blanket statements often get persons into trouble; tread lightly, friend.

 

[discrete*]

You'll think differently when somebody asks you to solve a nonhomogeneous partial differential equation.

While nonhomogeneous PDEs certainly satisfy the "hard" criterion, I wouldn't say they're particularly satisfying, but that's just me.

 

[usljoo]

yes but when you work harder you then get more pleasure out of it and i know math very well because i AM a mathematician and i don't see where the fun is in it, but physics really goes beyond your imagination and opens whole new views of the world and not views of some imagined set of objects witch has some rules that go with it. and another thing is that math would never evolve anywhere without the physics and the concepts developed there.

it is true that group theory for example evolved from solving polynomial equations but without physics it would be the only field where it would have been applied ... its just boring pfff

 

[disregardthat]

yes but when you work harder you then get more pleasure out of it and i know math very well because i AM a mathematician and i don't see where the fun is in it, but physics really goes beyond your imagination and opens whole new views of the world and not views of some imagined set of objects witch has some rules that go with it. and another thing is that math would never evolve anywhere without the physics and the concepts developed there.

it is true that group theory for example evolved from solving polynomial equations but without physics it would be the only field where it would have been applied ... its just boring pfff

You are arguing against yourself here. First you state that mathematics could not evolve without the physics to develop new concepts. Second, you give a prime example of a whole new area of mathematics that evolved without motivation from physics, but curiously then you dismiss it as boring, and wrongfully state that it cannot be applied anywhere else without physical motivation, which it obviously can. Your point seems to have been lost somewhere along the lines.

The matter of the fact is that much mathematics has evolved historically and still does without physical motivation. The applications tends to come in turn.

 

[discrete*]

yes but when you work harder you then get more pleasure out of it and i know math very well because i AM a mathematician and i don't see where the fun is in it, but physics really goes beyond your imagination and opens whole new views of the world and not views of some imagined set of objects witch has some rules that go with it. and another thing is that math would never evolve anywhere without the physics and the concepts developed there.

it is true that group theory for example evolved from solving polynomial equations but without physics it would be the only field where it would have been applied ... its just boring pfff

If, in fact, you are a mathematician, than I must ask: why? If you don't find mathematics interesting, than why is it your chosen field? For you to come into the math boards and refer to mathematics as "the easiest discipline" and imply that mathematicians are lazy is just a discredit to yourself (albeit a fallacy), if you truly are a mathematician.

Also, I'll say again -- everything you've said is a matter of opinion. You seem to think that only mathematics with application are interesting. That may be true in your case, but it's not true in all cases, especially mine.

You also seem to describe both mathematics and physics in an immature and unsophisticated manner. The two disciplines are completely separate entities; they differ in methodologies, conceptually, and in the goals that they set and accomplish. In my opinion, physics and mathematics ought not be compared to one another, it's simply comparing apples and oranges.

 

[gb7nash]

While nonhomogeneous PDEs certainly satisfy the "hard" criterion, I wouldn't say they're particularly satisfying, but that's just me.

Point well made.

 

[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!