A lot of people seem to be upset that this hack exists. It's used for evil, after all.

But that's not the point. Aren't you *glad* that you know this is possible? Now that we are aware this can be done, we can start trying to protect against it. The real crime here would have been for these hackers to see a vulnerability, and ignore it. Then anybody else who found the vulnerability could exploit it without knowledge of it even existing. That's a hundred times more dangerous.

Kudos to these guys on their brilliance, and ethical kudos on unveiling it. Without people like this, we would never know that we were in danger. Although, as they say, ignorance is bliss.

I reject that if your "senses do not reflect an underlying reality then the scientific method is useless." It just means that the scientific method tells us nothing about that underlying reality. In fact, given that said underlying reality (if it exists) is undetectable via any known method, one could argue that knowing anything about it would be useless - by definition, it could not be used.

On the other hand, so long as the scientific method tells us information about the world which we perceive (and many here have made that argument), then it is not useless, since it can be used. That is, we could perform an experiment, and predict results of future experiments. Even if this does not happen in some ultimate reality, the fact that we perceive it makes it useful, insofar as anything at all can be useful.

I hardly think this is true for everyone who uses a laptop for their work. With the exception of typing, tablets can do a large chunk of work - especially pretty basic stuff like email, scheduling, word processing, excel files, anything else people did with computers 5 years ago, etc. For many people who currently lug around a laptop, it might be better just to buy a dock for your tablet with a keyboard at work. Then you would only need to carry around the much smaller, lighter, and generally more carry-around-friendly tablet.

Most of the work I do on my computer is using LaTeX or the web. As neither of these is very processor hungry or niche, that could potentially be just about any tablet. My hope is to buy one, use it mostly for fun and portability, (like as an e-reader, gaming device, etc.), and then dock it at work and type. Definitely easier than a notebook.

What about the obvious: the data plan.

Yeah, having 3G is nice, being always (or at least often) connected. But when I run the tally - 92% of my day is spent somewhere with wifi. If a smartphone didn't require a data plan, I'd be all over it. But between the data plan and the two year contract, I'm spending over $720 just to fill in my commutes with internet connectivity (not too much, mind you, I wouldn't want to overflow my cap).

The only other justification I see for this expense is the GPS. Take it out, I don't care. That's a more impressive feature, in my mind, than the 3G, but still not something I need or want.

To sum up - cost sucks. Give me an android wifi-only phone and I'll be a very happy person. Or an android wifi-only tablet - this is my new hope, since it seems required data plans aren't going away anytime soon.

To some degree, this is true. 0.999... does represent the number closest to 1. That is, if we consider the interval (0,1) (which is all of the real numbers x such that 1 > x > 0 and x is not 0 or 1), 0.999... represents a the limit of a convergent sequence of numbers (0.9, 0.99, 0.999, ...) within this interval. Your thought is that since each of these approximations for 0.999... is less than *but not equal to* 1, 0.999... must also be less than (and not equal to) 1.

Sadly, this is not the case. For proof, let a and b be numbers. If b - a = 0, then by simple arithmetic, a = b. Hence by the contrapositive statement, we have that if a != b, then b - a != 0. Without loss of generality, assume that b is larger than a, so that b - a > 0. We can agree that 1 is not less than 0.999..., so in particular, we have that 1 != 0.999... if and only if 1 - 0.999... > 0.

Now, we know that for each approximation a_n of 0.999... (a_1 = 0.9, a_2 = 0.99, a_3 = 0.999, etc) 1 - a_n = .000...01 (where there are n-1 0s). And since 0.999... > a_n for each n, 1 - a_n > 1 - 0.999... for each n. In particular, this means that 10^{-n} > 1 - 0.999... for any natural number n.

Suppose for the sake of contradiction that 1 != 0.999.... Then 1 - 0.999... > 0, say 1 - 0.999... = e. Consider the reciprocal 1 / e. Because e > 0, 1 / e is some (possibly very large) number. Because 1 / e is a real number, we can find some n such that 10^n > 1 / e (This is the crux of the argument, which is sometimes called Archimedes's Principle. Most people agree that no matter how large a number you choose, I can choose another one that is bigger). But then e > 10^{-n}, so that 1 - 0.999... = e > 10^{-n} for some natural number n. This is a contradiction, since we found above that 10^{-n> > 1 - 0.999... for every natural number n. Because of this contradiction, one of our assumptions must be wrong, and our only non-obvious assumption was that 1 != 0.999.... Therefore 1 = 0.999...

The problem with your argument is that there is no number closest to 1 without being one. Just as I can always choose a number bigger than a given number, for any number less than 1 there is another number between that and 1.

What exactly is non-mathematic about symbolic computations?

What is the definition of sin? An elementary definition would be to say that the sine of an angle is the y-coordinate of the intersection point of a ray having that angle with the x-axis and the unit circle. We define pi as the circumference of the circle divided by the diameter, so in this instance, a ray from the origin having angle pi with the x-axis intersects the circle exactly halfway around, and as such has y-coordinate 0. That is, sin(pi) = 0. I don't understand what you mean by "real world," though. These are the definitions* - there is no other way to pursue mathematics than through precise definition.

If you, however, consider the definition to be based on a circle found in nature, then that is hardly a definition at all. You've already included in your definition all of the inaccuracy that we need (along with the ambiguity of deciding which circle found in nature should be used). This is why mathematics is not based on natural calculations, but is based in abstract logic and applied to calculations.

* - There are alternative definitions that could be used for sine, such as representing sine as a Taylor series. This definition can be shown to be equivalent to the one I've presented.

The problem with this approach is that you can no longer even define 0.999... The definition of this number is that it is the limit of the sequence sum_{i=1}^{i=k} 9/(10^k) as k "goes to infinity". If you have concluded that there is no infinity (in fact, I would agree with you here - infinity is not a number but a term meaning roughly larger than any number), and that infinity cannot be used in mathematics (with which I would disagree), then the number 0.999... cannot even exist.

So which is it? Either we can consider the concept of infinity, in which case 0.999... = 1, or we cannot, in which case the expression 0.999... has no meaning.

This is absolutely preposterous. Of course you can add 1/9 to 1/9. The answer is 2/9. You are failing to separate what can be added as floats in base 10 with what can be added at all. For example, there is no reason that we cannot represent 1/9 in base 9 as 0.1, then add 0.1 to 0.1 to get 0.2 in base 9. This can all be done without approximation using floating point arithmetic, just not in base 10.

The only reason infinity might come into play is because 1/9 has no finite representation in base 10. But this is not a problem. Consider the number 1/10. Would you say that you can multiply 1/10 by 10? Assuming that you would agree to this, consider what would happen if you performed this operation in binary? In base 2, 1/10 has no finite representation, so by your logic, we cannot multiply it by 10. In fact, given any finite decimal there is another base in which it has only an infinite representation, and hence by your logic we cannot multiply numbers by decimals ever.

You have imposed an artificial limitation on the basic concepts of arithmetic by limiting yourself to floats in base 10.

I also have a problem with you stating that you can't use the value of pi in math when you absolutely can. The inability to write down infinitely many digits (by which I mean write down more digits than any finite number of digits) does not preclude the fact that we can do math with pi. Math is in no way limited to calculations which can only be done with finite precision. Take this example from calculus: find the area under the curve 4/(1 + x^2) from x = 0 to x = 1. We can show that the antiderivative of 4/(1+x^2) is 4arctan(x), and so the answer is 4arctan(1) - 4arctan(0). By the definition of the tangent function, tan(pi/4) = 1 and tan(0) = 0, so 4arctan(1) - 4arctan(0) = pi. That is, the area described above is exactly pi, or exactly the same as the area of half a unit disk (even though these regions look nothing alike). Notice that the answer is not that these are pretty close, they are in fact so close that given a set of tools with any arbitrarily small margin of error, we could not tell them apart. This is just one example of doing math with pi and not a definable approximation of pi.

You got your Computer Science in my Math!

But seriously, since when do we define multiplication by how accurately we can perform it with computers? There is absolutely nothing wrong with multiplying (1/9) by 10. If I recall, the answer is 10/9.

Indeed, multiplying by positive integers at the very least has an easily definable (even in CS) meaning. Just add that many times. So 10 multiplied by (1/9) would be the same as (1/9) + (1/9) + ... + (1/9) ten times. Or are you saying that we can't add either?

As many others have said, this mostly comes down to the problem of definitions. People have trouble accepting that this is true because they do not understand the notation .999... This notation implies the existence of a limit, and so attacking this problem with a tool that has only finite precision will necessarily be flawed. It must be approached with the tools of analysis, and once these tools are understood, the proof is trivial and the fact is obvious.

Sorry, but as a graduate student in math, I can't agree with this.

Firstly, your terminology is wrong, infinity is not a number, much less an irrational number. An irrational number is defined to be a real number (loosely anything that can be expressed as an infinite decimal) which is not a rational number (a fraction). So infinite repeating decimals are not irrational. Their infinite repeating nature allows us to perform a trick similar to one mentioned above where we multiply by a suitable power of 10, subtract the original times a lower power of 10, and divide to get a rational number. Take for example x = .11205344344344344344... Then 10^8 x = 11205344.344344... and 10^5 x = 11205.344344344... Hence 10^8 x - 10^5 x = 11194139 so x = 11194139/99900000, a rational number.

Only numbers with infinite non-repeating decimal representations (like pi or e) can be irrational.

The spirit of what you are saying makes since - when we start to deal with infinite numbers strange things start to happen. However it appears that you lack the necessary background in analysis to understand what those strange things might be. The idea behind this is that an infinite decimal is actually a sum of infinitely (specifically countably) many rational numbers which converges. That is, they can only be viewed and examined with the technology of limits. Within the topology normally associated with the real numbers, the multiplication function is continuous, and hence interchangeable with limits. In particular, this means that the multiplication operators act on infinite repeating decimals in expected ways. So in this particular example, these "different things" that you claim happen really don't happen.