Showing posts with label maths. Show all posts
Showing posts with label maths. Show all posts

20080128

Repeating 9s

So, I've once again run headlong into more nerdy math stuff.

First, I'd like to address my dear brother's comment on my previous post on the subject of 1 = .999....

"This is because trailing the end of 9.999... is an additional 0 (due to multiplying by 10), while trailing the end of 0.999... is a 9."

Except that the number 0.999 repeating doesn't *have* an end for the zero to go on. Pick any 9 you want, but there's always a 9 after it, not a zero. It's like the proof of limits in calculus, where you prove it by showing that no one can specify a place where the function will suddenly deviate from its course towards your limit. 0.999 repeating is the same thing, except that it's by definition of what repeating is.

Your argument against the rule of nines is similarly ridiculous because, again, THERE IS NO EDGE FOR SOMETHING TO TRAIL ON. Infinity doesn't "end". That's why it's infinity, that's why infinities have to be dealt with completely differently than normal numbers. Just because .111 repeating is too long of a number for us to divide by brute force doesn't mean we have to resort to remainders to say that it's equal to 1/9. I'm pretty sure it's provable that 1.0 divided by 9.0 will be .111 repeating. You want to say that somewhere along the line you'll suddenly get a different correct answer when you do the division, 10/9 = 1 with 1 remaining. Obviously, it won't, as division is a function and will give the same answer when given the same input, and in this case the input for each small division in the long chain is the same as the output. 1/9 will divide into .111 repeating And that's what .111 repeating MEANS: 1/9. There IS NO TRAILING EDGE. IT GOES ON FOREVER.

There is no trailing edge. There is no trailing edge. There is *no* trailing edge. Are you saying there's some number which is at the top of infinity, which you can then put another number on top of? No, you can't do that, because infinity has no top. In the same way, repeating numbers HAVE NO TRAILING EDGE. -- To have an edge, an end, is to not be infinite --


I'll readily admit that my solution to this issue is equal parts philosophical as it is mathematical, and I think that's where we're coming into conflict.

Allow me to represent my argument a different way.

9.999... = 9/1 + 9/10 + ... + 9/(10^n)
0.999... = 9/10 + 9/100 + ... + 9/(10^n+1)


When we plug infinity into n, we have 10^infinity versus 10^infinity+1. In my mind, and maybe I'm being childish, there's a difference between infinity and infinity+1. If there's a proof to the contrary, then me and my philosophical meanderings are washed up.

That said, I have encountered a 1=.999... proof that I can't easily counter. That is the a/(1-r) proof. See: Geometric Progression.

0.999... = 9/10 + 9/100 + ...
0.999... = E ar^k (a = 9/10, r = 1/10, |r|<1)
0.999... = a/(1-r)
0.999... = (9/10)/(1-(1/10))
0.999... = (9/10)/(9/10)
0.999... = 1


So, either I'm a complete hack, or E ar^k (where |r|<1) != a/(1-r) is wrong. In fact, it's possible to represent the 9.999... - .999... in similar notation and blow my infinity versus infinity+1 argument out of the water. So, as any credible thinker should, I make my last stand and give my opponents a chance for a coup de grace.

Looking further into it, there is a step where we have limit (n->infinity) (ar^(n+1))/(1-r). Anyone who remembers their limits knows that this evaluates to zero as r<1. Desperate for an "out", I pose the following question.

Is ar^(n+1) zero or simply infinitesimally small to the point that we really can't keep track of it any longer and in 99.999...% of situations it doesn't matter? If it's the former, and mind you I would like a "because [explanation]" following any "It's zero", then my mental exercises have done little but given me a good, if futile, stretch. If it's the latter, then I would contend we've found the hole in another argument for 1 = .999...

Thus we end my grasping at straws.

20071204

1 ≈ .9999999999999...

For anyone who is familiar with math headaches, this is one of the major migraines. You can easily find all manner of argument over the internet as to the matter. I am not going to claim that in one short blog entry I can once and for all close the matter, but I can at least refute one argument.

Specifically, the following argument.

The precise mathematical proof consists of assuming first a number N which is defined by:

1) N = 0.999999999...

If we now multiply both sides of the equation by 10, we obtain:

2) 10N = 9.999999999...

Now substracting N from each side of the equation, we obtain:

3) 10N - N = 9.999999999... - N
4) = 9.999999999... - 0.999999999...
5) 9N = 9

or

6) N = 1 = 0.999999999...


It seems pretty conclusive, but there is a subtle mistake that is made at 5, specifically in the subtraction on the right hand of the equation. It should read as follows.

5) 9N ≈ 9

or

5) 9N = 8.9999999...9991

This is because trailing the end of 9.999... is an additional 0 (due to multiplying by 10), while trailing the end of 0.999... is a 9. That single digit difference is what unravels the issue. Either it must be admitted that step 5 is only an inaccurate approximation made because we are too lazy to actually travel down the path of infinity to find the "last" digit, or we have to account for the difference and display the equation appropriately.

And in case you were wondering, .8999...9991 / 9 = 0.999...

Another common argument is the table of nines.

1 / 9 = 0.111...
2 / 9 = 0.222...
3 / 9 = 0.333...
...
8 / 9 = 0.888...
9 / 9 = 1 = 0.999... = 0.111 * 9 = 1 / 9 * 9


Again, this is slightly disingenuous but in a slightly different fashion. Rather than ignoring the relationship of two separate infinites, this one ignores a fundamental concept learned in grade school. Specifically, the concept of remainders.

Trailing the edge of 0.111... isn't a pure 1, it's a 1 R(1/9). Evaluating the remainder is what gives us our next decimal place in infinity, but it will also, always have it's own remainder of 1/9. This remainder is unrepresentable in machine terms/thinking without evaluating it, hence infinity and the erroneous concept that 1 / 9 "ends" in a simple 1.

It is this remainder that, when multiplying 0.111... by 9, evaluates to 0.000...0001 and ticks the value over from 0.999... to 1.

Q.E.D. I am a nerd, thanks Dad.

20070618

1 + 1 = 3

The title of this post is true. I'm not making jokes about very large values of 1, I'm pointing out that people sometimes seem to forget a non-mathematical principle concerning combinations and aggregations. Specifically:

The whole is greater than the sum of its parts.


A game console is more than a price, a marketing team, a controller, a set of games, a TV, and a few hours to burn. You can't extract the attributes of one, compare them to another, and mathematically define which is intrinsically better. Attributes are not a homogenous group, they will interact with each other. Controllers interact with graphical power, prices interact with the aestetic look of the system, and games interact with the ease of use. Failing to realize the interconnectedness can lead to the updated principle:

The whole is rarely the exact sum of its parts, but is often less or greater.


It's the same reason a well written script, a dedicated staff, a great director, awesome actors etc. can somehow fail to make a great movie, while a group of lesser quality with the right synergy can make epics.

In whatever you do that involves combination or aggregation, remember this.

20061031

Maths

During my lunch break and other moments of rest at work today, I attempted to reengage a sagging and otherwise zombie like mind with some math. Math that many of you will find rather boring as it pertains to video game systems. These were posted in response to other inquiring minds in a revelant topic at slashdot.

There are two major entries here. One on the profitability of Nintendo versus Sony Computer Entertainment, and another on market share in the new generation (set to really get underway in November with the launches of the Wii and the PlayStation 3).

Enjoy, or ignore.

----------
Going to the actual Nintendo website and looking up their profits through each report from 2001 to 2005 I give you Nintendo's profits.

2005: $816,973,000
2004: $316,134,000
2003: $640,640,000
2002: $800,338,000
2001: $726,339,000

This is pure profit. Sales were often in excess of 4 billion dollars. 2004 is lower in profit due to costs incurred in the development of the DS.

None of this is assumption, this is straight numbers taken from Nintendo's fiscal reports free availible at:
http://www.nintendo.com/corp/annual_report.jsp [nintendo.com]

Going to Sony Computer Entertainment Inc(SCEI)'s page, I could only immediately find the numbers for 2005 and 2004 in yen.

2005: 35.5 billion Yen ($302,333,504 by today's conversion rates)
2004: 57.1 billion Yen ($485,916,092 by today's conversion rates)

SCEI made more than Nintendo in 2004, but Nintendo did far, far better than SCEI in 2005.

There's a point to be made that some income and expenditure for both companies during this period would come from the GBA, DS and PSP. However, I think my original point remains that Nintendo remains competitive even with the low market share of the Gamecube. In fact, in that two year period Nintendo soundly beats SCEI.

Quite simply, Nintendo was more profitable than SCEI this past generation despite the incredible market share the PS2 had.
----------

Japanese Allotment: 80k
American Allotment: 400k

Population of Japan: 127.42 million
Previous Japanese Market Volume: 30.31 million (Dreamcast + Gamecube + PS2 in Japan)
Number of PS3s per person: 1 per 1593
Number of PS3s per gamer: 1 per 379
Potential Market Share at Launch: 0.26% (Allotment / Previous Volume)

Population of America: 300.00 million
Previous American Market Volume: 70.8 million (Xbox + Gamecube + PS2)
Number of PS3s per person: 1 per 750
Number of PS3s per gamer: 1 per 177
Potential Market Share at Launch: 0.56%

Combined Potential Market Share: 0.47% (480k) [5.66% relative Market Share]
Current 360 Market Share: 5.93% (6 million) [70.07% relative Market Share]
Potential Wii Launch Market Share: 1.98% (2 million) [23.59% relative Market Share]

Projected March 07 Market Share: 5.93% (6 million) [27.28% relative Market Share]
360 March 07 Market Share: 9.89% (10 million) [45.46% relative Market Share]
Wii March 07 Market Share: 5.93% (6 million) [27.28% relative Market Share]

Addendum:

I used the official company goals/projections. Unofficially Nintendo may have 6 million availible by January, and Sony may fall short of 6 million in March. It seemed unobjective to factor such things in.

6 million for Nintendo is not bad at all, seeing as how the Xbox 360 has been out for almost a year now and has only recently crossed the 6 million mark. The rate at which they produce new consoles in their projections is quite steady and reasonable.

What concerns me are the PS3 forecasts. If taken in two month segments, Sony is practically calling for their production to double or triple itself twice over (.5 + 1.5 + 4.5 ~ 6). Given their continuing supply problems, I'm not certain of how reasonable their 6 million March 07 projection is. If they can manage to succeed in that goal it will be a good thing for them, but I remain skeptical.

20051206

Unadaptable

A recent past time of mine has been to spend about thirty minutes a day laughing at the complete idiocy of a very specific group of people. To be specific, the majority of people who play Paladins in World of Warcraft and post to the official forums.

So that at least part of this is interesting to the general public, I will start with a rant which is related to my pastime, but doesn't require intense knowledge of World of Warcraft or video games to understand. Following that will be boring video game related things few of you will care about.

I also am thinking of one person in particular who should probably be doing something else than reading this, especially given the subject. You know who you are, what you should be doing, and what will happen to you when I return home in eleven days and find out you neglected something for this.

Anyway, begin rant on "unadaptability".

Forenote: I liberally use some terms such as "idiot" in the following rant. I do note that while these terms are used I have an understanding that the internet turns people into jerks, including me, and that any idiotic tendencies in a person are highly exaggerated by internet exposure. As such, I would expect that the people I refer to and rather blatantly assault are better people in real life.

Now that the apology is out of the way, the rant can truly begin.

One of the major traits I cherish and am proud to possess even a little of is adaptability. It allows me to beat chess players who use a predictable strategy, to change my monopoly plans in a flash, and eat strange japanese foods I never knew existed. Adaptability is a nice thing to have.

That's not to say I'm perfectly adaptible. I enjoy consistency quite a bit, very much evident from my homebody and anti-social personality. If there's any situation where the unadaptable side of me is extremely evident, it is a social one. I rarely modify my behavior for different groups of people or situations. I am always me.

I think that being too adaptable is just as bad as being too unadaptable. While being too unadaptable obviously has its drawbacks, being too adaptable kills consistency in life. It's hard to "settle" on anything if you are always adapting "perfectly" to a new situation. However, I don't meet a lot of overadaptable people.

The vast majority of internet idiots I have encountered are extremely unadaptable. Whether in regards to other person's opinions, new trends, or the ever changing state of the internet. Quite simply, these people lack any ability to change as everything else does. Arguing with them is like arguing with a three year old, logic has nothing to do with it. They can not conceive of a different way of doing things than is already in place, and any suggestion that things could be done in a different way or might even have to be done a different way is the worst insult imaginable.

The story goes something like this.

1. X part of life of the internet changes.
2. Unadaptable person complains about X.
3. Adaptable person explains logically how it won't be very difficult to adapt to X, and will only require minor changes.
4. Unadaptable person "refutes" their opponent vigorously, making sure to tell them they are wrong and everything was better before X.
5. Adaptable person is confused as the logic and the new situation both back up the argument that adaption is both easy, and the result is better than life before X.
6. Unadaptable person continues to "refute" their opponent with "facts".
7. Adaptable person points out that the "facts" are entirely and solely related to the way the unadaptable person already does things, and how the same thing can be done with slight modifications, better.
8. Unadaptable person doesn't waver in their belief that X is a terrible injustice and that the world will now end.

And so on.

It really is quite striking how vicious the unadaptable can get about not adapting, only equalled by the retribution the adaptable sometimes mete out in return.

A nice example is a fierce interaction of me and my brother vs an entire forum. A person had posted the following problem from a math test:

"There is a town in which the inhabitants have no means of communication. There is a rule in this town stating that anyone with a mark on their forehead must kill themselves. The population met once a day, for three days, after which everyone who needed to be dead had killed themselves.

How many people had marks on their heads?"

Up until the point we entered the discussion, people had been arguing about how there wasn't enough information to solve the question, how there wasn't any way people couldn't communicate, or how people could have just waited for rain and looked in puddles or found a mirror somewhere. We joined a little ways into the discussion, with a very logical and mathematical solution.

Our solution stated the following.

"We must assume there is at least one marked person, because otherwise this problem would not state that it was after the third meeting that everyone who needed to be dead had killed themselves.

At the town meeting, everyone can see everyone else, but not themselves. If a person sees no marked people, they know that they must be a marked person (as there is at least one). However, if a person sees a marked person, they can not tell as there is at least one and not only one marked person. They might be marked as well. The same can be said for any number of marked people a person sees.

Scenarios:

If there is one marked person, they will see only unmarked people and know they are marked. This person will know to commit suicide after the first meeting.
Everyone else sees a marked person, and can not conclude whether or not they are marked.
Result: After the first meeting the marked person commits suicide and everyone who needs to be dead is. Everyone else comes to the second meeting and can conclude they don't have marks on their heads. If there was a mark, the marked person they saw would not have been able to conclude they were marked and would not have committed suicide.

If there are two marked people, they will see one marked person and can not conclude their status as marked or unmarked.
Everyone else will see two marked people, and also can not come to a conclusion.
Result 1: After the first meeting no one commits suicide, and everyone returns to the second meeting.
At the second meeting, the marked people each see that the other person did not commit suicide. They can conclude that the other therefore must have seen a marked person, and since they can see no other people they must also be marked.
Everyone else still can not come to a conclusion.
Result 2: After the second meeting the pair of marked people commit suicide. Everyone else goes to the third meeting and sees that the marked people are gone, and they now know they weren't marked.

If there are three marked people, they will each see two marked people. Because of that, they will act like the "everyone else" from the previous scenario, and wait until the third meeting before being able to confirm that they are marked. Everyone else will see three marked people and will wait until the fourth.
Result: After the third meeting the three commit suicide, and everyone who needs to kill themselves has.

There's the solution. For X number of people who are marked, X number of meetings must take place, after which everyone who needs to be dead has killed themselves.

Admittedly, neither this explanation nor the one I immediately gave is entirely flawless. However, the flaws are presentational only, with the logical not being expressed in pure perfection. The reaction to the solution was rather interesting.

Absolutely no one agreed with it. Not one other person thought we were right. It wasn't our logic that was wrong, we just apparently misunderstood the problem.

Any one of the following reasons were brought up for why we were wrong, and sometimes all at once:

1. The people probably could have seen themselves in a mirror or a puddle and known in the first place.
2. There's no possible way people can't communicate unless they can't see, hear, feel, taste, or smell.
3. How can people meet if they can't communicate?
4. How can people even know the rule if they can't communicate?
5. Why would a person commit suicide when they could just as easily forget the rule?
6. Better yet, they could purposely not do so, all the other sheep would commit suicide, and you could take their stuff, become rich, and go to some proper village where people can talk.
7. "After the third meeting everyone who needed to commit suicide was dead" doesn't mean they couldn't have been dead beforehand.

The general conclusion reached was that there was no solution to the problem.

We countered with the following:

1. The problem was on a Math test.
2. Math tests require concrete answers.
3. Therefore, this problem has a concrete answer.

It all went downhill from there. People brought up points such as, "The teacher might just be sadistic" or "People can make mistakes when making a problem". Despite the fact that we had a concrete solution, and the only refutations were entirely speculative in nature, no one changed their minds.

While I'll give some blame to my poor original wording of the solution, and the subsequent failure of my three revisions thereafter, I came to the eventual conclusion that the problem was largely in the inability of the other people to move out of the mindset they had started with and consider the problem from a different angle. The idea of actually listening to the problem when it said, "People can't communicate" just was too much of a reach. In short, by the time we arrived they had already become "set" into one way of thinking about the problem, and couldn't adapt into another.

The discussion burned down in flames.

I have, as I said, now taken particular interest in watching people show off their complete inability to adapt.

Thus ends the rant, and begins the second part which is even more boring and tedious than the first. Anyone who is unfamiliar with World of Warcraft should end here, and anyone else might want to for the sake of sanity. There is one person in particular I mentioned before who should be elsewhere to begin with, WHY ARE YOU READING THIS?!?!

Anyway...

Over the course of months the various classes of WoW have been being revamped to be more fun, more interesting, and more diverse. These changes are based half in changes made to their respective ways of fighting, and half in their talents.

The Paladin class has long been awaiting their revamp, having been rather upset at the unannounced change from the strike system to the seal system between beta and launch. A few things have been given to them here and there, but this patch is the one which will be bringing them up from the depths to which they plunged and into a world of fun.

That is, if you don't listen to the whining on the official forums.

The amount of froth being spat about the forums is incredible. Even in the most rotten of internet forums, I have never seen whining of such magnitude. It is so incredibly bad that what the actual facts say is almost completely unknown. Most of all, it is completely histerical, knowing some of the facts, watching it all.

Before I even begin, the changes were not perfect at the outset and still aren't (it wouldn't be being tested right now if they were complete). However, whining is not feedback, it is whining. The only people who listen to whining are bad parents.

Firstly, the non talent based changes are summarized as follows. Paladin seals cost a little more, and judgement costs a heck of a lot less. Debuffing judgements now last indefinately, so long as the Paladin is hitting the target. Additionally, Paladins can judge faster, at range, can stun faster, can now use their anti-undead spells against demons as well, and a particular skill (Seal of Command) now hits more often for less damage, but unmodified evens out in the end.

That all is pretty neat. However, there was general outcry about each of these changes. Paladins cried about seals costing more, declared loudly that judgement was already useless and being able to use it more often was worthless, sniffed at the indefinate debuffs, whined that judging at range defeated the purpose of being a melee fighter, called the new demon destroying powers pointless, and screamed that Seal of Command was now worse.

Despite the fact that with the exact same build as before, you'd be doing the same damage as before, for the exact same cost, people whined. Despite the fact that with a few easy tweaks you could be laying on more damage, more consistantly, and without the need to wait around for lady luck, people whined. Despite the numerous fine tunings and excellent small changes that added up to something big which I can not even begin to list, people acted like Blizzard had done as much work as a dead monkey on Sunday.

Specifically, Seal of Command had the exact same DPS, just more reliable. However, the extra power you could pour into just the seal allowed for greater DPS, and the new power of the Judgement added more on top of that. While the very last option mentioned cost (*gasp*) mana, the others didn't.

Even more hilarious, there's the possibility that higher ranks of the Seal do more damage, proc more often, or both. It's possible that such isn't the case, but as any thread asking people to try and find out is instantly swamped with naysayers and whiners who are just parroting whining to the contrary, we may never know.

The whines about the talent changes were even worse.

A good number of new talents were added, a lot of old ones were combined into one talent, gave the same or greater affects for fewer talent points, or both, and some more vital ones were made more accessible. The changes were extremely promising, and obviously still being reworked.

Whining ensued. People complained about every little change. They complained the combined talents were just combinations of useless talents into a single, useless talent. They complained that the accessibility of some of the more vital talents was still bad. They complained about each and every new talent. They whined incessantly about everything.

It was hilarious. A lot of whining centered around a particular palent called Pursuit of Justice, which increases Paladin running and mount speed somewhat. The decription is honestly poorly worded, as it states that it does not stack with "movement increasing effects", but testing shows that this nonstacking is limited to things such as potions and spells, as opposed to stats on gear (including enchants). The result? While it will not stack with potions and the offchance of a hunter running along side you with Aspect of the Pack, it will do more than replace an enchant on your boots.

You wouldn't know it if you listened to all the whining. Consistantly people cry out that it doesn't stack at all, and as such is useless (despite the fact that to achieve the same affect requires one enchant and four items).

It's absolutely incredible the amount of unadaptability being shown. Basically, every poster's complaints can be boiled down to this, "Blizzard didn't boost the talents I was already using, but instead made the ones I didn't use worthwhile and added other useful ones as opposed to making the already useful ones godly." The other whining can be similarly summarized, just replace "talents" with "skills".

It just strikes me as utterly hilarious, especially since half of the people complaining haven't even played the test server to find out what is up for themselves.

Certainly, everything isn't perfect. Sanctity Aura is still a little selfish, the judgement refreshes are currently bugged, and the strange bug which prevents one from using Seal of Command of a particular rank until you log out still persists. However, the remaining problems that may not be fixed are rather small, and not unfixable in the future.

If all the whiners actually follow through on their word, my Paladin's services will be in demand.

What ho!