I was mostly thinking about the integers (but perhaps also rational numbers or p-adic rationals like in games and numbers, I don’t know if that would make things very different).
Generally when you think about integers and write 1/0 = ∞, you are thinking of infinity in the sense 4 (there is no reason to put a sign in front of ∞ since 0 has no sign), so there is only one infinity here.
OK, I didn’t expect that.
Do I understand correctly then that -1/0 = ∞ if -1 is interpreted as a complex number or integer (case 4), but -1/0 = -∞ if -1 is interpreted as a real number (case 3)?
No I wasn’t clear. You can’t write things like -1/0 = -∞ either. Signed infinities (+∞ or -∞) are only used to write limits of real valued sequences or of real valued functions, like lim (1/sin(1/n)) = +∞ or lim (-1/sin(1/n)) = -∞.
The idea is that you are adding two extra points to the real line, and a sequence may converge to one of these points since each term of a sequence has a well-defined sign.
But an expression like 1/0 doesn’t represent a sequence, so you may view it either as a “meaningless expression” (NaN as you said), or you may want to view it as the limit of 1/un for any sequence un of nonzero real numbers which converges to 0. That makes sense only if you only add one point at infinity to the real line.
People tend to grab a random number of stones, in the absence of magic. There is no natural probability difference between an odd and even number. On the other hand, if a player is allowed to choose only a small number k of stones, such as K<4, then the drawing is clearly unfair.
It’s a pretty good way of choosing who gets black, and more pleasing to a go player than flipping a coin or rolling a die.
In the history of AI, Perceptrons were the first version of neural networks, and it proved mathematically impossible to train a Perceptron to distinguish the parity of a number of objects. Humans, also, cannot distinguish the parity of a large number of objects without some sort of counting or pairing procedure, with the possible exception of certain savants.
But they can be paired, there’s just no pairs If they couldn’t be paired, there would always be something left over, but there’s nothing in zero to begin with, so nothing to be left over.
But this begs the question why -1 is odd. If the even-ness of 0 is debatable, so is the odd-ness of -1.
In mathematics: if it makes whatever you do more easy, then yes, if it makes whatever you do harder, then no
To be precise, it’s not generally defined, but there’s nothing against defining it on the spot to make things work for you.
Second question is how to define it: clearly it’s the limit of 0=0/x with x getting smaller and smaller, so the value of 0/0 should be 0. But it’s also the limit of x/(x^2) = 1/x which grows infinitely large from the positive side so 0/0 should be positive infinity. But it’s also the limit of x/x = 1, so it must be 1. !?
A prime number is a number that is divisible by exactly two positive integers. 0 is divisible by every positive integer, so it’s more like an anti-prime.
Positive numbers are defined as numbers larger than 0, negative numbers as numbers smaller than 0 (or as additive inverses of positive numbers). That makes 0 neither positive nor negative. When including 0 with all positive numbers, in mathematics these are called “non-negative numbers”.
It really depends on definitions (as usual). Any high school teacher as well as any calculus teacher will repeat time and time again that infinity is not a number. The reason being that the standard arithmetic that we know from real numbers don’t work with infinite values.
You could define + / - infinity as some sort of number and define separate arithmetical rules, for example in the extended real line, but it’s crucial to realise that some arithmetical rules will fail. For example, A+B = A+C does not imply B=C anymore (take A infinite), there’s no unique additive or multiplicative inverses, and certain terms are still undefined, such as 0 * infinity.
You could consider infinite numbers that are more suitable to arithmetic, such as ordinal numbers, cardinal numbers, or surreal numbers. Especially the latter behave very nicely. However, these kind of infinities are very different from the type that we use in high school mathematics (e.g. the type where 1/x tends to + infinity when approached from above): limit infinities are not numbers.
I see you put Games in there, which is interesting, since the concept of a (combinatorial) game is closely related to the concept of a surreal number, except that in surreal numbers we kind of define a new number by filling a gap between two sets of “small” and “large” numbers (where each small number is smaller than each large number). In games we fill the gaps between any sets of games, so there’s no “small” / “large” distinction.
Maybe we can turn the question around. If you’re the one doing the nigiri, does black more often pick one or two?
I have a feeling people tend to do 1 stone because it’s nice. So maybe learning to grab even number of stones can help getting black color.
An even more fundamental question is, what result are you hoping for when you do nigiri? Depending on your personal preferences, whether or not you belief komi to be fair, which color you think komi might be biased toward, etc., I think there could be a diverse range of preferences, including (but not limited to):
- You want to play black
- You want to play white
- You want the colors to be assigned with 50-50 odds
- Other possibilities? (like 30-70 odds, etc??)
Maybe these preference might also depend on the context and who your opponent is? Maybe one opponent you perceive to be particularly weak as playing fuseki as black, while another you think is weaker in fuseki as white>
Let’s not forget that whilst “winning” the nigiri in most rulesets means that you play Black, in the Ing rules it means that you have the choice of playing Black or White.
If anything, at the current value of komi professionals seem to prefer White – see, for instance, that Ke Jie chose to take White twice against AlphaGo in 2018.
I think that preference for white is only the case with a “large” komi of 7.5+ (common with Chinese rules, AGA rules, ING rules etc), not with 6.5 komi (common with Japanese rules, Korean rules).
I was under the impression that 6.5 komi with territory scoring is also considered slightly good for White.
Perhaps I’m incorrect.
As far as I know, pro game statistics and KataGo give white about 51% win rate for 6.5 komi under territory scoring rules. Such a small advantage for white doesn’t seem to justify a clear preference from pros.
I don’t know of any pro game statistics for 7.5 komi under area scoring rules, but KataGo gives white 59% win rate, which is a significant advantage for white.
Note that black usually has to win by 9 points on the board to win with 7.5 - 8.5 komi under area scoring rules. With 7 komi under area scoring rules, KataGo gives white 51% win rate.
I tried to find out if white or black favors me but the tool https://avavt.github.io/gotstats doesn’t say anything about OGS games. So I looked at my IGS games (net social plaza): I have played 230 even games as black with winrate 58% and 237 even games as white with winrate 57%. The conclusion is that the color is indifferent.
Do you play under Japanese rules on IGS? How much komi do those games have?
Yes, I use Japanese rules, with 6.5 komi for even games.
Fujisawa Rina pulled 23 today. And Ueno Asami guessed one. I’m telling you Rina has oddly shaped hand.
First game of Gosei today. Ichiriki Ryo pulled 15 and made a tetris shape.
From BGJ #79 (Summer 1990), The Fujitsu Cup, Michael Macfadyen (pg. 6)
The Fujitsu Cup is a knockout tournament for top players, except that there are token foreigners to make up numbers. I was privileged to represent Europe in last year’s event (…) I was delighted to find myself matched against Otake. (…) He seemed to regard it as a great joke to have to play an amateur on evens, which was something of a relief as some of the pros would think it a bit of an insult.
The game started badly for me when I lost the nigiri and had to take White.
At this time komi was 5.5. After reading a comment in the BGJ from as early as 1980 suggesting that 5.5 komi was too low, I’ve started to wonder whether this opinion was actually widely held through the '80s and '90s, which would have lent more weight to the idea of “winning” or “losing” the nigiri.
As far as I remember, there was no debate about komi being too low in the '80s and '90s. But most players preferred black and 5.5 komi was not too high a price for the driver’s seat in the early opening.
But even with 6.5 komi, I think most players preferred black until about 2017-2018.
Only with the advent of hypermodern AI joseki, it became more difficult for black to push for a specific large scale opening. With san-ren-sei, pincers and the chinese opening losing popularity, black may have a more difficult time in recent years, especially with the overly expensive 7.5 komi that is becoming more common together with the Chinese rules.
In the BGJ #85 (Winter 1991), Brian Chandler seems to get in a sly dig at an earlier suggestion in the journal that one should always guess an odd number, there never being less odd numbers than even in any given finite set.
Although it may not have been sarcastic – it’s a strange article, from the time when tournaments were for Ladies, not Women, and Chandler spends much space describing their attire and appearance. Then again, it was a casual event, so who knows what the real context was…
Nigiri – the delicate Ms. Nakazawa [Women’s Honinbo] picks up a huge fistful of white stones, and Sakakibara-san [Women’s Kakusei], obviously not knowing the “more odd numbers than even numbers” rule guesses even, and wrong.