Numbers Aplenty | Hacker News Reader

Numbers Aplenty

numbersaplenty.com

48 pointsmsvan13 comments

See: http://en.wikipedia.org/wiki/Interesting_number_paradox

'The paradox states that all natural numbers are interesting. The "proof" is by contradiction: if there exists a non-empty set of uninteresting numbers, there would be a smallest uninteresting number – but the smallest uninteresting number is itself interesting because it is the smallest uninteresting number.'

Wouldn't the paradox have to be "all natural numbers are either interesting or uninteresting" thus creating a partition of natural numbers? Or at least "some (more than 2) natural numbers are uninteresting"?

But -- it's interesting in a rather different way.

BTW, I once decided that the smallest uninteresting number is 34.

Alas, I don't recall my exact reasoning.

> I once decided that the smallest uninteresting number is 34.

Doesn't that make 34 interesting? :-)

I love it, already learned some new stuff!

One comment though: the background almost made my eyeballs explode...

In reply to a dead comment: The most common definition of natural numbers is "positive integers." I do agree that 0 is important enough that it would have merited a mention at least.

It is somewhat controversial: according to Wikipedia

    There is no universal agreement about whether to include zero
    in the set of natural numbers. Today some textbooks,
    especially college textbooks, define the natural numbers to
    be the positive integers {1, 2, 3, ...}, while others,
    especially primary and secondary textbooks, define the term
    as the non-negative integers {0, 1, 2, 3, ...}.

Among mathematicians, the most common (by far) definition is the one given by Peano axioms, which include 0. Including 0 is very convenient for a number (pun intended) of reasons, for example it makes the naturals a monoid, and it is easier to define the integers.

I'm not sure if it is more common to say that 0 is a natural number. There's a math.stackexchange thread about this which seems to have equally many people saying that each is the more common one.

http://math.stackexchange.com/questions/283/is-0-a-natural-n...

(I count 4 people saying {1,2,3,...} is more common vs 3 saying {0,1,2,...} is more common. I also notice that N_0 is used by both sides to refer "unambiguously" to the convention of the other side.)

In my experience {1,2,3,...} is used more in number theory, and {0,1,2,...} is used more in logic.

Pedantic: they are still a commutative monoid without 0, under multiplication with identity 1. You need 0 as the identity for addition, which is fundamental in these constructions.

Absolutely love it - real great way to explore how all numbers are uniquely interesting.

This can be helpful in Project Euler challenges.

Another useful site for that purpose is the Online Encyclopedia of Integer Sequences (https://oeis.org/).

A pretty successful strategy for "cheating" at Project Euler is to implement a brute force solution to get the first few numbers out of a sequence and look it up in OEIS. If you're lucky, they've already listed a closed form expression for calculating the sequence at an arbitrary index. Drop the expression into your code, and you end up with something that tends to be orders of magnitude more efficient.

Obviously it depends on the structure of the exercise, but it's somewhat surprising how far it can take you.

Some of the OEIS sequences originated with Project Euler problems, so this strategy is closer to cheating than you might think.

Reply on news.ycombinator.com