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peterbirksAn idle mathematical question. Is the square of an irrational number always another irrational number? Or is it never an irrational number? Or is it sometimes one, sometimes the other? Just curious.
If you post this kind of thing on Facebook, I usually time it to see how long someone posts "Wikipedia says...." as if, duhhh, I had never heard of Wikipedia. There really ought to be a law forbidding any response to an online question where the response begins along the lines of "According to Wikipedia..."
++++++++
Which was a diversion before I even started. I was just thinking of paradoxes, contradictions, and the like. In the FT this morning Daniel Pimlott writes on how the savings ratio had become positive after a period in 2008 when we became net borrowers. He continues "While in the longer term this (a positive savings ratio) is seen as a necessary rebalancing of the economy away from consumption towards more production and investment, in the short term a too-rapid increase in savings could have devastating effects".
Hmm, I thought. And this situation would end, er, when?
In fact, what everyone wants is a higher savings ratio (so that people are prepared for their old age) and a lower savings ratio (so that the economy today keeps going). They get round this contradiction by saying that they want a higher savings ratio "in the long term", but higher consumption "in the short term".
It doesn't take a mastermind to spot that "in the long term" is something like "next month". It never actually arrives.
Eventually the pensions timebomb really does have to explode. And I'll even tell you where it will start -- in the local government sector, followed by the whole public sector. When you get to the stage that 80% of your local tax bills are going on pensions, people will get elected solely on the grounds that they will stop paying those pensions. When only the retired public sector is cushioned against inflation, people will be elected who want it to stop. And (worst case scenario), if all of those standing are in favour of keeping the status quo (because, of course, they benefit from it), then democracy will be in danger.
I have had chats with people who have been retired 10 years or more, and they look at me with total bafflement when I say that I am worried about my future. Most of these are so accustomed to a monthly stipend (often inflation-linked) that they just can't conceive of a life where that money was not coming in. That, as it were, is the basic. When I explain to them, slowly, that the amount in my pension pot would probably buy an annuity of about a tenner a month, it kind of dawns on them that people working today often have no safety net to look forward to at 60, 63 or 65. Such people are unlikely to look sympathetically on Council Tax Bills that pay for virtually nothing except someone else's inflation-linked deal.
It's all Lloyd George's fault, of course. He created the concept of the unfunded pension, and it's been Ponzi-ing away since 1909. When you look at the impact of the Madoff scheme, it's not difficult (although it is frightening) to imagine the impact of the implosion of a scheme that's been running 70 years longer than that.
Inflation in Russia in the 1990s virtually wiped out a generation of old people's savings. The "system" in the UK "protects" some retired people against even this. What is not realized is that inflation is, in a sense, a safety valve. When the numbers don't add up (which they often don't) inflation occurs to make them add up. So making a pension "inflation-proof" does not solve the problem. It just keeps the pressure building up until it explodes, more forcefully, elsewhere.
______________
If you post this kind of thing on Facebook, I usually time it to see how long someone posts "Wikipedia says...." as if, duhhh, I had never heard of Wikipedia. There really ought to be a law forbidding any response to an online question where the response begins along the lines of "According to Wikipedia..."
++++++++
Which was a diversion before I even started. I was just thinking of paradoxes, contradictions, and the like. In the FT this morning Daniel Pimlott writes on how the savings ratio had become positive after a period in 2008 when we became net borrowers. He continues "While in the longer term this (a positive savings ratio) is seen as a necessary rebalancing of the economy away from consumption towards more production and investment, in the short term a too-rapid increase in savings could have devastating effects".
Hmm, I thought. And this situation would end, er, when?
In fact, what everyone wants is a higher savings ratio (so that people are prepared for their old age) and a lower savings ratio (so that the economy today keeps going). They get round this contradiction by saying that they want a higher savings ratio "in the long term", but higher consumption "in the short term".
It doesn't take a mastermind to spot that "in the long term" is something like "next month". It never actually arrives.
Eventually the pensions timebomb really does have to explode. And I'll even tell you where it will start -- in the local government sector, followed by the whole public sector. When you get to the stage that 80% of your local tax bills are going on pensions, people will get elected solely on the grounds that they will stop paying those pensions. When only the retired public sector is cushioned against inflation, people will be elected who want it to stop. And (worst case scenario), if all of those standing are in favour of keeping the status quo (because, of course, they benefit from it), then democracy will be in danger.
I have had chats with people who have been retired 10 years or more, and they look at me with total bafflement when I say that I am worried about my future. Most of these are so accustomed to a monthly stipend (often inflation-linked) that they just can't conceive of a life where that money was not coming in. That, as it were, is the basic. When I explain to them, slowly, that the amount in my pension pot would probably buy an annuity of about a tenner a month, it kind of dawns on them that people working today often have no safety net to look forward to at 60, 63 or 65. Such people are unlikely to look sympathetically on Council Tax Bills that pay for virtually nothing except someone else's inflation-linked deal.
It's all Lloyd George's fault, of course. He created the concept of the unfunded pension, and it's been Ponzi-ing away since 1909. When you look at the impact of the Madoff scheme, it's not difficult (although it is frightening) to imagine the impact of the implosion of a scheme that's been running 70 years longer than that.
Inflation in Russia in the 1990s virtually wiped out a generation of old people's savings. The "system" in the UK "protects" some retired people against even this. What is not realized is that inflation is, in a sense, a safety valve. When the numbers don't add up (which they often don't) inflation occurs to make them add up. So making a pension "inflation-proof" does not solve the problem. It just keeps the pressure building up until it explodes, more forcefully, elsewhere.
______________
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no subject
Date: 2009-09-22 12:04 pm (UTC)no subject
Date: 2009-09-22 12:51 pm (UTC)My hunch was that pi squared would be irrational as well, and I tried wikipedia, and elsewhere, but fould lots of stuff on the square roots of various numbers, but nothing on the squares of irrational numbers.
It would be really cool if there were rational irrational numbers and irrational irrational numbers. Where's Bill Chen when you need him?
PJ
no subject
Date: 2009-09-22 02:50 pm (UTC)More interesting would be this lemma: the square root of every irrational number is also irrational. This is reasonably obvious, and leads to the following musings about pi^2 and e^2.
I have no proof that pi^2 is irrational, except by observation. Assume the contrary. If so, it should be relatively straightforward to write the value of pi^2 down, in any base you choose. If you can write pi^2 down, you can calculate pi to an arbitrary precision, simply by taking the square root. Square roots are algorithmically much quicker to calculate to an arbitrary precision than is pi, given any known present algorithm for pi.
Therefore either pi^2 is irrational, or the whole wide world of mathematicians is missing something...
I'd be extremely surprised if e^2 was rational, on a similar basis (substituting the calculation of natural logarithms for the above).
Wikipedia is actually quite good at the definitions for this sort of stuff (at least in comparison to factual statements for anything else, where it sucks), but it's not a magic wand and it can only really refer you to published articles and proofs for the actual workings. Given its record on attribution/references in other fields, it's not the first place I'd start looking.
Anyway, number theory doesn't really deal in standalone cases; more in enumerations and sets. The set of irrational numbers whose square is a rational number -- what I believe you are calling rational irrational numbers -- quite clearly has the same size as the cardinality of the set of rational numbers, ie aleph null. The set of irrational numbers whose square is an irrational number -- what I believe you are calling irrational irrational numbers -- is not aleph null, since the set of irrational numbers is not countable.
Or, to put it in betting terms, you're pretty safe to assume that the square of any irrational number is irrational.
no subject
Date: 2009-09-22 03:12 pm (UTC)Paradox by the dashboard
Date: 2009-09-22 03:44 pm (UTC)Damn.
no subject
Date: 2009-09-22 04:01 pm (UTC)PJ
no subject
Date: 2009-09-22 04:02 pm (UTC)PJ
no subject
Date: 2009-09-22 04:05 pm (UTC)What do you call a gorilla with a banana stuck in each ear?
no subject
Date: 2009-09-22 04:08 pm (UTC)If one shoe drops off the tree, but nobody hears it dropping...
Date: 2009-09-22 04:49 pm (UTC)See, I just know you'll call me a wuss for checking on Wikipedia (http://en.wikipedia.org/wiki/Transcendental_number)
But here we go:
"Any non-constant algebraic function of a single variable yields a transcendental value when applied to a transcendental argument." In other words, your theorem that there exists a rational member of the set (I^n) where I is irrational and n is a non-negative integer requires you to perform an ...
... er ... I'm not sure whether I should commit this horrible pun ...
Oh well. OK then.
You need to do a Reverse Fermat.
I plead necessity, m'lud, and the educational system in the UK is to blame.
Re: If one shoe drops off the tree, but nobody hears it dropping...
Date: 2009-09-22 05:00 pm (UTC)Christ, you used to be able to talk to a simple-minded second year undergraduate in Maths at a halfway decent redbrick such as Imperial College, back in the day, and they'd have bored the caps off a couple of bottles of Bells while they explained in great and accurate detail.
These days, of course, Maths PhDs are worthless quantitative cunts who know the derivative price of massively traded infinitesimals, but the value of nothing.
I'm sure they're great for discussing yesterday's episode of Strictly Come Wanking over the water-cooler, though, Mike. Stick with it, as directors of certain gonzo films in the Valley say.
no subject
Date: 2009-09-22 06:46 pm (UTC)The point about "make me save, lord, but not yet" is one which has been making me laugh for some time, and reinforces the sense that a lot of journalists and probably even finance professionals haven't a clue what they're talking about.
Also I like very much the final comment about inflation. In a pathological future, where the ratio of workers to non-workers goes through the floor, it really doesn't matter how much you have saved up (or what is promised to you) - it's going to to be the ability to labour to produce goods/food/whatever that becomes valuable, not savings. There's an interesting thought experiment about a nation overwhelmed by old people who dominate the democratic process by weight of numbers and a minority workforce who control the ability to actually produce stuff, and what happens. I believe Martin Wolf(e?) wrote an editorial in the FT about a year or so ago suggesting that democracy is a great governance system in times of economic surplus, but that it might be in for a hard time in a generation or two. I would recommend investing pensions in slaves, but since that seems to be politically no-go atm, perhaps the only option is to reproduce as much as possible and hope some of your kids feel guilty at poor old mum and dad wasting away...
no subject
Date: 2009-09-22 07:42 pm (UTC)Have you read Houellebecq? In, I think, Platform he talks about the heatwave in France a few years ago that killed around 30,000 elderly people. It was said by most that this showed France to be a barbaric country. He disagrees, thinking that it shows how advanced France is that it can let its old, unproductive people go so coolly.
And what an excellent readership you attract to your blog: the first eleven replies were to the mathematical musings, rather than to the matter that might to some appear more urgent.
no subject
Date: 2009-09-22 08:13 pm (UTC)Ben
Liverpool
no subject
Date: 2009-09-23 05:00 am (UTC)For all that the politicians will focus on "fat cat council bosses' pensions," the *average* local government pension is surprisingly low - I've seen figures quoted of under £3k a year. And the net cost to the taxpayer is even less, in that if the employer's pension was removed, much of this £3k would have to be made up from the basic state pension instead.
no subject
Date: 2009-09-23 12:46 pm (UTC)no subject
Date: 2009-09-23 03:44 pm (UTC)The economics of death is an interesting subject. I've got a mate down the pub whose answer to every little problem in life is "euthanase the bastards." Well, actually, he pronounces it "euthanise," and he doesn't really know what "euthanase" means, but it's a distinctly disturbing throwback to the 1890s, let alone the 1930s. I can just about cope with this when he's talking about child murderers ("You just have to disagree with me, don't you?" Well, yes), but I've got to admit that I was flabberghasted when he suggested euthanisia as a suitable penalty for winning a gold medal at the womens' 800m whilst in possession of internal testes. It wasn't a joke.
Nor, by the way, is Houellebecq, although his ouevres so far are (a) distinctly French, in the manner of Celine, and (b) clearly designed, for publicity purposes, pour epater les bourgeouis.
But, to the economics of death. Birks is beating his familiar drum concerning Lloyd George's ponzi scheme here, and I could go on about that, because it's interesting.
What is perhaps more interesting is the ponzi scheme which is the NHS. Now, don't get me wrong, I'm 100% behind the NHS. However, if you look at the non-linear growth in expense for drugs, equipment, procedures, and care homes, not to mention the imminent explosion in the Alzheimer's caseload (etc), you do have to wonder whether my mate down the pub is in fact right.
Oh well, he's over sixty five. First thing we do is, we kill all the old people...
And to be fair, in a democratic society, it shouldn't matter whether you have a pension or not. That would be elitism. First we kill you, then we take your savings.
no subject
Date: 2009-09-23 06:00 pm (UTC)I'm not sure they have a name as a class, but this may just be as a result of my ignorance on the matter. They could be Birksian numbers if you could find a use for the property!
no subject
Date: 2009-09-23 06:08 pm (UTC)I have a fear that the answer is "anything you like, he can't hear you", but I hope there's a better answer along the lines of the "Doug" / "Douglas" / "Warren" gags, for which I have a soft spot.
no subject
Date: 2009-09-24 04:16 pm (UTC)Just goes to show that off-the-cuff comments on even simple maths questions are ill-advised. It's not a very interesting set, though, is it? I don't know what we'd call it, but it seems to be "the set of all irrational numbers that are not transcendental..."
This may be what Birks was getting at, but I doubt it.
Sets
Date: 2009-09-24 07:16 pm (UTC)Now, since we know that there are far more irrational numbers than rational numbers, my question is: presumably there are as many positive irrational numbers which become rational numbers when squared as there are rational numbers that do not take a rational square root.
That implies that there are fewer irrational numbers that become rational numbers when squared than there are rational numbers
Now, it seems to me that you can continue this mapping (irrational numbers that become rational numbers after you raise them to a certain power) and it seems to me that you still don't get enough irrational numbers. Not by a long chalk.
That in turn means that there must be a much larger set of irrational numbers that never become rational numbers no matter how often you square them (i.e., the subset that is transcendental numbers) than there are nontranscendental irrational numbers.
So, do mathematicians know (i.e., have they proved) that transcedental numbers > non-transcendental irrational numbers > rational numbers?
Just idle ponderings, of course.
PJ
no subject
Date: 2009-09-24 07:19 pm (UTC)PJ
Re: Sets
Date: 2009-09-25 12:11 am (UTC)Proper mathematical notation would help at this point; but fortunately we both have degrees in the Humanities. (Oh, the Humanities!)
As to paragraph 3, I don't believe this is true, but then I get uncomfortable when reasoning about infinities. Obviously there are some rational numbers whose square root is rational. Placing these in a disjoint set (I think that's right), there are therefore gaps, if you like, in the set of rational numbers which are candidates for the property of having an irrational square root.
Your mapping in paragraph 4 is therefore the crux of the matter. I may be wrong, and the good lord Cantor preserve me from error, but this mapping is based on a false assumption: ie that there is a finite limit involved. There isn't. The cardinality of both the set of rational numbers and the set of irrational numbers whose square is a rational number is, I believe, aleph null. In other words, you can always (countably) find another one.
It's quite possible that I'm talking bollocks yet again here, which is why I'm uncomfortable without actual mathematical proofs. However, the cardinality of the set of transcendental numbers is most definitely not aleph null. (Researching mathematical proof of same is an exercise left to the internet-bereft reader.) As drivelled about above, the cardinality of your other two sets is aleph null. Or, to put it in the terms you suggest:
transcendental numbers > non-transcendental irrational numbers == rational numbers.
(For computer geeks, the operators above are left-associative.)
I wonder if Woodhouse has got back from chatting up his local quant yet?
Re: Sets
Date: 2009-09-25 12:16 am (UTC)transcendental numbers == non-transcendental irrational numbers > rational numbers.
I really must do that Open University course some time.
no subject
Date: 2009-09-27 09:22 pm (UTC)Wanna tackle the margins on Fermat's Last Theorem next?
no subject
Date: 2009-09-27 09:59 pm (UTC)PJ