[JMostert]

Did you read Dijkstra's article? He's making the case for 0, though he's doing it a little abstractly. Simply put, if you start numbering at 1, you are setting yourself up for more boundary problems and off-by-one errors than if you start at 0 (and by extension, inclusive lower bounds and exclusive upper bounds). That's not to say that some algorithms are not in fact easier expressed by numbering things from 1, just that they're not the majority.

Oh, and obviously 0 is even. Why? Because 0 mod 2 = 0, 0 is evenly divided by 2, and that's what "even" means. If you need more intuition, though: 1 is indisputably odd, and even and odd numbers alternate, so 0 is even. The rest is philosophy -- you can probably find definitions for "odd" and "even" where 0 is a problem. That's fine, but those don't help. Ignore them. Mathematically there's no problem whatsoever.

The trickiness only comes if you insist in thinking in terms of "the first element", rather than "element number 0". Some people use "zeroth", but this seems to invite more confusion because it induces two meanings for all the other ordinal forms -- if there's a zeroth element, does that mean the "first" element is in fact the second element? Best to avoid ordinals altogether -- you usually don't need any more than "first" and "last" anyway.

[lucumo]

> Did you read Dijkstra's article?

Yes, I did. Does it hurt to offer a different perspective? I feel it doesn't. Approaching a subject with an open mind or from a different angle tends to be good for discussion. Offering a contradicting opinion or observation helps everyone to understand an issue better.

> Oh, and obviously 0 is even.

I might counter this with a similar question you asked me: "did you read my post?" I didn't make a case that 0 was odd, I asked how I should look at it: as element 0, or as the first element. If you look at it as element 0, the first element of the vector is an even element. If, on the other hand, you look at it as the first element, you will think it is an odd element. This is the case I made, not that 0 is even.

[JMostert]

> Yes, I did. Does it hurt to offer a different perspective?

No, I was questioning whether you had considered his arguments at all, given the "why do we start counting at 0" question which he tried to answer.

> I didn't make a case that 0 was odd, I asked how I should look at it: as element 0, or as the first element.

You can look at it either way, as both are correct.

> If you look at it as element 0, the first element of the vector is an even element. If, on the other hand, you look at it as the first element, you will think it is an odd element.

Ah, I see your point now. To me this wouldn't be a question because everyone knows programmers start counting at 0 -- a machine instruction would operate on indexes, not ordinals. "The first element" is the element with index 0. It is not "element number one", or if you do want to see it that way, "number one" is the ordinal you use when you start counting on your fingers, which is something we deliberately ignore.

[lucumo]

> No, I was questioning whether you had considered his arguments at all

Ah, yes. No, I wasn't trying to disprove him. I just gave a different approach to the issue. My mind is not made up on this subject. I've been a programmer for too long: 0 is burned in my fingers and my mind.

> a machine instruction would operate on indexes, not ordinals

The interesting aspect isn't the difference between indexes and ordinals, but the difference between indexes and cardinals. The cardinals match the ordinals for normal people. For normal people, indexes have no meaning.

But the difference between indexes and cardinals aren't that easy for programmers either. In an example presented by Dijkstra a young programmer used 0 in everyday language as a cardinal replacing 1. Indexes have little meaning in real life, but cardinals do.

Good response, BTW. Thanks. It's interesting to link the concepts of ordinals and cardinals to the discussion.

[bhousel]

> I didn't make a case that 0 was odd, I asked how I should look at it: as element 0, or as the first element.

element 0, aka the 0th element..

[viraptor]

> you usually don't need any more than "first" and "last" anyway.

Apart from FP people - they need "the first" and "the rest" ;)

[probablycorey]

I often wonder if Dijkstra's arguments still hold sway, since algorithms written in modern languages differ greatly from those written when Dijkstra wrote this. Rather than writing hash functions or shortest route algorithms, a majority of my algorithms are centered around lists and or dictionary manipulations. There are still many situations where it makes more sense to start indexing at 0, but I wonder if it is still the majority of cases.

[lucumo]

Well, I think that there is a very important human reason to keep sticking to 0 as the start: convention. It has always been this way. Changing it would cause a whole lot of weird bugs.

To be honest, I think one reason why this grew is probably the way C handles array indexing. a[0] = *(a + 0) and this symmetry is a nice property to have in many cases. I think I use that symmetry in my MSc project a lot more than I use vec_mule/vec_mulo.

(Yeah, I'm not of a fixed mind in this matter :-))

[g_]

http://en.wikipedia.org/wiki/Evenness_of_zero

Learning that such article could exist (it DOES exist), and reading it was a shocking experience to me.

[maweaver]

Relevant quote from the article:

'A second-year was "quite convinced" that zero was odd, on the basis that "it is the first number you count"'

[dkersten]

"obviously 0 is even"

Sounds like a bit of an odd statement to me.