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Proving 0.999... Is Equal To 1

spaceturtle Making claims and taking names. (1260 comments)

It's tempting to say, well, let's subtract the fractional part of r, that is, the part whose absolute value lies in the range [0,1). But clearly that's not a unique construction either.

That would be floor(r). Why do you think it is not uniquely defined?

One can be used for counting. The other cannot.

Well, one could count "1.0", "2.0", if they really wanted to. In any case, subsets having properties that their supersets do not have is hardly unusual.

E.g. in an OO computer language we may have:
Class Integer inherits from Real
function Count()

We see that we can call function Count on an Integer, even though all Integers are of class Real. Since every Integer is a Real, we can do everything we can do to a Real to an Integer (though the result might not be an Integer). However since some Reals are not Integers there are some things we can do to Integers that we cannot do to (all) Reals.

more than 3 years ago
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Proving 0.999... Is Equal To 1

spaceturtle Re:All natural numbers are real numbers. (1260 comments)

I am not sure why you find it surprising that the integers are a subset of the reals. An uncountable infinity is a larger infinity than a countable infinity so it isn't any more surprising that a countable infinity would be a subset of an uncountable infinity than a finite set may be a subset of an infinite set. For example the set of single digit numbers {0,1,...,9} is finite and a subset of the infinite set of integers.

Also I am not sure what you mean by the nearest integer that approximates it. Why not just do something like let n = Round(r)?

about 4 years ago
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Proving 0.999... Is Equal To 1

spaceturtle In base 10 it is 0.14285714285711. (1260 comments)

In base 8, .11111111 = 1/10 + 1/100 + 1/1000 ... which in base 10 is 1/8 + 1/64 + 1/512 + ... = 0.14285714285711...., which multiplied by 7 is clearly 0.999... (which can be more succinctly represented as "1").

Also a terminology issue, the number of ones in 0.111111... is indeed "Countable".

about 4 years ago
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Indian Military Organization To Develop Its Own OS

spaceturtle Less likely than a $35 tablet? (466 comments)

India has announced a tablet that costs less to manufacture than the memory chips included in their tablet, though for some reason I can't seem to buy one yet. Once I read that the OS could run Windows, and was (to be) developed in India, I just thought "Ah another one of those announcements". I wonder why no government scientists outside India seem to be able to announce results?

about 4 years ago
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Word Processors — One Writer's Further Retreat

spaceturtle Cat is way simpler than a hexeditior. (391 comments)

Hex editors are too bloated. He should use cat instead (not the bloated monstrosity that is GNU cat of course).

about 4 years ago
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"Cumulative Voting" Method Gaining Attention

spaceturtle Everybody get 6 votes. (375 comments)

Tthey always got 6 votes. All that has changed is that before they had to vote for 6 different candidates, but now they can combine their votes.

So how does benefit minority groups? Well say there were 6+ white candidates but only one black candidate. Then voters could spend their votes only on white candidates, but did not have the option of spending their votes only on black candidates. So under the new system, if one sixth of the population wants a black representative, they get one. In principle this doesn't give them real political power, since the 5 white representatives could still out-vote them; however, for various reasons having a non-white representative gives some people warm fuzzies. For example a representative is meant to represent people as well as cast votes, so black people may be glad to have a black representative even if this doesn't directly increase their political power.

more than 4 years ago
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German Publishers Want Monopoly On Sentences

spaceturtle +3 and +5 Karma? (158 comments)

Well it seems like you got a +3 "Funny" and a +5 "Informative". Do *you* need the Karma? Maybe you'd like to lend your account to your little brother more often :P.

more than 4 years ago
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Google-Backed Wind-Powered Car Goes Faster Than the Wind

spaceturtle [OT] I wish I could deduplicate threads. (393 comments)

There are almost 300 comments but only really two arguments against this device: "The third law of thermodynamics says this can't happen!" and "How can you extract energy from the wind when you are travelling at the same speed as the wind?". As it is dozens of people are making these arguments and dozens of people of people are rebutting them. I think it would read a lot better if we could merge threads and see all rebuttals against the same argument in the same place.

more than 4 years ago
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Google-Backed Wind-Powered Car Goes Faster Than the Wind

spaceturtle Energy is not created, just transferred. (393 comments)

Energy is conserved. It is transferred from the wind to the vehicle. Consider a 1MW wind power station connected to tiny electric car. It is clear that that car would take off like a rocket.

The only question is how to generate energy from the wind when travelling as fast or faster than the wind. This has been discussed to death in the Slashdot comments as well as the comments on TFA.

more than 4 years ago
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Google-Backed Wind-Powered Car Goes Faster Than the Wind

spaceturtle Energy != Velocity. (393 comments)

But I see no reason why the drag from the wheels isn't exactly canceling out the benefit of rotating the propeller.

The Energy generated from the wheels has to match the Energy lost by the propeller. Thanks to gearing, the force is not the same.

Energy isn't the problem, a decent sized windmill can generate a megawatt of power. And it can generate the energy perpetually (assuming perpetual wind).

Consider if the vehicle was stationary, then we could easily generate the power from the wind: the force against the wheels wouldn't lose us any energy because E=mv^2 and so dE/dv=0 when v=0. Now imagine we are travelling at exactly the speed of the wind. Then our velocity relative to the wind is 0 so dE/dv=0. Thus we can push against wind without losing any energy, the same way a stationary windmill can push against the ground without losing energy. And so we can generate energy from the ground speed without losing kinetic energy (ignoring for the moment that the propeller doesn't have perfect grip on the air)

So we are currently travelling at wind speed, and generating energy from the ground. We now use that energy to push against the the wind to make us go even faster. Note that even a 50KW engine feels powerful when we are going slow and in first gear, and even a 200KWH engine can't burn rubber when we are going at 100KM/h. This comes back to E=mv^2, because Energy is proportional to the square of the Velocity, it takes more energy to speed up the faster we are going.

Note that we are still travelling faster relative to the ground than the air. Thus we can use the same trick as gears in an engine, we use a high gear relative the ground so have only a small force. We use a low gear relative to the air so we generate more force (for the same energy). We continue to speed up until the energy we gain from the different gearing ceases to make up for friction and other inefficiencies in the system (such as the propeller not having perfect grip on the air).

more than 4 years ago
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Microsoft's Free, Online Version of Office To Premiere This Week

spaceturtle There are emulators for the Apple ][ (264 comments)

Given that there are emulators for the apple ][, which is over thirty years old now, it seems likely that there will be an emulator capable of running say Windows 2000 a couple of decades down the track. But I guess it would be better to archive the ISOs of the installation media than the installed image. There are other methods they could use: convert to PDF (can't edit document easily); Convert to OpenDocument (likely to mess up formatting etc.); convert to plain UTF-8 (mess up formatting even more). It hard to tell what is best without knowing more about their requirements (perhaps a combination of all of the above).

more than 4 years ago
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Microsoft's Free, Online Version of Office To Premiere This Week

spaceturtle Use a Virtual Machine? (264 comments)

Have you considered archiving virtual machines with old versions of Word along with the documents? I think that you can get old versions of Word for about $50 each on ebay.

more than 4 years ago
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Scaling Algorithm Bug In Gimp, Photoshop, Others

spaceturtle As does mtpaint at 616K (368 comments)

Mtpaint has a "Gamma Correction" check box. If you check it you get the faint picture. If you leave it unchecked you get the grey box. Not bad for an editing program that fits entirely in one uncompressed 616KB executable.

more than 4 years ago
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OpenSolaris Or FreeBSD?

spaceturtle Can give a boost even with same instruction set. (405 comments)

You can also gain some performance by tweaking code for different processor types, even if they have the same instruction set. One example would avoiding XOR swaps on CPUs that have instruction pipelining, which is independent of the instruction set.

http://en.wikipedia.org/wiki/XOR_swap_algorithm

This maybe wasn't the best example since XOR swaps are rarely useful anyway. I suspect that other things like word (mis)alignment and varying cache miss costs may be a factor for different processors.

Gentoo claims that picking e.g. core2 over nocona can boost performance by 15% (which seems a bit much to me), so picking the right x86_64 variant is still something that is considered. Not something I worry about though, unless I am compiling from source anyway.

more than 4 years ago
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Engaging With Climate Skeptics

spaceturtle But still raw-data is good. (822 comments)

They story suggests that scientists should present all their raw data to improve "openness". I agree that it is unlikely to inform people who are unwilling to read more than a few key phrases from one textbook. However I think this should be the norm in *all* fields of science, controversial or not (where privacy is not a concern etc.).

For example, I once contacted a author of a paper basically saying "I read the paper you wrote on a utility to improve security. It seems to me that your utility could also be used to improve performance as well. Could I play with the utility?". Their response was "I wrote that a few years back. I think I lost the code." Other researchers have similar difficulties when trying to perform meta-studies based on other researchers data. This could have been avoided if submitting raw data and code was the norm. These days there would be almost zero-cost in submitting raw-data in electronic form along with almost every manuscript submitted for peer-review and publication.

more than 4 years ago

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