matt grime

It's not nasty: I did explain that the necessary tools to understand what the proof is are beyond the majority of highschool (and now college) courses (going with the US model of terminology). If you aren't going to look up these things that's fine by me. There's no real need to learn abuot them, any more than you need to learn about special relativity to drive a car (or even use GPS). That's all ok. But do please, do not tell people who are experts that they are wrong because they don't prove it in a way you can understand. It does embody difficult concepts that take more time to learn then is reasonable to expect. That's the trouble with maths: it's hard. However, if you read what I've written here in other posts all you need to realize is 1. decimal strings of digits are just representations of things that mathematicians call real numbers. 2. it is necessary in that definition to declare that the strings 0.999.. and 1.000... are different representations of the same real number. It is just a convention if you will. If you want to understand why it is that the real numbers are the thing that we have come (after many centuries of playing around with ideas) to use as our preferred mathematical system of measurement then that could take a long time. It is important to understand the culture of mathematics: things are just definitions; this is the definition we have come to use. Let me illustrate perhaps some of the justifications for you: 1/n, as n tends to infinty, well, we'd want that to tend to zero (read the other stuff in the previous few posts too to get an idea of where this came from) square root of two isn't a rational numbers. so lets suppose that i add in a symbol X and define it to be larger than all the positive rationals whose square is less than 2 and less than all those who square is large than 2, can i continue this construction? What do I get? what's the smallest space with all these 'gaps' filled in? what about taking limits? What happens if i put in all limits? But i'll want unique limits, right? I need the real numbers for this. how about your counter argument at the end. you say that 0.999.. differs from one cos it will always be 0.00001... short of 1. but it isn't. your string 0.0001 has a finite number of ones. adding that to even 0.999999999 produces something larger than 1, so adding it to 0.999.... which is surely larger than 0.99999999 must also produce an answer greater than 1. and that has to be true for any *finite* number of 0s then a 1. so you're verging towards the normal argument of 'well it's an infinite number of zeroes then a 1' but that does not make sense: decimal numbers do not have an infinite number of digits and then some more. So, what number are you adding to 0.999... to get 1?

this has nothing to do with maths, though No! 0.999... is a number, it is not approaching anything anymore than 1 or 1/2 or 0.25 is approaching anything. It *is* a limit. 0.999... is just a symbol, it doesn't approach, or attempt to reach anything. It is easiest to describe it as the limit of the following sequence: 0.9, 0.99, 0.999, 0.9999, etc and since the limit of that must be 1 in the real numbers these symbols must represent the same real number. Remember the real of real number is just a silly hang up from several centuries ago when people didn't understand how to rigorously deal with things like x^2=-1. The real does not indicate we are talking about the behaviour of real physical things.

They aren't arguing against the concept if they refuse to learn the definitions, they are arguing against their perceptions of the concepts (which is merely the archimidean principle that 1/n tends to zero; here's why the archimidean principle must be true in any reasonable place where we want to do analysis. 1/n is decreasing and bounded below, we want these things to converge (otherwise we can't do analysis at all), so it must tend to x for some x. But 1/n^2 must tend to x^2 if we are to have a reasonable analysis, and it must tend to x because it is also a subsequence, hence x=x^2 so x=0 or 1, it can't be 1, so it must be zero *if we are going to do analysis* and the real numbers are where we *do* choose to do analysis, they are constructed (and that is what is missing) as a place to do analysis, and they are a useful structure for measuring things like lengths.

You don't have to explain 'non-real' numbers at all. all you have to do is teach them that real numbers are *not* real. They do not exist in any physical sense. What exists in the physical world are things we wish to measure. Whole numbers do that wonderfully for many things. Fractions even begin to capture descriptions of quantity in quite a subtle way. Now, we can show that if the equation x^2=2 has a solution at all, whatever it is is not a rational number, so where do we go from here? It would be stupid to go into details about analysis to school kids, so the best we could do is the common white lie: we use decimals to represent things, and one of the *rules* of the representation is that we *declare* the symbols 0.99... and 1 to be the same number, just as we declare 1/2 and 2/4 to represent the same number. The reason for this is simple: it makes the decimal representations of real numbers consistent. If they weren't declared to be the same thing then we have all kinds of problems. In particular, and this is exact reason why they must be the same representation of a real number in base ten, 1/n would not tend to zero. Remember, there is no reason to suppose that 1/2 and 2/4 must represent the same thing unless we say what it is they're representing and what properties these symbols must have. Of course we want these symbols to be representative of the addition and multiplication of rational numbers, which is accurately summed up as a/b=c/d if and only if ad=bc

why? all of the 1+1=3 arguments are trivially wrong, and they all rely on dividing by zero: at some point one states xy=xz hence y=z, which is just not true if x=0. you state that those who assert 0.999...=1 are wrong, yet you don't seemingly know the reason why it is true (and it is in base 10 decimal representations of real numbers). look at the proof, learn the definitions of all the terms (I'll bet you don't know what the real numbers are; this is not ignorance, it is just that the real nature of the real numbers is not taught to school kids any more than quantum mechanics is taught to 11 year olds). there is no middle ground. this is *not* about opinion or debate. there is no debate. it is not a viewpoint. it is a demonstrable, provable statement. what really is annoying is people arguing with conviction from a position of ignorance against a proof that they do not, will not, understand, and refusing to acknowledge the mathematics. instead they rely upon 'gut instinct' or something: I know you wrote a proof but i just don't believe you, I can't refute it, or point out where it is wrong, but it is, you're wrong (insert fingers in ears and go La La La, I can't hear you at this point). 0.999..=1 (decimals) is true, and is true for *exactly* the same reason that 1/2=2/4, and I bet you have no problem accepting that. there are literally hundreds of explanations of this fact available all over the place. I must have run through the argument on this forum at least 20 times alone.

Why PM me? If i knew anything about Matlab I would have told you.

Get used to that feeling: almost no equations can be solved analytically. Of course there are handy formulae for quadratic, cubic and quartic equations (look up the cubic formula).

The _convention_ is that [math]\sqrt(x)[/math], when x is positive, is the positive number whose square is x. Thus it is correct to say that (x^2)^{1/2}=|x| when x is a real number.

The square root of a positive number is taken to mean the positive one. It is called picking a branch, and that is what we do.

Don't confuse the DNA tests were talking about here with a complete DNA analysis. Matching DNA tests do not mean matching DNA. The former just splits the DNA up into chunks and measures the lengths of the chunks. It produces approximately a 1/1,000,000 (or at least when I did probability 10 years ago that is the number bandied around; they may have a better test now) chance of matching you to someone else. A complete DNA analysis, a full sequencing would be essentially unique, but impractical to do (for time and expense reasons; one of these basic DNA tests takes a week to do, not 10 minutes a la CSI).

I think you have that wrong. The probability that you left that DNA evidence based on the DNA alone is 1 in 65. If you want to factor in other evidence then by all means do, however it is the correct probabilistic interpretation of the DNA evidence and is not a fallacy. It does not assume that each person with that matching sample was in the area, it assumes that they are equally likely to have been in the area, which is reasonable unless you can show otherwise.

Whenever I hear of something like this I tend to think of this case: (fallacy) There is a 1/1,000,000 chance of someone's DNA test returning as a match. If you match DNA of the criminal then the chances of a match being 1/1,000,000 mean you must be that criminal 999,999 times out of 1,000,000. Now, really, what you should think is this: there are (in the uk) 65 people then who match that DNA sample. So if that's the only evidence they have then the suspect really has only a 1/65 chance of being the correct person.

Examples? They won't really help: you know what functions from R to R are, and you know what a differential operator is already. How about this 1. something like dy/dx = x^2 is a differential equation 2. dy/dx is a function, equivalent to x^2 in this example, I can evaluate dy/dx at x=1 3 dy/dx tells me y is a function of x and that the derivative of y with respect to x is x^2 (in this example). It can be interpreted as the gradient function. 4. y(x) = (1/3)*x^3 +c for some constant c Now compare that to d/dx. 1. d/dx is a map from the space of functions (R to R) to the space of functions, and is not a function on R itself. 2. It maps x^3 to 3x^2 3. things like d/dx = x^2 don't make sense, you can't solve them.

dx/dy is a function (it takes in numbers and spews out numbers), d/dy is a differential operator (it takes in functions and gives out functions).

And i feel you didn't take very long to think about it, that you kept ignoring important points that I repeatedly had to make to attempt to get you to notice them, which would certainly seem like patronization. That would be ' inferred emphasis denoted'. I did not put that emphasis there, you did. I did not imply it, you inferred it. As I pointed out every irrational (or rational) number is gotten by stringing together only ten different strings that are 1 digit long. You kept ignoring that and apparently not thinking about it. My reasoning is frequently wrong. As can be demonstrated by the fact that the original explanation is now third in quality in the list of ones presented here. I hate the bloody tag. It doesn't mean I am anything official here, indeed I am not anything official here. Surely the fact that I kept responding should at least count for something. As you have now found sometimes 11 minutes (time between first post of thanks and next post of 'oh hang on..') is not sufficient to understand an explanation. You came back after lunch and it all seemed a whole lot easier. That is how I would expect it to work. And it might be worthwhile remembering that lack of patience rather than what you percieve as mine.

I think exasperated is a better word. Patronizing would seem to imply that I was explaining something to you that you understood in an overly elaborate manner. Still, if you want to insult me after the help I voluntarily gave then feel free; I shan't bother to help again, then, if you find it patronizing.

What I defined does that already: start with all possible strings if you must then take those that define irrational numbers. These are in bijection with all irrational numbers so there must be an infinite number of them: there is no need to count them at the beginning. There is clearly an uncountable number of possible gluings and only countably many of them are rational, this in itself is enough of a proof now I come to think of it. There that is three proofs for you: all of the stuff I did after this point in the original is just proving that there are indeed uncountably many possible gluings in painstaking detail. I'm just making irrational numbers in base 10^10, and it is exactly the same as making irrational numbers in base 10. I'm not going to go through the details anymore if you don't see them because 1. they're clear, if you don't see it replace strings of '10 digits' with strings of '1 digit' and think about *all* parts of the replies I've given you pointing out what your line of reasoning implies. 2. I've given you two better proofs.

But why are you only allowing yourself to glue them together in that obviously repeating fashion? I didn't say anything about cycling through them like that. You have to allow all possible ways of gluing the 10^10 - 1 strings together, just like you have to allow for all possible ways of gluing the digits 0,1,..,9 together to get all decimal expansions of all numbers. You don't think that the only decimal number is 0.123456789012345678901234567890... do you? But that is equivalent to what you've written.

You don't see how to get irrational numbers from gluing together strings of finite length? Every irrational is gotten by gluing together strings of length 1 and there are only 10 of those.

Here's a far far better explanation of the same thing. Get your string of ten digits r. Pick some digit that is not the first or last digit of r, well, what the heck, we can suppose it is zero. Consider this injection from the irrationals to T. take an irrational, insert 11 zeroes between each digit in the decimal expansion. The result is irrational and in T (because any string of ten digits must start or end with a zero and we're assuming that r neither starts nor ends with a zero) and it is an injection.

And? There are 10^10 - 1 such, but what does that have to do with anything?

Of course the assertion: any string of numbers can be found in the decimal expansion of any irrational number is false. I don't see why anyone would think otherwise. What is true is that pi is thought to be normal, and thus any finite string of digits should appear at some point in the decimal expansion almost surely (i.e. with probability 1, which is *not at all* the same thing as saying it must happen). And in answer to the final question, fix an r and let T be the irrational numbers in which r does not appear in the decimal expansion, then T is definitely uncountable, as is the complement of T in the set of irrational numbers. To show the former here's a messy way of doing it. Take you string r, and let x be a digit that is not the first nor last digit of r, and let y be a string of that digit repeated 10 times. Construct the following set: S:={ s in R: s = 0.s(1)ys(2)ys(3)y....} where s(i) is any string of 10 digits except r, and such that s is irrational. 1. S is a subset of T: any substring of ten digits is one of the s(i) or starts or ends with x, and is then not equal to r. 2. For s in S consider the set X:= { x : x= 0.s(1)s(2)s(3)...} ie now delete the y strings. S and X are in bijection. But X is just the set of irrationals written in base 10^10 with one string of 10 digits omitted from the expansion, and now it doesn't matter what the string is so we may as well assume it is 9999999999, and X is now clearly in bijection with the set of irrational numbers written in base 9999999999 hence we have a bijection from a subset of T onto the irrationals in the interval (0,1) which is uncountable. 3. note these shenanigans were purely to avoid the possibility that r might occur as a string overlapping any splitting of a number into substrings.

Depends on what the set of irrational numbers is. Why don't you say?

Quite simple really: you'd have to be a total moron not to understand the distinction between freedom of expression (to prefer to wear clothing that is refered to as feminine) and a desire to supress other people's rights (to breathe, in some cases). Unless that is you are somehow likening the support of wish of a man to wear a dress to to supporting someone who denies that several million people of many cultures, creeds, sexual preferences etc were slaughtered for their beliefs and a wish to promote racial hatred. You appear to be confusing 'decent foks' with kneejerk tossers who shouldn't be allowed to procreate or vote. An eyesore? Why, because he wore a dress? I rather hope I don't live in your country whereby it is necessary to consider that offensive in order to be considered decent. Ok, you might not be able to empathize, but no one is asking you to do so. But that does not preclude you from understanding and accepting.