I am not talking about arbitrary large natural number. I am talking about infinite natural number, as opposed to a finite natural number. I do not see any reason why an infinite natural number should not have a successor.
I would not necessarily agree with that. 1+1=2 talks about integers, there is no infinite summation involved, you are really just referring to Peano axioms here. In fact, that's probably a definition of 2. Completely formal definition of 2.
I don't really understand the analogy. Surely, if the number of 9's in the decimal expansion is an infinitely large natural number, then you can always increase it by 1, can't you?
I don't think that would work, but i think 0.999... + (1-0.999...)/2 would!
But even if you could not find a number between them, does it mean they have to be equal? Does your real number system (either by an axiom, or as a result of construction) require that between any two real numbers, there must be another real number?
I think the rpobem is that most people do not understand real numbers. In fact, most people have probably never seen a definition of real numbers.
I would say that it is somewhat informal in the sense that there are at least two terms that need to be defined, and could easily admit different definitions.
I think everybody would agree that 0.999... = \sum_{n=1}{\infty} \frac{9}{10^n}. The two symbols that IMHO need clarification are \infty and =. In ZF, I think \infty would be \omega, but in other set theories you may end up with different things. The = is definitely an equivalence relation on some set, but details would depend on the way you construct real numbers. It could for example mean the two are infinitely close. The sum on the left could also be a problem, it could be one real number, or whole set of numbers, all infinitely close to each other. Ultimately, the informal meaning would be the same, but the details and therefore proofs could be quite different.
The point is that the three points in 0.999... is short notation for an algorithm.
That's a very constructivist point of view. I believe that (an extreme) constructivist would then go on and say that since you cannot actually carry that algorithm out, as it requires infinite number of steps, 0.999... does not actually exist.
For most mathematicians, \sum_{k=1}^\infty \frac{9}{10^k} is not an algorithm, but a number. But they could still argue whether the number is 1, or merely infinitely close to 1.
And series or limit is not a number? Provided it converges?
0.999... is an infinite series. It is one of (infinitely) many infinite series that are all equal to 1. Or, as some would say, infinitely close to 1. How close? That depends on how many nines are actually there. We can always add one more nine, and get little closer, but once we have infinitely many 9's there, then no matter how many of them are actually there, all the numbers will be infinitely close to each other.
If 1=.999... then infinitesimals don't count... if that's true, can we skip calculus altogether?
Actually, it is the other way around. We need calculus exactly because infinitesimals don't count. If you can freely work with infinitesimals, most of calculus becomes arithmetic.
They honestly think of infinity as being a really large finite number, so they believe that no matter how many digits you add to 0.333..., it never quite reaches 1/3.
It never quite reaches 1/3, but it will be infinitely close to it.
Actually, many of us do gather. Getting a good quality mushrooms in a store is nearly impossible in most of USA. You can mail-order dry mushrooms, but that's not the same as fresh. In addition, because mushrooms are generally extremely picky about where they grow, most of the ones you can buy are gathered in the wild anyway.
I personally believe that every kid should experience going to a forest and eating berries straight from the bush. For me, that is just a part of childhood. I can see how some people can feel the same way about hunting and fishing.
The fact that many hunters (as well as berry pickers) are irresponsible assholes is another problem.
There could be a number of organizations setting up their own servers. For example, an university could have their own server, that would be tweaked to fit visually into the larger IT infrastructure (loco, colors), hook into other university IT services (course management system, registration, advising,...) and in general be customized for students at that particular school. They could also host accounts for alumni, which would make them better connected to their former school (which would be good for stuff like fundraising).
A good nail gun does not make it easy to build a bad house. At least it does not make it easier to build a bad house than to build a good house. Think about a nail gun that makes it easy to put a nail in every 15 feet or further, but hard to put them any closer. A nail gun that makes it very easy to use 1/4 inch nails, but significantly harder to use any larger nails. A nail gun that makes it significantly easier to put nail into a boring gray plasticky looking materials than anything nice looking. Imagine a nail gun like that, and imagine that 95% of all builders use that type of nailgun. That would certainly explain lot of the construction I see around the last 10 or 15 years.
First, as freetype license is BSD, they are not stealing anything. They have complete right to use it.
Second, they don't do it to get better fonts. They are using a library that 1) emulate their own cleartype engine well enough for their purposes and 2) already exists on OS X. There is no indication anywhere that freetype is better than their own cleartype. It may be, and when we see them ditching cleartype and using freetype on Windows, we may conclude that it is, but until that happens, MS is simply doing the obvious and using already existing library on OS X instead of wasting time and money porting their own.
The interesting part is that they chose freetype over the native OS X engine.
Even more interesting would be if they contributed back to freetype. They are under no legal obligation to do that, but it would certainly make their recent proclamations of love for open source sound more sincere.
I agree that the summary is completely idiotic, especially since TrueType was patented by Apple, not Microsoft, as the summary claims, and the patents has since expired.
However, there is something interesting about it. MS could have used the native font rendering engine on OS X. They had at least three options: native renderer, freetype, or port cleartype. I am not surprised that chose not to port, but their choice seems to indicate that freetype does good enough job emulating cleartype rendering, better than the native renderer. This does not mean that freetype is better than the native renderer, it just means that it is better at emulating MS's cleartype.
Actually, MS got access to it the same way any Mac developer does - by linking to/usr/X11/lib/libfreetype.dylib.
Wow! How did they manage to do that on a Windows machine back in 1989? I know MS programmers are beast, but linking to a library from a completely different operating system, especially a library that has not even been written yet, that's quite an achievement. They must have used some sort of magic, that's how they circumvented Apple's TrueType patents, too.
Actually, none of your three options is good by itself. In order to transfer substantial amount of information to the crowd, you need to combine all three of them. Besides, all three of the individual options can be done with PowerPoint. I have seen may talks where someone put basic outline of their talk on the screen, and then talked and talked and talked. At the end, most people remembered the outline, if anything at all. I have seen talks where the speaker had gobs and gobs of text on each slide, and read them along. Interestingly enough, I have seen a number of talks where this method worked extremely well, but in these cases the text was extremely complicated and needed very close attention to detail, and the audience was extremely motivated, carefully followed the text with the speaker, and asked questions when they did not understand some part. Finally, I have seen presentation full of multimedia, very entertaining, but with very little actual content.
On the other hand, each of the tree options can be easily done without PowerPoint.
In most cases, you need to strike the right balance between the three options you listed, Whether you use PowerPoint or not is largely irrelevant. PowerPoint and similar tools can make things easier, on the other hand, they seem to often "lead people the wrong way" to easily.
You're SUPPOSED to put your points in bullets because you're there giving the presentation to elaborate upon said bullet points.
No, you are NOT! This is in my opinion the very worst thing about PowerPoint, it sort of forces everything into a list of bullet points. There is some, in my experience rather small percentage of talks that really are mainly about a list of bullet points, with a little bit of explanation for each of the points, and PowerPoint is very useful for these talks. In most talks, the structure of the talk, the "table of contents", is much less important than the contents itself. If in one part of my talk I concentrate on linear systems, people will generally remember that if I announce it clearly, and will not need a giant slide projected onto the screen to remind them. It's *what* I am saying about linear system, their properties, relations to non-linear system, linearization etc. that's important. NOT the fact that I am talking about these things in some particular order, but WHAT I am saying about them. I can include a "talk plan" somewhere in a corner of a screen or whiteboard and every once a while point to it in order to make it easier for listeners to realize where I am and how things relate to each other, but it should not be the main focal point of my presentation.
Unfortunately, PowerPoint makes it extremely easy to create presentations that are either just lists of bullets with no contents whatsoever, or, when people try to include contents, they end up with the opposite extreme, which you yourself described as someone putting three paragraphs of text onto the slide, reading them to the audience during the presentation.
It is perfectly possible to make a good PowerPoint presentation, and I have seen a number of them, but it is not easy. It is not entirely a PowerPoint's fault, making good presentations is hard, no matter which tool you use. The problem with PowerPoint is that it makes it so easy to make a bad presentation. In addition to that, if you look at a random PowerPoint tutorial, somewhere on the web or even in the official documentation, they tell you "this is how you make bullet points, this is how you add irrelevant clipart, this is how you make things fly in and bounce around in an extremely irritating way, and this is how you choose an ugly theme for your presentation." So that's what people do.
Yeah, I try to do the same thing, in my case they are math classes. Unfortunately some bright person designing our classrooms decided to place the pull down projection screen in *every single classroom* on the entire campus directly in front of the whiteboard. So I have about 2 feet of whiteboard on each side of the screen available for detailed derivations and examples.
Slashdot Mirror
Actually, it does not. .999... is not an infinite natural number. And if it indeed is equal to 1, then its successor would not be itself, but 2.
I am not talking about arbitrary large natural number. I am talking about infinite natural number, as opposed to a finite natural number. I do not see any reason why an infinite natural number should not have a successor.
I would not necessarily agree with that. 1+1=2 talks about integers, there is no infinite summation involved, you are really just referring to Peano axioms here. In fact, that's probably a definition of 2. Completely formal definition of 2.
I don't really understand the analogy. Surely, if the number of 9's in the decimal expansion is an infinitely large natural number, then you can always increase it by 1, can't you?
I don't understand. Even if I have infinitely many 9's, i can still add one more 9, can't I. Unless of course there is already \Omega of them.
Duh. 0.9999... and a half!
I don't think that would work, but i think 0.999... + (1-0.999...)/2 would!
But even if you could not find a number between them, does it mean they have to be equal? Does your real number system (either by an axiom, or as a result of construction) require that between any two real numbers, there must be another real number?
I think the rpobem is that most people do not understand real numbers. In fact, most people have probably never seen a definition of real numbers.
I would say that it is somewhat informal in the sense that there are at least two terms that need to be defined, and could easily admit different definitions.
I think everybody would agree that 0.999... = \sum_{n=1}{\infty} \frac{9}{10^n}. The two symbols that IMHO need clarification are \infty and =. In ZF, I think \infty would be \omega, but in other set theories you may end up with different things. The = is definitely an equivalence relation on some set, but details would depend on the way you construct real numbers. It could for example mean the two are infinitely close. The sum on the left could also be a problem, it could be one real number, or whole set of numbers, all infinitely close to each other. Ultimately, the informal meaning would be the same, but the details and therefore proofs could be quite different.
The point is that the three points in 0.999... is short notation for an algorithm.
That's a very constructivist point of view. I believe that (an extreme) constructivist would then go on and say that since you cannot actually carry that algorithm out, as it requires infinite number of steps, 0.999... does not actually exist.
For most mathematicians, \sum_{k=1}^\infty \frac{9}{10^k} is not an algorithm, but a number. But they could still argue whether the number is 1, or merely infinitely close to 1.
And series or limit is not a number? Provided it converges?
0.999... is an infinite series. It is one of (infinitely) many infinite series that are all equal to 1. Or, as some would say, infinitely close to 1. How close? That depends on how many nines are actually there. We can always add one more nine, and get little closer, but once we have infinitely many 9's there, then no matter how many of them are actually there, all the numbers will be infinitely close to each other.
If 1= .999... then infinitesimals don't count... if that's true, can we skip calculus altogether?
Actually, it is the other way around. We need calculus exactly because infinitesimals don't count. If you can freely work with infinitesimals, most of calculus becomes arithmetic.
They honestly think of infinity as being a really large finite number, so they believe that no matter how many digits you add to 0.333..., it never quite reaches 1/3.
It never quite reaches 1/3, but it will be infinitely close to it.
do I have to read this?
Seriously, someone needs to control eldavojohn's grammar and spelling in outgoing slashdot posts.
lynx shall be spoken of with curses forever!
Not necessarily, you can choose to speak of lynx in slang instead.
Actually, many of us do gather. Getting a good quality mushrooms in a store is nearly impossible in most of USA. You can mail-order dry mushrooms, but that's not the same as fresh. In addition, because mushrooms are generally extremely picky about where they grow, most of the ones you can buy are gathered in the wild anyway.
I personally believe that every kid should experience going to a forest and eating berries straight from the bush. For me, that is just a part of childhood. I can see how some people can feel the same way about hunting and fishing.
The fact that many hunters (as well as berry pickers) are irresponsible assholes is another problem.
There could be a number of organizations setting up their own servers. For example, an university could have their own server, that would be tweaked to fit visually into the larger IT infrastructure (loco, colors), hook into other university IT services (course management system, registration, advising, ...) and in general be customized for students at that particular school. They could also host accounts for alumni, which would make them better connected to their former school (which would be good for stuff like fundraising).
I think I have pretty good basic understanding of geometry, calculus and physics, but I still don't understand the first paragraph of the summary.
Is this latest vulnerability related to scripting? The article is somewhat short on details.
A good nail gun does not make it easy to build a bad house. At least it does not make it easier to build a bad house than to build a good house. Think about a nail gun that makes it easy to put a nail in every 15 feet or further, but hard to put them any closer. A nail gun that makes it very easy to use 1/4 inch nails, but significantly harder to use any larger nails. A nail gun that makes it significantly easier to put nail into a boring gray plasticky looking materials than anything nice looking. Imagine a nail gun like that, and imagine that 95% of all builders use that type of nailgun. That would certainly explain lot of the construction I see around the last 10 or 15 years.
Fail!
First, as freetype license is BSD, they are not stealing anything. They have complete right to use it.
Second, they don't do it to get better fonts. They are using a library that 1) emulate their own cleartype engine well enough for their purposes and 2) already exists on OS X. There is no indication anywhere that freetype is better than their own cleartype. It may be, and when we see them ditching cleartype and using freetype on Windows, we may conclude that it is, but until that happens, MS is simply doing the obvious and using already existing library on OS X instead of wasting time and money porting their own.
The interesting part is that they chose freetype over the native OS X engine.
Even more interesting would be if they contributed back to freetype. They are under no legal obligation to do that, but it would certainly make their recent proclamations of love for open source sound more sincere.
I agree that the summary is completely idiotic, especially since TrueType was patented by Apple, not Microsoft, as the summary claims, and the patents has since expired.
However, there is something interesting about it. MS could have used the native font rendering engine on OS X. They had at least three options: native renderer, freetype, or port cleartype. I am not surprised that chose not to port, but their choice seems to indicate that freetype does good enough job emulating cleartype rendering, better than the native renderer. This does not mean that freetype is better than the native renderer, it just means that it is better at emulating MS's cleartype.
Actually, MS got access to it the same way any Mac developer does - by linking to /usr/X11/lib/libfreetype.dylib.
Wow! How did they manage to do that on a Windows machine back in 1989? I know MS programmers are beast, but linking to a library from a completely different operating system, especially a library that has not even been written yet, that's quite an achievement. They must have used some sort of magic, that's how they circumvented Apple's TrueType patents, too.
Actually, none of your three options is good by itself. In order to transfer substantial amount of information to the crowd, you need to combine all three of them. Besides, all three of the individual options can be done with PowerPoint. I have seen may talks where someone put basic outline of their talk on the screen, and then talked and talked and talked. At the end, most people remembered the outline, if anything at all. I have seen talks where the speaker had gobs and gobs of text on each slide, and read them along. Interestingly enough, I have seen a number of talks where this method worked extremely well, but in these cases the text was extremely complicated and needed very close attention to detail, and the audience was extremely motivated, carefully followed the text with the speaker, and asked questions when they did not understand some part. Finally, I have seen presentation full of multimedia, very entertaining, but with very little actual content.
On the other hand, each of the tree options can be easily done without PowerPoint.
In most cases, you need to strike the right balance between the three options you listed, Whether you use PowerPoint or not is largely irrelevant. PowerPoint and similar tools can make things easier, on the other hand, they seem to often "lead people the wrong way" to easily.
You're SUPPOSED to put your points in bullets because you're there giving the presentation to elaborate upon said bullet points.
No, you are NOT! This is in my opinion the very worst thing about PowerPoint, it sort of forces everything into a list of bullet points. There is some, in my experience rather small percentage of talks that really are mainly about a list of bullet points, with a little bit of explanation for each of the points, and PowerPoint is very useful for these talks. In most talks, the structure of the talk, the "table of contents", is much less important than the contents itself. If in one part of my talk I concentrate on linear systems, people will generally remember that if I announce it clearly, and will not need a giant slide projected onto the screen to remind them. It's *what* I am saying about linear system, their properties, relations to non-linear system, linearization etc. that's important. NOT the fact that I am talking about these things in some particular order, but WHAT I am saying about them. I can include a "talk plan" somewhere in a corner of a screen or whiteboard and every once a while point to it in order to make it easier for listeners to realize where I am and how things relate to each other, but it should not be the main focal point of my presentation.
Unfortunately, PowerPoint makes it extremely easy to create presentations that are either just lists of bullets with no contents whatsoever, or, when people try to include contents, they end up with the opposite extreme, which you yourself described as someone putting three paragraphs of text onto the slide, reading them to the audience during the presentation.
It is perfectly possible to make a good PowerPoint presentation, and I have seen a number of them, but it is not easy. It is not entirely a PowerPoint's fault, making good presentations is hard, no matter which tool you use. The problem with PowerPoint is that it makes it so easy to make a bad presentation. In addition to that, if you look at a random PowerPoint tutorial, somewhere on the web or even in the official documentation, they tell you "this is how you make bullet points, this is how you add irrelevant clipart, this is how you make things fly in and bounce around in an extremely irritating way, and this is how you choose an ugly theme for your presentation." So that's what people do.
Yeah, I try to do the same thing, in my case they are math classes. Unfortunately some bright person designing our classrooms decided to place the pull down projection screen in *every single classroom* on the entire campus directly in front of the whiteboard. So I have about 2 feet of whiteboard on each side of the screen available for detailed derivations and examples.
I don't think awesome runs on Windows.