622: Haiku Proof
Title text: After somewhere around 40 hours, there's no academic reason to go to the class. Only go for the hallucinations.
In this comic Cueball attends a math class after having been awake for two full days (48 hours). After that he begins to hallucinate and dreams that the teacher Miss Lenhart (a professor in this comic) answers Megan's question, about a proof that there are an infinite number of prime numbers, in haiku. After the first line she floats up and during the third and final line she flies over the students heads. Basically it is a dream, as also indicated with Cueball's thought that are "outside" this lucid dream where it could be seen as it was the entire comic, except Cueball's thoughts, that are inside the dream bubble. (Dreams being a recurring theme in xkcd). Note also that where Cueball looks up after the flying teacher when she takes off, Megan never moves her head to watch her teacher fly, because it's not happening in her world, and Cueball only dream the teachers moves.
Euclid's theorem states that there are an infinite number of primes, prime numbers being numbers that are only divisible by themselves and 1. The most notable proof of this theorem, and the one presented in this comic, was first given by Euclid himself in his Elements. A more traditional form of this proof follows:
- If we suppose that there are a finite number of primes, then they must have a product, i.e. p1p2...pn = q. Now consider q + 1. If this number is prime itself, then we have discovered a new prime number, contrary to the assumption that we had listed them all. If it is not prime, it must have a prime divisor. Since all of the pk are a factor of q, they cannot be a divisor of q + 1. So q + 1 is divisible by a prime not on the list, which again is a contradiction. Therefore, there must be infinitely many primes.
At the last line of the haiku, Miss Lenhart says "Q.E.D., bitches!", Q.E.D. stands for "Quod Erat Demonstrandum", a latin phrase which is used to show a proof is over. Ironically, the proof is not complete.
The comic essentially takes this proof and states it in the form of a haiku, which is a traditional form of Japanese poetry which is in Japanese broken up into patterns of morae (or syllables), a unit that measures the length of sound. A haiku consists of three lines with 5, 7 and 5 syllables respectively per line. An English Haiku has 5, 7 and 5 syllables per line. And the proof poem goes like this:
- Top prime's divisors'
- Product (plus one)'s factors are...?
- Q.E.D., bitches!
Which can be divided in syllables like this:
- Top - prime's - di - vi - sors'
- Pro - duct - (plus - one)'s - fac - tors - are...?
- Q. - E.- D., - bit - ches!
The haiku proof given is slightly off, as the first line talks about the "top prime's divisors," which makes no sense because the top prime doesn't have any divisors besides itself and one. You need to take the product of all primes, not just one. But, hey, it's a hallucination.
Haiku was also referred to before in 554: Not Enough Work.
The comic and title text conclude that going to class while sleep-deprived is an interesting, but entirely noneducational, experience. So do go for the sake of the hallucinations.
- [Miss Lenhart teaching a class gestures with both hands up as Megan, sitting at the first desk on a stool, raises a hand and asks a question. Cueball sits at the desks behind her supporting his head in both hands with the elbows on the desk.]
- Megan: How do you know there are an infinite number of primes?
- Miss Lenhart: I'll answer in haiku!
- [In a frame-less panel, Miss Lenhart lifts a hand up while answering. Both students sit upright on their stools.]
- Miss Lenhart: Top prime's divisors'
- [Miss Lenhart floats into the air with three lines beneath her legs. Cueball looks up. Megan does not change position.]
- Miss Lenhart: Product (plus one)'s factors are...?
- [Miss Lenhart flies over the students heads with a curved line behind her. Neither student look up. The bottom frame of the panel is a curving thought/dream bobble that goes through the middle of the panel at a height just below the desk tops. Two thought circles goes from Cueball's head down to this frame, and Cueball's thoughts are shown below outside of the panel - without any frame around.]
- Miss Lenhart: Q.E.D., bitches!
- Cueball (thinking): Wow, after the 48-hour sleep-dep mark, lectures get really interesting.
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