Famous Theorems in Math #1 Trivia Quiz

Answer: The sum of the squares of the two other sides

The Pythagoreans were a strict secret society in Ancient Greece. It is not known whether Pythagoras himself discovered this famous theorem, since the Pythagoreans (even after his death) attributed all results to him.

Answer: The cosine law

The cosine law states that c^2=a^2+b^2-2ab*cos C (where capital C is the angle). Note that this reduces to the Pythagorean Theorem when C=90 degrees, because cos 90 = 0. There are sine and tangent laws, they describe ratios in the triangle. There is no "triangle law" that I know of.

Answer: The "reduction method" for the solution of the general cubic equation

The method was for the general solution of polynomial equations of degree 3. The general solution for degree 4 curves was found not long after, and in the 1800s Abel proved that equations of degree 5 and higher cannot be solved in general. The "exhaustion" method was used by Archimedes (a primitive version of the integral calculus), "fluxions" was Sir Isaac Newton's invention, and the "Erlangen Programme" was stressed by the German Felix Klein in the late 1800s to unify group theory and geometry.

Answer: (pi squared)/6

This is one of the most beautiful and interesting results in infinite series: that the sum of the reciprocals of the squares "closes in" on the irrational number (pi squared)/6. This result can also be used to provide a proof that there are an infinite number of prime numbers.

Answer: Elliptic curves & modular functions

The method of infinite descent was used by Fermat in order to solve lower degree specific cases of the theorem, and is essentially a proof by contradiction. This method was proven not to work in general. Direct proofs also exist for specific cases (I have seen n=3 and n=4) but no general direct proof is known. Wiles finally cracked Fermat's enigma using modular forms of elliptic curves.

Answer: The four-colour theorem

To this day there are mathematicians that debate the validity of the proof, as it cannot be manually checked by man. The computer program used, however, can be checked and reproduced. Rolle's theorem is a simple result from the calculus, the marriage theorem is a combinatorial result that describes matchings, and Kuratowski's theorem is a test for planarity of graphs.

Answer: All closed curves have an "inside" and an "outside"

Seems obvious, doesn't it? The proof, however, is not. The reason is due to topological generalizations of the notions of "closed" and "open".

Answer: Ideals

Kummer called a number "ideal" if it behaved like a prime number in any arbitrary ring. The other three answers are algebraic terms, but were around long before Kummer.

Answer: All of these

The other three answers are variations of the same result. Cantor's most violent opponent was Kronecker, who strongly disagreed with the notion of different "sizes" of infinity. Cantor had several nervous breakdowns and eventually died in an asylum.

Answer: Brouwer fixed-point theorem

If we were to paramatize the curves, the graphs of ascent and descent would be automorphic (since the time for each of the ascent and descent is the same). Brouwer's theorem says that any plane automorphism has a fixed point. Chebyshev's theorem deals with number theory, the Minkowski theorem deals with optimization, and the Lebesgue theorem refines the notion of an integral.