# Theorems every mathematician should know

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 $$\int_{M} d\omega$$ = $$\int_{\partial M} \omega$$

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gb7nash
Homework Helper
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
Gold Member
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).

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

How meta is this joke?

gb7nash
Homework Helper
How meta is this joke?

Yes.

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!

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

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...

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

gb7nash
Homework Helper
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
Gold Member
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
Homework Helper
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.

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

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

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.

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

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.

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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.

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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.

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!!

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.

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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.