I thought the MIT fight song went..."MIT, PHD, M-O-N-E-Y"
They must have changed it, since I heard last.
A Mathematical Dilemma
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kborg wrote:
I though the MIT fight song went..."MIT, PHD, M-O-N-E-Y"
They must have changed it, since I heard last.
No No. You're confusing the graduate fight song with the undergraduates fight song.
The MIT undergrads are more clever, in any case. Mea culpa.
I know im coming into this late, but dividing by zero is "undefined", not infinity.
Division as the inverse of multiplication
The concept that explains division in algebra is that it is the inverse of multiplication. For example,
- 6/3=2
since 2 is the value for which the unknown quantity in
- ?x3=6
is true. But the expression
- 6/0=?
requires a value to be found for the unknown quantity in
- ?x0=6
But any number multiplied by 0 is 0 and so there is no number that solves the equation(including infinity).
Therefore, dividing by zero does not give you infinity, it is undefined.
If you are going to use ratios, use the amount of material remaining on the board instead of what is captured.
I remember in 8th grade algrebra, I insisted that division by 0 was 0. Mr. Miller, it a fit of frustration, took all the change out of his pocket and dumped it on my desk. "Divide that into 0 piles!" he said.
I swept the change off my desk and put it into my pocket.
That is what happens when we try to boil down a complex situation into simple metrics. It's just a question about which one is more useful or less useful to know. They can be used when a student is learning and maybe that is the only time they should be used. It relates to politics too, since we can say the GDP was this many billion dollars vs. that many, but then there are still very many details left undetermined and unknown.
kborg wrote:
"Not Real," like the rest of the pieces on the OP's original chess board in his first post.
The OP has now largely abandoned his thread. Thankfully.
LOL.
To your utmost sadness and disgust, I still lurk around this post everyday, reading your (and others') comments. And nice to see you managed to get everyone against me.
Sincerely,
The OP who "largely abandoned" this topic.
PenguinsKnight wrote:
I know im coming into this late, but dividing by zero is "undefined", not infinity.
Division as the inverse of multiplication
The concept that explains division in algebra is that it is the inverse of multiplication. For example,
6/3=2since 2 is the value for which the unknown quantity in
?x3=6is true. But the expression
6/0=?requires a value to be found for the unknown quantity in
?x0=6But any number multiplied by 0 is 0 and so there is no number that solves the equation(including infinity).
Therefore, dividing by zero does not give you infinity, it is undefined.
I agree with your point at ?x0=6.
But look at this,
2x3=6
That means, 2+2+2 (added to itself thrice) gives 6.
But here,
0x?=6
That means you have to keep adding 0 to itself until you get 6. Since that is impossible, it is considered "infinity". Why, have you ever thought that it is strange that in Physics, an object kept at the focus of any lens gets formed at "infinity", i.e. it is never formed. In that case, do you oppose that it shouldn't be infinity? Infinity stands for a "never-ending" number whose extent you cannot reach. Zero added to itself an infinite times gives 6 (or for that matter, any real number).
How can 5x = 7x for any x other than zero? What does this equation mean??
The OP who "largely abandoned" this topic.
There are multiple infinities. That's why I pointed you in the direction of Morris Kline's book, regarding Georg Cantor's work on transfinite numbers, in post # 11. But to no avail.
And you're still stuck on exchanging "two Rooks for a Bishop" as a doorway to the mathematical universe and the secrets of chess?
So be it. Knock yourself out. More power to you.
fburton wrote:
How can 5x = 7x for any x other than zero? What does this equation mean??
Answer to Question #1: It can't. Many equations have a rather short list of values for x which make the equation "true". Example: for equation x(x-1) = 0, the only sol'ns are x = 0 and x = 1
Answer to Question #2 (Part 1): See answer to Question #1. That is x = 0 is the only sol'n to this equation.
Answer to Question #2 (Part 2): and I hate to have to be the one to break it to you, you are apparently "algebraicly challenged".
StrategicPlay wrote:
PenguinsKnight wrote:
I know im coming into this late, but dividing by zero is "undefined", not infinity.
Division as the inverse of multiplication
The concept that explains division in algebra is that it is the inverse of multiplication. For example,
6/3=2since 2 is the value for which the unknown quantity in
?x3=6is true. But the expression
6/0=?requires a value to be found for the unknown quantity in
?x0=6But any number multiplied by 0 is 0 and so there is no number that solves the equation(including infinity).
Therefore, dividing by zero does not give you infinity, it is undefined.
I agree with your point at ?x0=6.
But look at this,
2x3=6
That means, 2+2+2 (added to itself thrice) gives 6.
But here,
0x?=6
That means you have to keep adding 0 to itself until you get 6. Since that is impossible, it is considered "infinity". Why, have you ever thought that it is strange that in Physics, an object kept at the focus of any lens gets formed at "infinity", i.e. it is never formed. In that case, do you oppose that it shouldn't be infinity? Infinity stands for a "never-ending" number whose extent you cannot reach. Zero added to itself an infinite times gives 6 (or for that matter, any real number).
Excuse me, but either you have never gone to school or don't get mathemathics at all. With all respect I must say, that 0 will always be zero, even if multiplied/added infinite times. However, since "infinite" is not a number, rather a directional expression, statements with x being multiplied, added, substracted from infinity are illegal.
You may ask, why 0 infinite times is not 6, 7 or any other real number. The answer is simple:
01+02+03+04+...+0n = 0, since 0+0 = 0. therefore (0+0)*n is still zero, even as n approaches the infinity.
The expresions 1/0, 2/0 etc. have no meaning, since, as the other commentators correctly said, all numbers multiplied by zero will always give zero. If we would allow such expressions, proving that 2 * 2 = 5 would be no big deal.
Regarding the OP's first post, he has made a mistake. You should calculate piece points only by addition and substraction. You shouldn't use ratios, since these methods are not compatible with how the chess game works and are therefore invalid. You will never hear a person say: "my position is 1,67 times better than my opponent's". You usually hear: "I'm 5 points ahead of my opponent". Therefore, ratios should not be used when calculating chess problems, because it is not how a chess game works.
The problem you've stated is simple and one needs little math knowledge to solve it. If a bishop, which is accompanied by a queen takes your rook, you shouldn't retake that bishop since you will lose more points than if you had not taken it. Your first argument, concerning addition and substraction is alright.
a/0 is undefined, but the limit of a/x as x -> 0 from the right = infinity, which is what Kingpatzer said I believe. a/x is not defined as anything.
@OP: Your idea of ratios might work if you only consider the pieces left on the board.
browni3141 wrote:
Ricardo_Morro wrote:
The fallacy is that you can't divide by zero. This is not allowed by the laws of mathematics. It doesn't give the result of "infinite;" it gives you the result of "meaningless." If you allow the expression 1/0 in algebra, it is easy to prove that 1=2. So your "ratio" of 5:0 yields no fraction and is null and void.
How's that 1=2 thing work?
@StrategicPlay: What type of math are you studying right now?
When you try to solve anything that reduces to something like 4x = 20x and at some point divide out your x then you end up with a nonsensical expression like 4 = 20. But of course because x was actually 0 you weren't able to treat it as a factor.
Oh, looks like I'm late to the party lol.
0 times infinity is useful as an "indeterminate form" in the study of limits, in the mathematical basis of differential and integral calculus, but I don't think it has anything to do with chess and definitely not with the post that began the thread unless you want to consider an infinite chessboard with infinitely many chess pieces on it. Any particular game would take place in a finite area, but the union of all the games would take up the entire infinite board (I'm thinking it would be 8 ranks and infinitely many files.)
Take the limit of Houdini's eval function as knight --> captured = ???
On second thought lets not go to chess-calculus, tis a silly place 
You'd think that even with a BA in math that paul would know the correct value of pi. Of course, maybe in the "system" (or language?) he is using, his value is correct.
LESS WE FORGET the M.I.T. fight song:
SIN - SIN - COSINE - SIN
3 - POINT - 1 - 4 - 1 - 5 - 9
GOOOOOOOO TEAM!