Fun Trivia : Theorems

Structure Sub-topic of: Math Quizzes Interesting Questions, Facts and Information There are a total of 10 general entries. Special Topics Pythagoras's Crazy Theorem Quod Erat Demonstrandum Basic Math by the Definition Fermat's Last Theorem

Interesting Questions, Facts, and Information

Theorems

According to the Pythagorean Theorem, the square of the hypotenuse of a right triangle is equal to what?

Famous Theorems in Math #1


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.



The Pythagorean Theorem only applies to right-angled triangles. However, there is a more general "law" that governs all triangles in a relationship similar to that of the Pythagorean Theorem. What is the name of this law?

Famous Theorems in Math #1


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.



Moving on several hundred years, in Italy in the 1500s there arose a great dispute between two leading mathematicians named Cardano and Tartaglia over a new method. What was this method?

Famous Theorems in Math #1


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.



Moving on to the great Swiss mathematician Euler, he was one of the first to really explore infinite series. It has been known for a long time that the harmonic series (1+1/2+1/3+1/4+...) diverges, but that the infinite sum of the reciprocals of the squares (1+1/4+1/9+1/16+1/25...) converges. Euler was the first to determine the exact value of convergence, what is it?

Famous Theorems in Math #1


(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.



Fermat's Last Theorem is undoubtedly the most famous theorem in all of mathematics, first being proposed in the 1600s but not fully solved until the mid 1990s. What method was used to finally solve this centuries-old problem?

Famous Theorems in Math #1


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.



Another famous theorem was first conjectured in the 1800s, but was not solved until 1976 in a highly controversial way: the proof depends on the use of a computer. To which theorem am I referring?

Famous Theorems in Math #1


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.



An example of a deceptively simple-stated theorem with an extraordinarily difficult proof is the Jordan curve theorem. What is the fundamental result of this theorem?

Famous Theorems in Math #1


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".



Cantor made waves in the traditional mathematical community by proving what?

Famous Theorems in Math #1


All of these (There are different "sizes" of infinity, There are "more" real numbers than rational numbers, There exists a bijective map between the counting numbers and the fractions). 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.