# The 100 Greatest Theorems of Mathematics

October 26, 2011 2 Comments

As a mathematics teacher, I am often asked what I believe is the single greatest theorem in all of mathematics. And, depending on my mood, I could claim any one of a dozen theorems to be the greatest. Talk to other math people and you will probably get a completely different dozen. In fact, there are probably as many different opinions as there are theorems. To me that is one of the beautiful things of my subject.

With that being said, I guess there is no point in anyone ever trying to construct a list, right? Not really. Enter mathematicians Jack and Paul Abad. In 1999, they set forth on the arduous journey of generating the list of the 100 Greatest Theorems. In making the list, they used 3 criteria.

- the place the theorem holds in literature
- the quality of the proof
- the unexpectedness of the result

What did they come up with? Below is their top 12. For the complete list, click here.

- The Irrationality of the Square Root of 2 by Pythagoras (500 B.C.)
- Fundamental Theorem of Algebra by Karl Frederich Gauss (1799)
- The Denumerability of the Rational Numbers by Georg Cantor (1867)
- Pythagorean Theorem by Pythagoras (500 B.C.)

- Prime Number Theorem by Jacques Hadamard and Charles-Jean de la Vallee Poussin – separately (1896)

- Godel’s Incompleteness Theorem by Kurt Godel (1931)

- Law of Quadratic Reciprocity by Karl Frederich Gauss (1801)

- The Impossibility of Trisecting the Angle and Doubling the Cube by Pierre Wantzel (1837)

- The Area of a Circle by Archimedes (225 B.C.)

- Euler’s Generalization of Fermat’s Little Theorem by Leonhard Euler (1760)
- The Infinitude of Primes by Euclid (300 B.C.)

- The Independence of the Parallel Postulate by Karl Frederich Gauss, Janos Bolyai, Nikolai Lobachevsky, G.F. Bernhard Riemann – collectively (1870-1880)

What do you think? Did they get it right? Enjoy the debate!

This is a very good list.

The theorems listed are truly among the most interesting results in mathematics. As a student, I thought Godel’s Incompleteness Theorem was both surprising and interesting. The implications of this one theorem are huge for epistemology and computer science.

I do find it strange how infrequently I actually use the 12 theorems above directly. The theorems I actually use are #2 FTOA, #4 Pythagorean, and #9 area of circle. The rest I rarely use despite the fact that I am a mathematician and an engineer.