yA9Tx5pH

1const responses = [
2    'There are infinitely many distinct infinities. For example, there are exactly as many positive integers as there are integers (positive and negative). However, there are infinitely many more real numbers than integers.',
3    'a^2 + b^2 = c^2 has infinitely many solutions in nonzero integers, but a^3 + b^3 = c^3 has none. In fact, if a^n + b^n = c^n and n>2, then abc=0.',
4    'Gödel\'s incompleteness theorems are similar to the "uncertainty principle" in quantum mechanics: they prove that every axiomatic system for mathematics is either complete or consistent, but not both.',
5    'The Millenium Prize problems are seven research problems defined in 2000, each having a $1M bounty for a solution. Only one problem has been solved, and the mathematician declined the prize.',
6    'One of the most famous open problems today is the `abc` conjecture, related to prime numbers and a deep connection between addition and multiplication. In 2012 a mathematician claimed to have a solution, and because of its complexity, mathematicians are still unsure whether it is correct.',
7    'Remember the quadratic equation, "x equals negative b, plus or minus the square root..."? There are also equations for 3rd and 4th power polynomials, but there is no equation for 5th degree or higher",
8    'The Halting Problem in computer science states that we cannot, in general, know whether a program will finish running. The reason this is true is closely related to the reason that the real numbers are uncountable ("infinitely more reals than integers").',
9    'Évariste Galois was a French mathematician who invented group theory, a field that studies symmetry. He died at 20 in a duel related to a broken love affair.',
10    'The square has 8 symmetries: 4 rotations and 4 reflections. In fact, a regular shape with n vertices has 2n symmetries. The Rubix cube has 2.1*10^24 symmetries.',
11    'The Monster Group is an enormous mathematical object that, in a sense, is the most complex symmetry possible in nature.',
12    'Graham\'s number is an unfathomably large integer, and is one of the largest ever used in math research. The digit representation cannot fit in the universe, even if every digit is a Planck volume (the smallest meaningful size possible).',
13    'pi is a real number: it cannot be written as a fraction. It is also transcental: it cannot be a solution to an algebraic equation. We suspect, but don\'t yet know, that pi is also normal: every possible sequence of digits exists _somewhere_ in its infinite decimal representation.',
14    'If you recite the digits of pi, the sequence "999999" occurs at the 762nd decimal place. Seven 9\'s occurs at the 1,722,776th decimal, and ten 9\'s occurs at the 564,665,206th.',
15    'Mathematicians have made several errors recently. After he was frustrated by a mistake, from 2006-2009 Vladimir Voevodsky began a new research program to formalize mathematics in a way that we can write computer programs to avoid these errors.',
16    'In 1913, the mathematicians and philosophers Alfred Whitehead and Bertrand Russell attempted to formalize mathematics in logic. It takes 360 pages to prove 1+1=2.',
17    'There is a sort of "periodic table" of symmetry: every possible symmetry in nature is a combination of 16 simple groups and 26 sporadic groups.',
18    'The Intermediate Value Theorem in calculus can be used to prove many surprising facts. For example, there are always two points on opposite sides of the world with the exact same temperature. Also, for any square wobbly table, you can rotate it until it is stable.',
19    'Topology is a field of geometry that studies connectivity of objects without distance. For example, to a topologist a coffee mug and a donut are the same object, because they both have exactly one hole.',
20    'Fractal geometry is a field that studies "roughness" of objects. For example, the coastline of Britain has a fractal dimension of ~1.25.',
21    'Several mathematicians are now working to create fair systems of political map-drawing, to prevent gerrymandering. These attempts use techniques from both pure and applied mathematics, like Monte Carlo methods and compactness.',
22    'The Fields Medal is the highest recognition in mathematics. Legend has it that math is excluded from the Nobel Prize because Alfred Nobel\'s partner cheated on him with a mathematician; this is probably a myth.',
23    'One of the most famous problems in math is Fermat\'s Last Theorem (a^n + b^n ≠ c^n). Stated in 1637, Fermat claimed that he had a proof but it was too large to fit in his notebook; it then took 358 years of research for a correct solution to be published.',
24    'There is an easy formula to add the first n integers: 1+2+...+n = n(n+1)/2. Gauss discovered this when he was seven, to avoid a schoolteacher\'s punishment of adding 1+2+...+100.',
25    'There are exactly as many integers as there are fractions. To see this, you can write the rational numbers in a 2x2 infinite grid, and count them via a zig-zag pattern from the top-left corner. However, it\'s impossible to count the real numbers.',
26    'If you want to prove a statement about numbers, you can use a strategy called mathematical induction: first you prove your fact for n=0, then you prove that if the fact is true for n, then it is also true for n+1. This is similar to dominos: if you can knock over the first domino, and you know that every domino will knock over the next one, then you know that every domino will eventually be knocked over.',
27    // Analysis
28    'e^(pi*i) + 1 = 0. This is a remarkable equation: note how it uses some of the most important numbers in math, 0,1,e,pi, and i, and it uses some of the most important operations, addition, multiplication, and exponentiation.',
29    'The opening chord to The Beatle\'s "Hard Day’s Night" was debated for many years. In 2008, a mathematician published a paper using Fourier Analysis to reverse-engineer the exact notes every bandmember played.',
30    // Proofs
31    'Why are there infinitely many prime numbers? Well, if not, we\'d have n primes, say p1,...,pn. But then if you multiply them all and add one, X = p1*...*pn + 1, then X is neither a prime (greater than pn), nor divisible by any of the primes (it always has remainder of 1). This is impossible, so our assumption of n primes is incorrect.',
32    'It isn\'t very hard to see why sqrt(2) is irrational. If it were rational, sqrt(2)=a/b in lowest form for some integers a,b, with b≠0. Then, squaring both sides and multiplying, a^2 = 2b^2. Because a^2 is even, then a must be even, and so a^2 is a multiple of 4. Because a^2 = 2b^2, b^2 must be even, and hence b must be even. This contradicts our assumption that a/b is in lowest form, so our assumption must be false: sqrt(2) cannot be written as a/b.'
33];
34responses[Math.floor(Math.random() * responses.length)];