Also, why is 0.9 repeating not the number nearest to one?
Because there's no such thing as the number nearest to one in the reals. There is no greatest real number less than 1. Likewise there's no smallest real number greater than 1, either. And this isn't even a unique property of the reals. The rational numbers have this same property. 1 is a rational number and there is no other rational number that is "nearest to" 1. For any rational number you give me that's not 1, I can find a rational number closer to 1. Therefore, there is no rational number closest to 1. The same argument applies to the real numbers.
However, 0.(9) is saying it's repeating forever, infinitely.
Yes, it's a short-hand notation for the sum I gave in my previous post:
In fact all decimal expansions are a short-hand notation for a geometric series. The value of that geometric series is the value of the decimal expansion we wrote.
0.(9) is that infinitely repeating number that inches closer and closer to 1 but never reaches it.
Does 1 inch closer and closer to 0.9...? No? Maybe that's because decimal expansions aren't limits, they are single values. They don't move or change, they are a single value. 0.9... doesn't approach anything. The sequence {0.9, 0.99, 0.999, 0.999, ...} does approach something because it's a sequence. But 0.999... isn't notation for a sequence, it's a notation for a sum and that sum has exactly one value if it converges. It does indeed converge. It converges to 1. So 0.999... and 1 represent the same value. Thus we call them equal.
This "infinite incomprehensible number" that is supposedly non-existent in the space between 1 and 0.9(8), is nicely shortened into the convenient 0.(9).
What are you even saying? There's lots of actual real numbers between 0.9(8) and 1. 0.99(8), 0.999(8), I could go on all day. But like I pointed out in my previous post, 0.999... is apparently both greater than all real numbers less than one but simultaneously less than 1. This does not make sense in the real numbers. No such real number can exist by definition of the real numbers.
Are you familiar with the concept of open sets?
Sorry, this is a bit of a difficult concept to explain
That's probably because you don't know what you're talking about.
Why do people assume that because mathematical equations say they're equal, they're equal?
Because this isn't philosophy. Equality has a specific mathematical definition, it's a particular binary relation on a set. Within the real numbers, the notations 0.999... and 1 both refer to the same real value. They are equal. If you want to go off and deal with other number systems where they might be defined otherwise (like the hyperreals), you can, but we're not talking about the hyperreals, because they're even weirder than the reals.
I'll tell you right now, if you come up with a number system where 0.999... != 1 (which you can), you're probably not going to be satisfied with the properties of those numbers. They're going to act in an even weirder way. But you cannot treat them as different within the reals and end up with a consistent system. All of this arises from the specific axioms accepted when constructing the reals.
What if the math we know only works to a certain extent, like how relativity essentially fails on a quantum level.
Math is not physics. Math has almost nothing to do with the real world. It is the area of human knowledge that is probably the most divorced from reality. As much as you can realistically separate something from reality. Stop thinking of math as a model of reality. It's not. Math is one of the tools that we use to model reality, but it does not, itself, model reality. It's something far more abstract.
People always get their panties in a bunch over this for some reason. But when you throw the Banach-Tarski paradox at them there never seems to be a counter-argument. Or, hell, an actual response.
Here's the thing: your brain sucks with infinities. I'm sorry, but it's true. When you start working with infinities all sorts of weird and counter-intuitive things start happening. But that's the beauty of math, it has to be logically consistent and it is.
Within the set of real numbers 0.999... is a short-hand notation for the following sum:
Stop trying to think of its value in terms of its partial sum. 0.999... is not "approaching" 1, it is exactly 1, because it can only have one single value. The only real number that makes sense to assign to that sequence in 1. Let's do a bit of a proof by contradiction, it won't really be a proof, but hopefully it should illustrate something:
First let me introduce a new notation: 0.9{n}. I will use this to indicate a decimal number with n 0s after the decimal point. As a corollary this is also true: 0.9{n} = 1 - 10-n, so we can actually extend this to negative values for n, weird though that may be.
It should be clear that 0.9{n} < 0.999... for all natural numbers n.
Now, assume that 0.999... < 1. This is equivalent to saying that 0.999... = 1 - ε for any ε > 0.
Let's consider for a moment the equation: 0.9{n} > 1 - ε
Now we can rewrite this as: ε > 1 - 0.9{n}
ε > 10^-n
Now comes the actual argument: no matter how small we make ε, there will always be a finite integer n that will satisfy that the above inequality. That means that for every real number less than 1, there is a finite n that makes 0.9{n} bigger than that real number. Therefore, 0.9{n} is greater than every real number less than 1. So, 0.9{n} cannot be less than 1, because it is greater than every real number less than 1. So basically, this proves that 0.9... >= 1 (should be evident from here that it must be equal to 1).
This is essentially the same argument put forth earlier with the number line. Two real numbers are distinct if and only if there exists a real number between them.
If we applied the same naming standards to our solar system that we apply to extrasolar systems, then Earth would be Sol b, since it was the first planet discovered orbiting Sol.
T'was saying, the solar system is a generalized name in my eyes... Theirs millions upon millions of "solar systems" (I know millions is an understatement) out there, why not name ours to differentiate ours from our neighboring one easier?
Theirs millions of people, but where all different, that's why we're not all individually called people. Is that a decent comparison? (sincere question haha)
I guess using the word "scientifically" in my last post was apparently a mistake of sorts.
You've got it backwards. Scientifically, other systems are referred to as star systems or planetary systems not solar systems, because "solar" is the adjective associated with the sun, which is our star.
1. Yes. I work for a company that builds flight simulators and some of our customers do use them for military training. The majority of applications are civilian, however.
2. Yes. In the case of flight simulators specifically it provides a safe environment to learn procedures and systems and, to a slightly lesser extent, the behavior of the aircraft itself.
3. Yes, for the same reason as 2.
4. Yes, simulators are cheaper to operate and maintain and provide more control over the training. Simulations permit complete control of the training environment.
5. I've already listed the advantages. It's cheaper, safer, and provides more control over instruction.
6. Simulators will never perfectly mimic the system they're designed to model. Interaction with cockpit systems can be modeled very accurately and this is an important aspect of flight training (essentially learning where all the buttons are), however flight dynamics and more complicated physical systems are too complex to model precisely or depend heavily on flight data that may not be available, so the performance of the simulated aircraft may not match the performance of the actual aircraft. There are various other limitations related to feeling the motion of the aircraft (motion bases can help with this, but cannot be perfect) or having a high definition visual system (which is time consuming to develop).
7. I cannot say, I don't know how much funding the government puts into this area.
8. No.
9. That depends on what kind of training. For flight training, no. For systems training, the simulator is effectively indistinguishable from the actual system (e.g. learning to operate cockpit instruments) and so they could replace real life training for systems since these can be modeled precisely.
10. This is far too vague. How often training is required depends on the person and what they're training for. Are you expecting something like "every other day"? Because that'd be silly. A person should get as much training as they need when they feel they can. The amount, duration, and frequency of training is far too variable between topic and individual.
Anyone who doesn't see or understand the value in recreational pursuits is a naive fool. Attempting to be productive every waking hour will only result in decreasing your actual productivity and burning you out.
In the days of digital photography, how is it there are still people that don't know how this could be done?
In any case it involves taking more than one picture from the same location. There are two basic techniques for combining them. One is to simply crop both images and stitch them together, and the other is technique involves simply comparing two images and separating the background from subjects within the image.
People use the latter technique all the time to much greater effect than what your friend has done (which looks like just a crop job, since there's a visible seam).
Never thought about either of those because the professor didn't mention either one that I could remember. I see how re-deriving arc-length works now that you mention it, but I'm still at a loss on the center of mass. I forgot how to take a weighted average.
However, I was also told that center of mass calculations become much easier this semester once we learn double integrals (or was it arc length? Or something else completely?)
Center of mass is very simple when expressed as a multiple integral. In fact, I think it's kind of backwards that they teach it to you starting with the single integral since they hide the fact that you are actually doing multiple integration (which is no more difficult than integrating once, you just integrate something multiple times).
So the center of mass is simply the multiple integral of position times density (which is typically defined to be 1 everywhere, but it doesn't have to be) divided by the multiple integral of density (and how many integrations you need to do depends on the number of dimensions you're working in). Just like a weighted average is the sum of weights times values divided by the sum of the weights.
Of course, the hard part of this is coming up with the limits of integration, but that's not something that can or should be memorized anyway.
2
Because there's no such thing as the number nearest to one in the reals. There is no greatest real number less than 1. Likewise there's no smallest real number greater than 1, either. And this isn't even a unique property of the reals. The rational numbers have this same property. 1 is a rational number and there is no other rational number that is "nearest to" 1. For any rational number you give me that's not 1, I can find a rational number closer to 1. Therefore, there is no rational number closest to 1. The same argument applies to the real numbers.
Yes, it's a short-hand notation for the sum I gave in my previous post:
http://www.wolframalpha.com/input/?i=sum 9/10^n for n from 1 to infinity
In fact all decimal expansions are a short-hand notation for a geometric series. The value of that geometric series is the value of the decimal expansion we wrote.
Does 1 inch closer and closer to 0.9...? No? Maybe that's because decimal expansions aren't limits, they are single values. They don't move or change, they are a single value. 0.9... doesn't approach anything. The sequence {0.9, 0.99, 0.999, 0.999, ...} does approach something because it's a sequence. But 0.999... isn't notation for a sequence, it's a notation for a sum and that sum has exactly one value if it converges. It does indeed converge. It converges to 1. So 0.999... and 1 represent the same value. Thus we call them equal.
What are you even saying? There's lots of actual real numbers between 0.9(8) and 1. 0.99(8), 0.999(8), I could go on all day. But like I pointed out in my previous post, 0.999... is apparently both greater than all real numbers less than one but simultaneously less than 1. This does not make sense in the real numbers. No such real number can exist by definition of the real numbers.
Are you familiar with the concept of open sets?
That's probably because you don't know what you're talking about.
Because this isn't philosophy. Equality has a specific mathematical definition, it's a particular binary relation on a set. Within the real numbers, the notations 0.999... and 1 both refer to the same real value. They are equal. If you want to go off and deal with other number systems where they might be defined otherwise (like the hyperreals), you can, but we're not talking about the hyperreals, because they're even weirder than the reals.
I'll tell you right now, if you come up with a number system where 0.999... != 1 (which you can), you're probably not going to be satisfied with the properties of those numbers. They're going to act in an even weirder way. But you cannot treat them as different within the reals and end up with a consistent system. All of this arises from the specific axioms accepted when constructing the reals.
Math is not physics. Math has almost nothing to do with the real world. It is the area of human knowledge that is probably the most divorced from reality. As much as you can realistically separate something from reality. Stop thinking of math as a model of reality. It's not. Math is one of the tools that we use to model reality, but it does not, itself, model reality. It's something far more abstract.
4
Here's the thing: your brain sucks with infinities. I'm sorry, but it's true. When you start working with infinities all sorts of weird and counter-intuitive things start happening. But that's the beauty of math, it has to be logically consistent and it is.
Within the set of real numbers 0.999... is a short-hand notation for the following sum:
http://www.wolframalpha.com/input/?i=sum 9/10^n for n from 1 to infinity
Stop trying to think of its value in terms of its partial sum. 0.999... is not "approaching" 1, it is exactly 1, because it can only have one single value. The only real number that makes sense to assign to that sequence in 1. Let's do a bit of a proof by contradiction, it won't really be a proof, but hopefully it should illustrate something:
First let me introduce a new notation: 0.9{n}. I will use this to indicate a decimal number with n 0s after the decimal point. As a corollary this is also true: 0.9{n} = 1 - 10-n, so we can actually extend this to negative values for n, weird though that may be.
It should be clear that 0.9{n} < 0.999... for all natural numbers n.
Now, assume that 0.999... < 1. This is equivalent to saying that 0.999... = 1 - ε for any ε > 0.
Let's consider for a moment the equation: 0.9{n} > 1 - ε
Now we can rewrite this as: ε > 1 - 0.9{n}
ε > 10^-n
Now comes the actual argument: no matter how small we make ε, there will always be a finite integer n that will satisfy that the above inequality. That means that for every real number less than 1, there is a finite n that makes 0.9{n} bigger than that real number. Therefore, 0.9{n} is greater than every real number less than 1. So, 0.9{n} cannot be less than 1, because it is greater than every real number less than 1. So basically, this proves that 0.9... >= 1 (should be evident from here that it must be equal to 1).
This is essentially the same argument put forth earlier with the number line. Two real numbers are distinct if and only if there exists a real number between them.
1
That's true, so Sol would be named Earth b.
0
If we applied the same naming standards to our solar system that we apply to extrasolar systems, then Earth would be Sol b, since it was the first planet discovered orbiting Sol.
0
You've got it backwards. Scientifically, other systems are referred to as star systems or planetary systems not solar systems, because "solar" is the adjective associated with the sun, which is our star.
0
"The Solar System" is the scientifically correct name for the planetary system that is gravitationally bound to the Sun.
0
2. Yes. In the case of flight simulators specifically it provides a safe environment to learn procedures and systems and, to a slightly lesser extent, the behavior of the aircraft itself.
3. Yes, for the same reason as 2.
4. Yes, simulators are cheaper to operate and maintain and provide more control over the training. Simulations permit complete control of the training environment.
5. I've already listed the advantages. It's cheaper, safer, and provides more control over instruction.
6. Simulators will never perfectly mimic the system they're designed to model. Interaction with cockpit systems can be modeled very accurately and this is an important aspect of flight training (essentially learning where all the buttons are), however flight dynamics and more complicated physical systems are too complex to model precisely or depend heavily on flight data that may not be available, so the performance of the simulated aircraft may not match the performance of the actual aircraft. There are various other limitations related to feeling the motion of the aircraft (motion bases can help with this, but cannot be perfect) or having a high definition visual system (which is time consuming to develop).
7. I cannot say, I don't know how much funding the government puts into this area.
8. No.
9. That depends on what kind of training. For flight training, no. For systems training, the simulator is effectively indistinguishable from the actual system (e.g. learning to operate cockpit instruments) and so they could replace real life training for systems since these can be modeled precisely.
10. This is far too vague. How often training is required depends on the person and what they're training for. Are you expecting something like "every other day"? Because that'd be silly. A person should get as much training as they need when they feel they can. The amount, duration, and frequency of training is far too variable between topic and individual.
0
Because they're teenagers.
26
2
In any case it involves taking more than one picture from the same location. There are two basic techniques for combining them. One is to simply crop both images and stitch them together, and the other is technique involves simply comparing two images and separating the background from subjects within the image.
People use the latter technique all the time to much greater effect than what your friend has done (which looks like just a crop job, since there's a visible seam).
2
Gee, I can't imagine why.
0
Why would you be interested in things that came straight from his ass?
0
"Fredrick shat on the bookcase."
I'm not sure which one sounds better.
0
Center of mass is very simple when expressed as a multiple integral. In fact, I think it's kind of backwards that they teach it to you starting with the single integral since they hide the fact that you are actually doing multiple integration (which is no more difficult than integrating once, you just integrate something multiple times).
So the center of mass is simply the multiple integral of position times density (which is typically defined to be 1 everywhere, but it doesn't have to be) divided by the multiple integral of density (and how many integrations you need to do depends on the number of dimensions you're working in). Just like a weighted average is the sum of weights times values divided by the sum of the weights.
Of course, the hard part of this is coming up with the limits of integration, but that's not something that can or should be memorized anyway.
0
Chocolate rain...
I'm 25, but I haven't gone through puberty because I have the Benjamin Button disease.