Up numbers

[ar_boblad]

So sometimes I invent some maths, though never anything too smart or complex and I very strongly doubt ever anything original. Lacking a full maths degree is handy here because there is a greater* space of maths to be discovered the less you know in advance. And now I've got a place to share such I occasionally will, starting with the subpar

Comments

[undyingking]

I kind of feel it's the floor that lets it down. It does seem a bit arbitrary, and with something different you might be able to avoid some of the brokenness. Dunno what though (and I am a mathematical ignoramus really).

[ar_boblad]

The tricky part is you must round somehow. 7↑0.5 can't exist because it would be between 7 and the smallest number greater than 7, but 14↑1 * 0.5 ought to equal something. If that something takes the ↑ part into consideration you seem to always hit an inconsistancy where division isn't the inverse of multiplication. If you don't take the ↑ part into consideration you end up with the a+a≠2a issue. But hopefully I've missed something.

[undyingking]

Either that or you've proved that Up numbers as described can't exist...

[bateleur]

My feeling about this (having only just found this LJ) is that the first issue with extending the real with up numbers which I would want to address is what happens when I do this:

(2 + 2↑) / 2 = ???

The verbal definition of 2↑ makes it sound like it belongs on the number line somewhere, albeit not as a real number. But this calculation makes it seem like there must be a thing of some kind "between" 2 and 2↑. I am therefore inclined to conclude that the set of {reals plus up numbers} is not closed under arithmetic.

[Incidentally, I realise it's against the implied rules of the game, but purely in case you're ever curious, what you're on the verge of inventing here is Nimbers, which are in fact pretty cool and interesting. You win a shiny kudo! :-)]