First years say the darndest things.

I have just finished marking 1/4 of the questions of a first year exam. But before I begin on what little meat there is in this story¹, perhaps a little explanation is in order about the paper² in question. It is a 'discrete mathematics and linear algebra' course. There are two such courses (covering exactly the same material) running simultaneously. One takes a year, and the other one half a year. The half year version begins half way into the full year version. The reader is perhaps under some misapprehension that the latter course, covering the same material in half the time, is full of the really bright and/or hard-working people. This is, sadly, not the case. In fact it is full of individuals who originally enrolled in the previous course who then took a long hard look at their assessment results and, looking down the barrel of an epic fail, decided to drop out of the full year version and enrol in the half year version instead. The reader is perhaps also under the misapprehension that they tend to put more effort into their second attempt.

Anyways, I want to talk to you about question 3 a): Write down the definition of a rational number. Most people answered this incorrectly, and here are some of the responses I received:

1. Any number that is not a fraction.
2. Any number which can be expressed as p/q where p,q ∈ ℝ and p ≤ q. ³
   
3. A number that exists rationally.
4. Any number that is not irrational.
5. A number that makes scence(sic).
6. Any number that doesn't believe in fairy tales⁴.
7. Definition of a rational number. There is a chap in class who I always thought was an ace but he has written less than I. I guess I am not the only person who was either stumped by this course or who didn't work on it hard enough & is now compelled to write Ramayamas in the answer sheet. Yay! 15 minutes to go. Actually 20 mins but I think I am going to make a run for it. Have no clue what I am writing and I think am simply allowing all thoughts to spill out onto paper. Possibly my constant scribbling is leading the guy next to me to get very worried as he isn't writing anything either. I wish I could go home for a bit. Really wish I could go home if only it wasn't 18 hours away.
   

¹Which may be safely characterised as being to world literature what a ham bone stew is to world cuisine. P.S. Fuck you, yes, this is a footnote. Despite your endless pay-outs on this front, I still like the fucking things.

²In New Zealand Universities 'courses' are called 'papers'. This caused some confusion for me when in casual conversation a great number of people started casually talking about papers they did in first year. I thought I was surrounded by geniuses. P.S. Yes, this is another footnote. Go to hell.

³I had to at least acknowledge internal consistency here. When asked to prove that √3 was irrational they said: √3 = 3/√3 and, since 3 > √3, √3 is irrational. The correct answers were kind of cute, too, actually (& not the way I've ever seen this answered). They reasoned as follows: Let p,q ∈ ℤ. Then the prime factorization of both p² and q² contain an even number of terms. Thus p²≠ 3q ² and so √3 ≠ p/q.

⁴This guy got marks.