Why is the diagonal of a square not "2"

[Aidoneus]

And yet, it's a widely documented fact that on average we're collectively getting smarter. This is across all countries, developed and developing nations. See http://en.wikipedia.org/wiki/Flynn_effect. So your conspiracy theory of a deliberate dumbing down of the population doesn't square the evidence.

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed. But beyond that, it is not really relevant because I was not referring to the average level of the entire populace, but to the average level of those attending university. This also is consistent with my claim of dumbing down: people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them - it merely means that they have been trained to pass standardized tests instead of think for themselves.

I can't type fast enough to stay in this conversation.

But I am in absolute agreement. Schools now focus so much on metrics that students seem to learn more test-taking skills than critical-thinking skills.

[oca]

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

[RBerenguel]

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

When starting in maths you eventually get some definition of the reals where this is one of the most stricking things... And then you just kind of "get it." I.e. for me the reals were introduced as completion by Cauchy sequences (well, I got them introduced as Dedekind cuts earlier, but on my degree they were as Cauchy sequences.) For the non-maths, a sequence {x_n} is an infinite, ordered set of numbers (or things, doesn't matter) like 1,2,3,4... A Cauchy sequence is one such that if we pick some arbitrary elements of it, say, x_n and x_m such that n and m are very large (say, n and m larger than N) then x_n and x_m are very close. When formalised, this has epsilons and conditions, but essentially "once you look very far, elements are incredibly close."

This would seem to imply Cauchy sequences have a limit (i.e. elements are closer and closer and closer... isn't there an "ending" element, then? Such an element X should satisfy that as n grows larger, X-x_n is smaller and smaller) and they do in complete spaces (i.e. a metric space where this happens is called *complete*.) But turns out that Cauchy sequences of rational numbers don't have as limit a rational number (:o! this is tricky to prove, but there are many counterexamples.) Since to get the reals you fill up the rationals by completing these sequences so all have "ending points," 0.9999... for me is the limit of {0.9, 0.99, 0.999, etc } which has clearly as limit 1 (since 1-any of these elements is smaller and smaller)

[oca]

I also like to compare N* (positive integers,without 0) and Z* (integers, without 0)

So :

for each number n in N*, Z* contains that n and -n, so... Z* should be bigger than N* ...
but their will always be a number in N* to count each number of Z*

That's so weird when put into words...

[leichtloeslich]

I also like to compare N* (positive integers,without 0) and Z* (integers, without 0)

I think you're confusing notation here. If R is some ring, then R* usually denotes the multiplicative group of that ring.

For the integers Z that would be Z* = {1, -1}.

(You probably got confused by the fact that if F is a field, then F* is indeed just F without 0 (which is kind of the definition of a field, really))

[lemmata]

We are now getting many U.S. students into extreme debt for non-STEM degrees (science, technology, engineering, math)--degrees that have little marketable value, but which have done wonders for the bottom line of public and private institutions through government guaranteed loans and grants.

Government college loans: Putting more young people into debt with unrealistic and unprofitable career fantasies. Sad. Furthermore, when those people can't pay up, unborn babies are on the hook for the bill, too.

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed.

Indeed! Some IQ test questions actually require you to take invalid leaps of logic.

[lemmata]

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

I believe that this sort of proof is often taught it schools. I learned to hate it as teaching tool. The proof focuses on symbolic manipulation and often leaves students with the impression that there is some mystical math voodoo.

The kind of "meaning" that RBerenguel demonstrates in his post is what we should aim to transfer to students when we teach that 0.999...=1 to high schoolers, but the current standards are terribly inadequate for that purpose.

[John Fairbairn]

I think the biggest maths mystery in the world has not been mentioned here, and furthermore whatever explanation it has seems to fly in the face of the usual snobbery about IQs, Mensa and top universities.

It is this. A significant proportion of the populace who would normally not be regarded highly if talk was restricted to IQ and degrees, and who would look blank if you asked them whether they were interested in maths, have the ability to stand in front of a dart board and INSTANTLY, as soon as a dart lands, calculate all scores backwards, as well as calculating what further arcane combinations of scores (also backwards) are required to finish. They can further do this when addled by alcohol in a noisy pub while carrying on normal and robust conversations. Furthermore, they can do this while keeping in mind whose round it is and who is drinking what, and checking accurately that the barman has not diddled them with their change.

I have never been able to work out which I admire more: that or Ginger Rogers' ability to dance everything Fred Astaire (supposedly Hollywood's best dancer) did, but also backwards and in high heels.

[RBerenguel]

John, this kind of ability is essentially lots and lots (loads and loads ?) of practice. If every night you play darts at the pub, eventually you'll "know" all the numbers, just out of practice and a lot of repetition. But of course, in many fields lots of this kind of practice can be misunderstood for "intelligence," and the lack of this practice by a lack of it.

Personally, I'm astonished when I see a basketball (Petrovic) or soccer player (Zidane) that seems to "navigate" among the pitch like time works differently for them.

[oca]

Here's another one of modern math: 0.999... = 1.

I like that one too, very hard for me to accept that 0.999... = 1, 0.999... should be smaller, but ...

0.9999... = x | multiply by 10
9.9999... = 10x | substract x
9 = 9x

or

1 / 3 = 0.333...
0.333 * 3 = 1 (or 0.9999999... so 0.99999... = 1)

I believe that this sort of proof is often taught it schools. I learned to hate it as teaching tool. The proof focuses on symbolic manipulation and often leaves students with the impression that there is some mystical math voodoo.

The kind of "meaning" that RBerenguel demonstrates in his post is what we should aim to transfer to students when we teach that 0.999...=1 to high schoolers, but the current standards are terribly inadequate for that purpose.

I agree, I would say that helped me to accept, not to understand...

[quantumf]

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed.

I realize you didn't necessarily focus on this in your reply, but I'd still like to comment on this. As far as I know, there is no other well-known, widely-agreed measurement of "smarter", apart from IQ scores. As you imply, IQ scores are pretty well correlated with success, at least in terms of what we usually measure success by, such as educational level, income, status and longer life, at least compared to any other single metric.

people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them

I'm not really sure what you mean by that. Presumably one analyzes the world around you in order to best decide how to navigate a successful path through it. If IQ is correlated with success, then I would suggest that it's also correlated with an ability to analyze the world around you. All other things being equal, the smarter (by IQ) person is more likely to make the better choice in most dilemmas.

[DrStraw]

That link does not say we are getting smarter. It say IQ scores are increasing. IQ scores measure a specific subset of those skills required to succeed.

I realize you didn't necessarily focus on this in your reply, but I'd still like to comment on this. As far as I know, there is no other well-known, widely-agreed measurement of "smarter", apart from IQ scores. As you imply, IQ scores are pretty well correlated with success, at least in terms of what we usually measure success by, such as educational level, income, status and longer life, at least compared to any other single metric.

You got this completely backwards. That is not what I meant at all. What I meant was that of all the things which are necessary to succeed only a small part of that set is related to IQ. EQ is probably more important than IQ.

people's ability to perform certain tasks as tested by IQ tests does not imply those same people have the same ability to analyze the world around them

I'm not really sure what you mean by that. Presumably one analyzes the world around you in order to best decide how to navigate a successful path through it. If IQ is correlated with success, then I would suggest that it's also correlated with an ability to analyze the world around you. All other things being equal, the smarter (by IQ) person is more likely to make the better choice in most dilemmas.

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

Even though I am supposed to have a very high IQ as tested (+3 sd), I am not a big believer in it as a measure of much more than the ability to pass IQ tests. There are millions of people more successful than I who have much lower IQs. And, yes, I am quoting a sample of one there, but I obverse the same with others as I look around me. My problem is that I don't think my EQ is much, if any, above average.

[Bill Spight]

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

That question came up in one of my undergraduate courses. Studies indicated that IQ was positively correlated with income (IIRC, to take a common measure of success), but only up to an IQ of 125 or so. Among people with higher IQs there was no correlation between IQ and income.

[DrStraw]

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

That question came up in one of my undergraduate courses. Studies indicated that IQ was positively correlated with income (IIRC, to take a common measure of success), but only up to an IQ of 125 or so. Among people with higher IQs there was no correlation between IQ and income.

I thought we were talking about success, not income. The two are not at all the same. Success is, in my opinion, a measure of happiness. Many people with high incomes are not happy, and therefore not successful, because they spend too much time acquiring that income.

[Bill Spight]

But IQ is not correlated with success. Well, it is, but not in the sense you mean. I have no figures but I doubt the correlation is higher than 0.5, but based on observation. No arguments that more people with high IQs will be successful, but there are a lot of successful people with average IQs. As I said above EQ is very important, as is simple perseverence and street smarts.

That question came up in one of my undergraduate courses. Studies indicated that IQ was positively correlated with income (IIRC, to take a common measure of success), but only up to an IQ of 125 or so. Among people with higher IQs there was no correlation between IQ and income.

I thought we were talking about success, not income. The two are not at all the same. Success is, in my opinion, a measure of happiness. Many people with high incomes are not happy, and therefore not successful, because they spend too much time acquiring that income.

Oh, I quite agree that income does not equal success.

As for any correlation between IQ and happiness, my guess is that there is a small negative correlation. Ignorance is bliss.