I got bored, and I didn't see any mathematician threads since october. So in a fit of boredom, I made this one! Joy!

Got any questions about the way math works? Just need help on a homework problem? Just ask me, here! What could go wrong? :mischief:

Note: There's only one question I'll ban, and that is questions like this:

"What is 48/2(9+3)?"

Why? Because this question starts flame wars.

Does .(9)=1 and why or why not?

Does .(9)=1 and why or why not?

Ooh! I've actually got the answer to this in a post I made on a different forum.

Short answer, yes.

Long answer, yes, and here's why: Why .999... = 1, A Treatment With Infinite Series (http://www.physicsforums.com/showthread.php?p=3207507)

What is infinity?

What is your mathematical background? :)

Why is the Square root of 2 is considered a irrational number?

What is infinity?

Infinity is really rather strange. It's not a number, but in some cases we can treat it as one. (Note that this "some cases" does not extend to arithmetic!) The best way to think about infinity isn't as a number, but as a limit, in my opinion. Of course, every math person thinks different things about infinity!

Personally, I use the Projective Real Line, which is like the line of real numbers that's so familiar, but with an "endcap" put at each end of the line, such that the line forms a circle. This "real circle" has one end zero and the other end "infinity". You still can't really do arithmetic with it, though.

What is your mathematical background? :)

Registered math major in college. My latest completed courses are Vector Calculus and Linear Algebra, and I'm working on Differential Equations right now. After that, I'm off to Real Analysis!

Why is the Square root of 2 is considered a irrational number?

Because it was proven to fit the definition of an irrational number. The proof is simple, and so I can reproduce it here. First note, however, the definition of an irrational number: a number that cannot be expressed as a ratio of two coprime numbers a and b. So let's use this!

First, assume that sqrt(2) is rational. From this assumption we'll derive a contradiction, showing that the assumption is untrue.

Then, if sqrt(2) is rational, it can be expressed as a ratio of a and b, i.e. sqrt(2)=a/b. This can be rearranged to give b sqrt(2) = a. Square both sides and we get 2 b2 = a2 . What this tells me is that a2 is even. Since a2 is even, a must be even (this is easy to prove, try it!) and a = 2c for some number c. So we can rewrite this as 2 b2 = (2c)2 = 4 c2 . Divide left and right side by two to get b2 = 2 c2 . This tells us that b2 , and thus b is even. But wait! We said that a was even. a and b cannot BOTH be even, because two even numbers are not coprime. Thus, contradiction! Therefore, sqrt(2) is irrational.

A definition: For those of you who do not know what "coprime" means, i.e. the non-math crowd, two numbers are coprime if they do not share any factors. For example, 3 and 14 are coprime, because the only factor of 3 is 3, and the factors of 14 are 2 and 7. 2 and 6 are NOT coprime, as the factor of 2 is 2, and the factors of 6 are 3 and... 2

Hope that long-winded explanation helped!

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

I simply remind myself that what I'm doing is the purest of all possible professions.

How did you grasp mathematical thinking? Do you have any tips on how to do so?

whats your favorite polynomial? show us a pretty graph.

I simply remind myself that what I'm doing is the purest of all possible professions.

And what do you mean by that? Also, what are your thoughts on the following comic?
http://zs1.smbc-comics.com/comics/20091116.gif

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

Hey man, nothing is harder than signing a piece of paper or firing someone. Those CEOs work harder than the rest of us combined! :gripe:

Thank you, for the explanation. I am definitely Subscribing to this.

Registered math major in college. My latest completed courses are Vector Calculus and Linear Algebra, and I'm working on Differential Equations right now. After that, I'm off to Real Analysis!

It's a fun ride. I majored in math in college (along with economics).

How did you grasp mathematical thinking? Do you have any tips on how to do so?

I can second this question, even now. I "think" topologically and do fine with analysis, but for the life of my I could not do abstract algebra. I think it was a mental block or something.

My answer: it took me a long time to learn how to think like an analyst. I did lots of practice and wrote lots of proofs, many of them poorly. Eventually you build a repository of examples, counterexamples, and "standard proofs" (both strategies and results) and get into a rhythm so that things become intuitive. Again: intuition is relative and grows with you, and for me it took a lot of practice.

You learn tricks along the way. A good chunk of real analysis is judiciously adding 0 or multiplying by 1 in a clever way (at the undergrad level).

I survived topology by having a long list of examples and counterexamples in my head, which also came with practice.

I never learned to think like an algebraist. :(

How did you grasp mathematical thinking? Do you have any tips on how to do so?

Gah. I had a nice long paragraph here, but then my internet froze and I lost it all. Oh well, something for the Rants thread, I guess.

In short, my main strategy throughout most of my learning was threefold: I visualized what I was trying to find. (This often doesn't work!) I memorized handy little rules to help me through. And finally, I tried to go above and beyond whatever the professors were teaching, all the time. Talk about finding limits? I looked up the epsilon-delta method. Talk about the product rule? I proved it, which went a long way toward remembering it.

Basically, I guess my best piece of advice could be to try to go above and beyond whatever you're learning. Once you know the underlying theory, the formula suddenly becomes cake (or perhaps pi :mischief:) to remember!

whats your favorite polynomial? show us a pretty graph.

Hmm. I haven't gotten asked that question before! My favorite polynomial would probably be the simple x^2, always the first thing we differentiate, the first thing we integrate. But that's not a very pretty picture. So let's look at y=x4-5x2+5, instead. It has a cool W-shape!

http://i863.photobucket.com/albums/ab200/IdiotsOpposite/wolframalpha-20120130214051257.gif

As for non-polynomial things, my favorite is the Riemann Zeta function of 1/2 + i x. Why? Well, it's extremely important (particularly in finding the zeros), and it just plain gives a cool graph! For example, here's a graph of the Riemann Zeta function in polar coordinates:

http://upload.wikimedia.org/wikipedia/commons/thumb/7/7f/Zeta_polar.svg/300px-Zeta_polar.svg.png

And what do you mean by that? Also, what are your thoughts on the following comic?
http://zs1.smbc-comics.com/comics/20091116.gif

Well, I consider mathematics to be pure. You don't need to work with reality or experimentation at all with mathematics. And moreover, mathematics is the only scientific discipline where you can, with only a sheet of paper, prove that something is irrefutably true! That's what I mean by the purest possible profession.

As for my thoughts on the comic, it certainly does paint mathematicians in a better light, doesn't it? But while it's true that not many people will know what you do, I wouldn't say that none of the people who DO know are able to fire you. I should hope the head of the math department wherever you work knows what you're doing!

Thank you, for the explanation. I am definitely Subscribing to this.

Thanks for taking the time to read it! Man, this whole thread is an ego-boost.

It's a fun ride. I majored in math in college (along with economics).



I can second this question, even now. I "think" topologically and do fine with analysis, but for the life of my I could not do abstract algebra. I think it was a mental block or something.

My answer: it took me a long time to learn how to think like an analyst. I did lots of practice and wrote lots of proofs, many of them poorly. Eventually you build a repository of examples, counterexamples, and "standard proofs" (both strategies and results) and get into a rhythm so that things become intuitive. Again: intuition is relative and grows with you, and for me it took a lot of practice.

You learn tricks along the way. A good chunk of real analysis is judiciously adding 0 or multiplying by 1 in a clever way (at the undergrad level).

I survived topology by having a long list of examples and counterexamples in my head, which also came with practice.

I never learned to think like an algebraist. :(

Ah, yes, the infamous multiplying by 1. That's how I learned the integral of sec(x), and I've never forgotten that! (Although I wish there was a better method of doing it!)

I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

Oh fine. Math major then.

I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

Whence come this arrant pedantry?

Whence come this arrant pedantry?

Just kinda irritates me to see people pretending to be something that they're not (I was also kinda annoyed when you started a thread claiming you were a rocket scientist). I already know a lot of college students, and some of them are or were math majors, so I don't have much to ask about that. If there's any basic math questions that I need answered I can just look them up on wikipedia.

In general I think our society gives students too much false hope that getting a bachelor's degree in a subject will automatically make them a professional in that subject, when in fact there's a huuuuge gap that they'll still need to cross. For most subects, being a professional in that subject virtually requires them to become a college professor.

Just kinda irritates me to see people pretending to be something that they're not (I was also kinda annoyed when you started a thread claiming you were a rocket scientist). I already know a lot of college students, and some of them are or were math majors, so I don't have much to ask about that. If there's any basic math questions that I need answered I can just look them up on wikipedia.

Well, from Wikipedia:

A mathematician is a person whose primary area of study is the field of mathematics.

So IdiotsOpposite is quite literally a mathematician, at least at this juncture.

By that definition, a 6 year old who's focusing really hard on learning arithmetic is also a mathematician. In a philosophical sense that might be true, but that's not the usual meaning of the word.

By that definition, a 6 year old who's focusing really hard on learning arithmetic is also a mathematician. In a philosophical sense that might be true, but that's not the usual meaning of the word.

Let's not do that. It says "primary area of study," not "momentary focus."

Let's not do that. It says "primary area of study," not "momentary focus."

OK, I tell you what. Go to a party and tell everyone there that you're a Rocket Scientist or Mathematician or whatever. See if they believe you.

All of this is kind of only tangentially related to the main point of the thread, isn't it? No, I do not get paid to work mathematics (yet). Yes, I do still have quite a bit of mathematical skill, and can answer the questions most people come up with.

Don't forget my thread in sci/tech.

I do maths every day as well, being a games programmer. Was doing some matrixy stuff yesterday.

Wandering in as a math major as well, graduating in may so I have (Calculus, Linear Algebra, Real Analysis, Abstract Algebra, Number Theory, Probability & Statistics, etc.) I also work as a math tutor and have applied to grad schools to pursue a PhD in the fall. I still wouldn't quite claim to be a "mathematician" but I'm not far off either.

.9...=1, the simplest proof is algebraic.
consider 1/3=0.3.... This is known to be true and easy to show. Now multiply by three on both sides. 3 * (1/3) = 3*0.3...
so 1=3*0.3... => 1=0.9... Q.E.D.

As for mathematical thinking it can be hard. I suck at 3D geometry, Calc 3 was awful for me I got an A but I didn't understand much of what I was doing. I just ignored what it meant and followed the algebraic representations. I find Calculus/Analysis irritating, but fairly easy, but I love Algebra and Number Theory and that type of thinking comes really easily to me. From my tutoring experience I feel like often times people just refuse to learn or understand, and they could do it if they just accepted and legitimately tried. Instead of freaking and being frightened and emotional about it. That goes even for other math majors who I see falling apart in theory classes, it's more of a mental block than anything else.

What is the application of the field of mathematics you plan/think you'll study?

Well, from Wikipedia:


Studying something as an expert and studying something as a student are two different things.

Four points are chosen at random on the surface of a sphere. What is the probability that the tetrahedron formed from those four points will contain the center of the sphere?

What is the application of the field of mathematics you plan/think you'll study?

I'm planning to aim toward the study of partial differential equations as a focus. As I've seen lately, there's still a lot of work that can be done in that area! Failing that, a friend of mine is trying to urge me toward category theory. So who knows? Maybe that will strike my fancy.

Four points are chosen at random on the surface of a sphere. What is the probability that the tetrahedron formed from those four points will contain the center of the sphere?

Well, first, I should urge you to remember that I haven't taken an official probability course yet. But after a bit of research, I do believe the answer to that would be .125, or a 1 in 8 chance.

Well, first, I should urge you to remember that I haven't taken an official probability course yet. But after a bit of research, I do believe the answer to that would be .125, or a 1 in 8 chance.

Yes, that is the correct answer. :goodjob: Due to symmetry, the proof actually doesn't require probability theory (beyond 1/n) and can be reasoned out.

Woot!

Now to celebrate. Root beer float!

2 + 2 = 4.

Why?

..oh and I agree with the arrant pedantry too! An individual studying medicine should not make an "Ask a Doctor Thread"!

2 + 2 = 4.

Why?

..oh and I agree with the arrant pedantry too! An individual studying medicine should not make an "Ask a Doctor Thread"!

Well, 4 is defined as 3+1, 3 is defined as 2+1, and 2 is defined as 1+1, and addition is associate sooo....
2+2=2+(1+1)=(2+1)+1=3+1=4. Obviously this uses an inductive definition of the numbers, but it makes sense its difficult to put together an elementary definition without that, or at least that's all I've seen.

Well, 4 is defined as 3+1, 3 is defined as 2+1, and 2 is defined as 1+1, and addition is associate sooo....
2+2=2+(1+1)=(2+1)+1=3+1=4. Obviously this uses an inductive definition of the numbers, but it makes sense its difficult to put together an elementary definition without that, or at least that's all I've seen.

I was just about to talk about the successor function! Ninja'd, I guess.

I'm planning to aim toward the study of partial differential equations as a focus. As I've seen lately, there's still a lot of work that can be done in that area! Failing that, a friend of mine is trying to urge me toward category theory. So who knows? Maybe that will strike my fancy.


Differential equations have many diverse applications. What about category theory? From skimming the wiki article I think it's for computer science and programming problems.

Do you see mathematics as a science or an art form? Or is it neither but somethintg uniqe?

Currently studying Mathematics at my local university, 3rd year.


I feel like you really shouldn't claim the title of Mathematician (or anything else) until you've at least finished the degree, and really not until you've gotten hired in a permanent job doing math. Which realistically, only like 1% of math majors ever will.

I understand that your last sentence was probaly a hyperbole, but either way as it may intrest people here to know what mathematicians actually do after their studies: of the people who finish their math dergee here (that is, after at least 5 years studying, master diploma) around:
- 25% doctorate (research)
- 25% will go teach math in the last years of high school (so 16-18 year old pupils)
- 25% end up in the finances and business sector (game theory, actuary, etc)
- 25% end up in other kind of firmes, this is the broadest category and includes stuff like algoritms, physics, statistics, cryptografy, etc.
(according to my university)

Of course, this is just the situation here. It's by no means universal. Anyway, it's a misconception that it's hard to find a job doing maths (<1% unemployment 1 year after studies as well).

edit: since I'm talking about job prospects, this is somewhat relevant as well: http://www.careercast.com/jobs-rated/10-best-jobs-2011

I've never heard anyone with a math degree have trouble finding jobs, hell I know physicists who work for banks and make mad money.

How are you at combinatorics?

I took a lot of math in university and hated a lot of it (statistics, calculus, etc.) but loved stuff like algebra (I really loved proving things).. Anyway, for some reason I excelled in combinatorics.. I kinda miss it! Do you dabble in that at all?

Studying something as an expert and studying something as a student are two different things.

I wasn't aware that definition contained a distinction. :huh:

It's not like Mathematician is a concrete title. As long as he is honest about his status and serious in his study of it, let him call himself what he wants.

Eh, no. Let me see the paper that gives someone the right to call themselves that, otherwise no. Mathematician in training, sure, I'll go for that.

"honest about his status and serious in his study of it" Would you let a 1st year med student call themselves a physician? Let someone studying to become a pilot call themselves a pilot before they get their license? No, of course not.

Eh, no. Let me see the paper that gives someone the right to call themselves that, otherwise no. Mathematician in training, sure, I'll go for that.

"honest about his status and serious in his study of it" Would you let a 1st year med student call themselves a physician? Let someone studying to become a pilot call themselves a pilot before they get their license? No, of course not.

Okay, but "mathematician" isn't a concrete, legal term like "physician" or "trained pilot" or even "lawyer." It's a little abstract and with definitions like "person whose primary field of study is mathematics" the room for interpretation is greater than that with physicians and pilots.

fake edit: especially when AFAIK nobody has ever received a piece of paper that says something to the effect of "Certified Mathematician."

I think we're getting a little far afield here. Can we return to the original discussion, that is, math people answering questions? (Math person is a word defined by me to mean "someone with interest and a learning in math far above the average person")

Do you see mathematics as a science or an art form? Or is it neither but somethintg uniqe?

I can't resist giving the Mathematician's Answer to the first question here: yes. Yes I do.

To be exact, I do not consider mathematics a science. Why? Because to me, science depends on one thing: experimentation. But you don't experiment in mathematics, you prove. You prove that if X is true, then Y is true for some conditions X and Y. In effect, math is logic. And considering how beautiful they both can be sometimes (math and logic), I'd definitely put math on the level of an art.

Differential equations have many diverse applications. What about category theory? From skimming the wiki article I think it's for computer science and programming problems.

Yes, it's definitely useful in those, but my friend (a Post-graduate from Belgium; the internet gives you a large network!) says that it's also used to form a basis for the rest of mathematics, though it's entirely possible he's bragging.

How are you at combinatorics?

I took a lot of math in university and hated a lot of it (statistics, calculus, etc.) but loved stuff like algebra (I really loved proving things).. Anyway, for some reason I excelled in combinatorics.. I kinda miss it! Do you dabble in that at all?

Combinatorics is fun! I like to experiment with things like that in my spare time, and although I haven't gone into combinatorics much, the times I have gotten into it were fun and enlightening. Kind of like my dabbling in probability!

EDIT: Gah. Unclosed parenthesis. That'll just tear me apart if I didn't edit it.

fake edit: especially when AFAIK nobody has ever received a piece of paper that says something to the effect of "Certified Mathematician."
I was thinking of something like this (random pick from image search)

http://www-personal.umich.edu/%7Ehnarayan/images/pages/degree-math.jpg

Are you heartset on entering academia, or have you considered using your degree for other employment opportunities?

Are you heartset on entering academia, or have you considered using your degree for other employment opportunities?

Ya know, I'm not sure yet. I'm about 85% sure I want to go into academia, but I might want to pay off my debts from a four-year degree first. I try not to set the future too much in stone; that's when karma comes and breaks up all your plans!

Ya know, I'm not sure yet. I'm about 85% sure I want to go into academia, but I might want to pay off my debts from a four-year degree first. I try not to set the future too much in stone; that's when karma comes and breaks up all your plans!

Good luck. I hope you realize how insanely difficult it is to get a job as a professor these days. Please consider statistics like these (https://career.berkeley.edu/Major/Math.stm) and make sure you have a backup plan, like learning computer programming or taking the tests to be an actuary. Note that one of those Berkeley math grads is "employed" as a Starbucks barista.

For what it's worth I minored in math so I guess that qualifies me to answer questions in this thread.

Good luck. I hope you realize how insanely difficult it is to get a job as a professor these days. Please consider statistics like these (https://career.berkeley.edu/Major/Math.stm) and make sure you have a backup plan, like learning computer programming or taking the tests to be an actuary. Note that one of those Berkeley math grads is "employed" as a Starbucks barista.

This is an excellent suggestion. There is high demand for actuaries (largely because it isn't easy) and it has good job security and pay.

Even if you decide that you really want to pursue a career in academia, it's something that is worth putting in your back pocket. If you can make it through the hell of several years of testing, you're as close to "set for life" as a mathematician can get.

I'll have to keep that in mind. Thanks for letting me know!

re: academia:

I'm a grad student in economics. The prospects for academic employment go down pretty quickly once you leave the top-20, and even in the upper tier academic jobs are difficult to get. I can't imagine that mathematics is any easier.

To be clear: in my field, about half of students drop out with the Master's and of the remainder, half to two-thirds eventually get academic positions. So there is real risk involved, however I think it's worthwhile (otherwise I wouldn't be doing it!).

Math and econ both have good outside options (actuarial, government, statistics, corporate finance). Just keep them in mind.

:)

which do you prefer? Applied or pure mathematics.

which do you prefer? Applied or pure mathematics.

I prefer pure mathematics. But I recognize that applied mathematics is a LOT more likely to get me a job.

Interesting tidbit: Pokemon was actually what originally got me into mathematics. Besides the number-crunching of calculating stats, there's also the amount of interesting things that can be found by looking into the programming of the game. (Especially since it looks like it's only half-finished!)

How are you at combinatorics?

I took a lot of math in university and hated a lot of it (statistics, calculus, etc.) but loved stuff like algebra (I really loved proving things).. Anyway, for some reason I excelled in combinatorics.. I kinda miss it! Do you dabble in that at all?
I've done a decent amount with it, but I've never really enjoyed combinatorics.

Are you heartset on entering academia, or have you considered using your degree for other employment opportunities?
I have no intention of entering academia, I want to work for either the NSA or a bank still need a PhD or at the very least a Masters for that though. o academia is a backup/retirement plan I suppose.

Is there any particular proof that you admire more than the others?

Combinatorics was one of the rare math classes at my uni whose problem sets made seasoned math majors cry.

Is there any particular proof that you admire more than the others?

Cliche Answer (http://en.wikipedia.org/wiki/Fermat's_Last_Theorem)

I really like the proof of their being infinitely many prime numbers, mostly because it is exceedingly easy so I love using it as an example of proofs to non math people. I also like the proof of Euler's Theorem/Fermat's Little Theorem done using Group Theory.

Is there any particular proof that you admire more than the others?

Yes, and it's not the one you're expecting!

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

Why this proof, you ask? (Well, probably not, but I'll imagine you asking)

Well, simple. It establishes the density of primes, it's a proof that a great many people including my own personal favorite mathematician, Georg Riemann, and I just read a book explaining every step of the history of the proof!

A couple of questions, anyone may answer (I don't care what people call themselves, it's their accomplishments that matter more):

First, what's your Erdos number?

Second, I'd like to go back to Reimann's zeta function and the zeros. I've read a little about this (Marcus Du Sautoy's Music of the Primes (http://en.wikipedia.org/wiki/The_Music_of_the_Primes)) and I really don't understand it.

The way it was described in the book (assuming I remember it correctly!!), the zeros are areas of a complex landscape that are i distant from the y axis and never cross the line. The problem everyone was working on was finding a prove that no zero was ever off the line through i. Reimann's housekeeper burned many of his personal documents after he died, so any proof that he had figured out must be rediscovered.

Can you explain to me how that graphs of the zeta in polar coordinates relates to my description above? Again, I'm probably mangling parts of this - so I apologize if the question is ill-formed or a frustrating waste of mathematical times ;)

A couple of questions, anyone may answer (I don't care what people call themselves, it's their accomplishments that matter more):

First, what's your Erdos number?

Second, I'd like to go back to Reimann's zeta function and the zeros. I've read a little about this (Marcus Du Sautoy's Music of the Primes (http://en.wikipedia.org/wiki/The_Music_of_the_Primes)) and I really don't understand it.

The way it was described in the book (assuming I remember it correctly!!), the zeros are areas of a complex landscape that are i distant from the y axis and never cross the line. The problem everyone was working on was finding a prove that no zero was ever off the line through i. Reimann's housekeeper burned many of his personal documents after he died, so any proof that he had figured out must be rediscovered.

Can you explain to me how that graphs of the zeta in polar coordinates relates to my description above? Again, I'm probably mangling parts of this - so I apologize if the question is ill-formed or a frustrating waste of mathematical times ;)

:D The Zeta function! My favorite function of all, and the focus of the book I was reading. First, it's important to note that there are two categories of zeroes for the Riemann Zeta function. The first are called trivial zeros, and occur at every negative even integer. -2, -4, -6, etc. The others are called non-trivial zeroes, and so far as we know, they all seem to be on the critical line Re(s) = 1/2. Now, the Riemann Zeta function exhibits some symmetry around this critical line thanks to the functional equation. If there's some zero between 1/2 < Re(s) < 1 and Im(s), then there's going to be an equivalent zero, with Im(1-s)! Couple this with the fact that the Riemann Zeta function's zeroes are also symmetric over the real line, this means that any zero that does not lie on Re(s)=1/2 will imply the existence of 3 other zeros that also do not lie on Re(s)=1/2. But that's not important now, is it? I sort of completely lost track of what question you were asking.

All right, so the function in polar coordinates I put up is a polar representation of Zeta(1/2 + i t), which would normally be represented as a complex number x+iy, is shown using the polar representation of complex numbers instead. Using this representation, every time the function that's moving around in its series of circles crosses the point (0,0), that's a zero of the Zeta function. I can't remember the exact wording, but I believe that there's a proof that if the Zeta function makes a complete circle WITHOUT crossing the point (0,0), then the Riemann Hypothesis will be disproven. I'll have to look that up when I get home.

I wasn't aware that definition contained a distinction. :huh:

It's not like Mathematician is a concrete title. As long as he is honest about his status and serious in his study of it, let him call himself what he wants.

It's a bit pretentious, which is why I am bothered by it. It's like calling myself a "researcher" or "scientist" just because I've worked for a month in a cell biology laboratory. I do not feel that I have earned either designation yet and as such I will refrain from doing so. I don't think I'll be calling the business students who have internships at major downtown financial institutions "bankers" too. I could go on for a lot of other examples too.

How many of the old classical "unsolved problems" have been solved in the past hundred years? I'm thinking mostly of things like the Goldbach Conjecture, Fermat's Last Theorem, etc.

Which solved one do you feel is the most signifigant? Which unsolved one?

@BlueEmu - are you thinking of the Millennium Problems proferred by Hilbert?

Combinatorics is fun! I like to experiment with things like that in my spare time, and although I haven't gone into combinatorics much, the times I have gotten into it were fun and enlightening. Kind of like my dabbling in probability!

It was really fun, especially graph theory. I wish they introduced me to combinatorics back in high school, I would have probably taken more courses in the field had I known about it earlier.

I was thinking of something like this (random pick from image search)

Heh, you don't need a degree to be a mathematician.

I've done a decent amount with it, but I've never really enjoyed combinatorics.

Blasphemy!

Combinatorics was one of the rare math classes at my uni whose problem sets made seasoned math majors cry.

:lol: At Waterloo what made people cry were first year "weeding out" courses like linear algebra. My brain really liked combinatorics for some reason, it just *made sense*.

@BlueEmu - are you thinking of the Millennium Problems proferred by Hilbert?

I wasn't thinking of those seven problems in particular... I had in mind the older, "classical" problems in number theory such as Goldbach's conjecture.

I always loved this one:

If a hotel with infinite rooms, each room already occupied, gets another guest ... can he get a room? :)

:lol: At Waterloo what made people cry were first year "weeding out" courses like linear algebra. My brain really liked combinatorics for some reason, it just *made sense*.

"Combinatorics? It's counting! How hard could counting be?"

*dies*

I always loved this one:

If a hotel with infinite rooms, each room already occupied, gets another guest ... can he get a room? :)

Yup. Hilbert ran an infinite hotel, and even if an infinite number of new guests showed up when he was already fully booked, they all got rooms.

Yup. Hilbert ran an infinite hotel, and even if an infinite number of new guests showed up when he was already fully booked, they all got rooms.

And the follow up question is, of course: How can he do that while keeping his guest book straight (which is infinitely large, of course).

And the follow up question is, of course: How can he do that while keeping his guest book straight (which is infinitely large, of course).

The infinite monkeys in the back room do it for him.

--

On a more substantive note, I discovered a theorem today that makes me happy.

Let f be defined on an interval I. If f is monotonic, then it is differentiable almost everywhere on I.

--

A few of my favorite theorems:

1. The Kuhn-Tucker generalization of Lagrange's theorem, for obvious reasons.

2. Berge's maximum theorem, which has a host of useful results for maximization problems.

3. Kakutani's fixed point theorem, a set of sufficient conditions for the existence of a fixed point for set-valued functions.

1. The Kuhn-Tucker generalization of Lagrange's theorem, for obvious reasons.

Would a non-mathematician be interested in it? I'm hoping I don't come across as snarky, but it's not at all obvious to someone who doesn't know any of the people mentioned, nor the theorem referenced...

Thomas Kuhn?

Lagrange?? as in the Lagrange points of an orbit (which I've heard of, but don't know anything about - perhaps relate to geosynchronous orbit or specific point ahead of and behind an orbiting object sharing that same orbit)

I wasn't thinking of those seven problems in particular... I had in mind the older, "classical" problems in number theory such as Goldbach's conjecture.

Well, there are plenty of theorems that have been proven since before 1850 (which I'm just setting as a baseline here), and plenty that haven't been yet. For example, Poincare's Conjecture in 3-dimensions has been proven, and that's somewhat important of course. But Poincare's Conjecture in 4 dimensions hasn't been proven yet.

Would a non-mathematician be interested in it? I'm hoping I don't come across as snarky, but it's not at all obvious to someone who doesn't know any of the people mentioned, nor the theorem referenced...

Thomas Kuhn?

Lagrange?? as in the Lagrange points of an orbit (which I've heard of, but don't know anything about - perhaps relate to geosynchronous orbit or specific point ahead of and behind an orbiting object sharing that same orbit)

I'm being a bit glib, sure. :)

And it is the same Lagrange who found the Lagrange points in astronomy. Busy guy!

Lagrange's theorem provides a set of necessary conditions for solving constrained optimization problems. Problems like, "maximize the function F(x,y) subject to the constraint that you have to be on the unit circle, x^2 + y^2 = 1."

Of course it works for any general function, and for any (finite) number of equality constraints. Not only does he give you necessary conditions, he gives you a method for finding the optima you so desperately seek!

The method is this.

1. Use the function, and the constraint, to write down a "Lagrange function":

L(x,y,T) = F(x,y) + T*(x^2 + y^2 - 1)

where T is a brand new variable called the multipier. Note that this L is an unconstrained problem.

Maximize L. So, take derivatives of L with respect to your original choice variables (x,y) and with respect to the multiplier (T).

You now have a set of 3 equations in 3 unknowns, and can solve them. This gives you the maxima and minima for L.

Here's the magic: the points you have found are also candidate optima for the original, constrained problem.

The induced variable T has some neat economic interpretations when applied to economic problems.

The Kuhn-Tucker generalization broadens Lagrange's theorem by allowing for inequality constraints. "Maximize F(x,y) subject to x >=0, y>=0, and px + qy <= 10". We encounter these types of problems all the time in economics; indeed the one I just stated is the classical optimization problem for a person, with a budget of 10, trying to consume x and y at prices p and q.

Lagrange/Kuhn-Tucker is the cornerstone of mathematical economics. You literally cannot get anywhere without it!

Interestingly, Lagrange first proved his theorem back in the 1700s, but the Kuhn-Tucker generalization was not proven until 1951. It's a surprisingly "recent" result.

Would a non-mathematician be interested in it? I'm hoping I don't come across as snarky, but it's not at all obvious to someone who doesn't know any of the people mentioned, nor the theorem referenced...

Thomas Kuhn?

Lagrange?? as in the Lagrange points of an orbit (which I've heard of, but don't know anything about - perhaps relate to geosynchronous orbit or specific point ahead of and behind an orbiting object sharing that same orbit)

The lagrange points of an orbit are somewhat connected to Lagrange's Theorem, in that Lagrange's theorem is used to find minima and maxima of multivariate functions subject to a constrant, and the Lagrange points of an orbit, I believe, are points where the gravity in a system at an instant is zero. (i.e. an object at that point is experiencing null gravity)

Something that you often hear mathematicians talk about: beauty. Is all of mathematics beautiful or is it only some concepts, theories and proofs? What makes them beautiful? And is it a subjective assessment that some mathematicians can disagree with, or is it objective and undeniable by all?

First, what's your Erdos number?


11 or smaller.

Does that mean I can answer questions, although I certainly do not claim to be a mathematician?

What's your favorite number?

Do you own a Rubix Cube?

What's your favorite computational aid?

How many digits of pi have you memorized?

Who is your favorite mathematician?

What do you think of statistics?

What do you think of non-standard analysis?

What do you use for writing equations with on a PC?

What's your favorite kind of triangle?

Do you own a slide rule?

Which is best, M&Ms, Reese's Pieces, or Skittles?

What's your favorite even prime number?

Please, even I know that one, too.

;)

Something that you often hear mathematicians talk about: beauty. Is all of mathematics beautiful or is it only some concepts, theories and proofs? What makes them beautiful? And is it a subjective assessment that some mathematicians can disagree with, or is it objective and undeniable by all?

Mathematical beauty is connected to timelessness, certainty, abstraction, generality, and clarity.

Timelessness is always present, mathematical concepts are defined without relation to time so the theorems that describe them will remain true forever. Certainty and rigorous methods have been associated with mathematics since ancient times, it is by far the most rigorous of all the sciences. In the latter 19th century mathematicians reached a new level of rigorous methods based on set theory and they pushed for axiomatic perfection, but by the 1930s Godel's incompleteness result showed that such a perfect axiomatic system was impossible. Since then the focus has shifted away from logical foundations, towards more practical levels of certainty.

Abstraction and generality are closely related: a good abstract concept will be very general, applying to many different particular situations. Abstraction and generality are what give power to mathematical methods. Mathematicians face an infinite wilderness of infinitely strong beasts, and the only way forward is by finesse - leveraging our finite human intelligence - not brute force.

The opposite of abstraction and generality is an ad hoc solution, a term which means "for the problem at hand", whereas concepts, methods, and results are more beautiful if they solve a large number of disparate problems at once. For example, 19th century mathematician C.F. Gauss said he was not interested in Fermat's Last Theorem, as one could easily write down many similar equations and statements that were equally hard to prove, and that he would only be interested in the problem if there were a general theory encompassing all of these (which is what happened with Wile's proof over a century later based on the algebraic geometry of elliptic curves).


Not all mathematics would be regarded as beautiful. Newer proofs tend to be raw, undigested and unpolished, which makes them less beautiful by the five attributes above. Some subjects are inherently unwieldy and are left for future generations to hopefully tame. A lot of mathematical research is generated to fill grant quotas or supply students with problems, these unimportant results may never be polished to be particularly beautiful.

In judging whether a work is beautiful, it's a matter of subjective taste. There are many sub-areas of mathematics and everyone has preferences. Still, just like with movies, books, wines, etc it is possible with practice to evaluate the excellence of a mathematical work based on objective features, unclouded by personal preference. This happens all the time because research is peer reviewed.

Something that you often hear mathematicians talk about: beauty. Is all of mathematics beautiful or is it only some concepts, theories and proofs? What makes them beautiful? And is it a subjective assessment that some mathematicians can disagree with, or is it objective and undeniable by all?

Meromorph answered it better than I ever could. I'll just direct you to his answer above.

What's your favorite number?

For some reason, 37. It's been my lucky number for years, even though I can't even remember WHY I like it so much.

Do you own a Rubix Cube?

Yup! In 3, 4, and 5 dimensions.

What's your favorite computational aid?

My laptop. :)

How many digits of pi have you memorized?

Uhh... 5? That's really all I need for any serious hand calculation. Any more than that and you're just getting picky.

Who is your favorite mathematician?

Tossup. Riemann because he formulated much of the starter analysis about my single favorite function, and Gauss because, um, he's Gauss. Have you SEEN the amount of things named after him, and all for good reason too?

What do you think of statistics?

Statistics is kind of fun. Not nearly as fun as things like calculus, but a lot more fun than things like linear algebra.

What do you think of non-standard analysis?

I don't like it. I'm sure it works very well and all that, but I just... I don't like it.

What do you use for writing equations with on a PC?

Do you mean just equation writing software? I use LaTeX, because that's what my favorite knowledge forum uses. If you mean an actual program, Mathematica.

What's your favorite kind of triangle?

The kind with three 90 degree angles. (Yes, it DOES exist)

Do you own a slide rule?

Nope! I'm about... 20 years too young for that. Maybe 30?

Which is best, M&Ms, Reese's Pieces, or Skittles?

Reese's Pieces. By a mile.

I've never heard anyone with a math degree have trouble finding jobs, hell I know physicists who work for banks and make mad money.

I survived a graduate program and escaped with a degree [1] in "applied mathematics". Shortly thereafter (this was in 1998) got a job as a computer programmer, which has been my career since. My math skills are awfully rusty by now.

Out of the classmates I've kept in touch with, and the ones I met at the 10-year reunion a while back, I'd say less than 25% are actually working primarily with mathematics. The rest are all over the place, although mostly in somewhat technical fields (hell, one guy even went to medical school and is now a GP).

[1] A now-obsolete Norwegian degree roughly equivalent to a Master's, which is what it has been replaced with in the current (post-Bologna Process) system.

The kind with three 90 degree angles. (Yes, it DOES exist)


Now you've got me intrigued :) Could you show one?

How do you mathematians look at us physicists? We're bunch of non-rigorous players that don't care about divergence, and tries to think that infinity is not a problem.

Now you've got me intrigued :) Could you show one?
I suppose it involves non-Euclidean space ;) (Think about drawing a triangle on the inside of a sphere.)

Yup! In 3, 4, and 5 dimensions.dimensions :confused:


I don't like [non-standard analysis]. I'm sure it works very well and all that, but I just... I don't like it.Why not?

The kind with three 90 degree angles. (Yes, it DOES exist)What's your favorite Euclidean Triangle? :ack:


Nope! I'm about... 20 years too young for that. Maybe 30?I have two :p


Reese's Pieces. By a mile.Good man.

"Combinatorics? It's counting! How hard could counting be?"

*dies*

50% of the course was "counting", the other half was graph theory.

It's not that easy to wrap your head around the "counting" aspect of it - at least a lot of very smart people in the class had a hard time with it. It's not really fair to describe it as counting either.. It's like counting with your penis, and a blindfold. Interesting, but not as simple as simple "counting".

Now you've got me intrigued :) Could you show one?

I suppose it involves non-Euclidean space ;) (Think about drawing a triangle on the inside of a sphere.)

Leoreth was on the ball with this one. I'm talking of course about the triple-right triangle on a sphere. It's non-Euclidean, but that just makes things more fun!

http://www.math.cornell.edu/~mec/tripleright.jpg

How do you mathematians look at us physicists? We're bunch of non-rigorous players that don't care about divergence, and tries to think that infinity is not a problem.

I don't think I could talk for the general "you mathematicians", but I personally think of you guys as just applying all the fun math to the real world.

dimensions :confused:

Yup! Wanna try?

http://www.superliminal.com/cube/cube.htm

Why not?

I don't know that, really. I think it's just because they use a definition and practice of infinitesimal that rubs me the wrong way.

What's your favorite Euclidean Triangle? :ack:

A right triangle with angles 90, 22.5, and 67.5 degrees. It allows me to use the far-more-interesting-than-other-values sine and cosine of pi/8.

Mathematical beauty is connected to timelessness, certainty, abstraction, generality, and clarity.

Timelessness is always present, mathematical concepts are defined without relation to time so the theorems that describe them will remain true forever. Certainty and rigorous methods have been associated with mathematics since ancient times, it is by far the most rigorous of all the sciences. In the latter 19th century mathematicians reached a new level of rigorous methods based on set theory and they pushed for axiomatic perfection, but by the 1930s Godel's incompleteness result showed that such a perfect axiomatic system was impossible. Since then the focus has shifted away from logical foundations, towards more practical levels of certainty.

Abstraction and generality are closely related: a good abstract concept will be very general, applying to many different particular situations. Abstraction and generality are what give power to mathematical methods. Mathematicians face an infinite wilderness of infinitely strong beasts, and the only way forward is by finesse - leveraging our finite human intelligence - not brute force.

The opposite of abstraction and generality is an ad hoc solution, a term which means "for the problem at hand", whereas concepts, methods, and results are more beautiful if they solve a large number of disparate problems at once. For example, 19th century mathematician C.F. Gauss said he was not interested in Fermat's Last Theorem, as one could easily write down many similar equations and statements that were equally hard to prove, and that he would only be interested in the problem if there were a general theory encompassing all of these (which is what happened with Wile's proof over a century later based on the algebraic geometry of elliptic curves).


Not all mathematics would be regarded as beautiful. Newer proofs tend to be raw, undigested and unpolished, which makes them less beautiful by the five attributes above. Some subjects are inherently unwieldy and are left for future generations to hopefully tame. A lot of mathematical research is generated to fill grant quotas or supply students with problems, these unimportant results may never be polished to be particularly beautiful.

In judging whether a work is beautiful, it's a matter of subjective taste. There are many sub-areas of mathematics and everyone has preferences. Still, just like with movies, books, wines, etc it is possible with practice to evaluate the excellence of a mathematical work based on objective features, unclouded by personal preference. This happens all the time because research is peer reviewed.

Cool. Very good answer (way beyond my expectations for CFC ;)) Thank you! :goodjob:

I don't think I could talk for the general "you mathematicians", but I personally think of you guys as just applying all the fun math to the real world.

Wait, then what do engineers do?

How do you mathematians look at us physicists? We're bunch of non-rigorous players that don't care about divergence, and tries to think that infinity is not a problem.

It's kind of important to distinguish between science as a field of study and Science as in the method used for the natural sciences. The rigor achieved in mathematical disciplines is done through formal logic, and not so much observation, experimentation, and theoretical explanation. Physicists try to use math that has some actual application, and not be concerned with hypothetical constructs (ie Nth dimensional space) without sufficient prompting. If divergence/infinity are involved in the physicist's research, there's probably some interesting physical activity to care about interpreting.

Physicists tend to want to describe the universe we are currently living in, and avoid the ones we are not in.

Wait, then what do engineers do?

Linear approximations. :D

/facetiousness

Wait, then what do engineers do?Eat pizza, and define the most desirable social skill as touch-typing.

Edit: /me was an engineer for 35 years and is allowed to say that. :mischief:

Physicists try to use math that has some actual application, and not be concerned with hypothetical constructs (ie Nth dimensional space) without sufficient prompting.


In statistical mechanics we have to be concerned about N-dimensional spaces. And N is so large that, being physicists, we just set huge number=infinity and go with infinte-dimensional spaces


If divergence/infinity are involved in the physicist's research, there's probably some interesting physical activity to care about interpreting.

We encounter infinity often enough. But usually we just plug it into our equations and see if anything useful comes out. Because despite the mathematicians' protests we will do arithmetic with infinity.

Have you ever wanted to scream at a physicist while watching him doing physicist math?

Leoreth was on the ball with this one.
Hehe.
Yup! Wanna try?

http://www.superliminal.com/cube/cube.htm

And you actually own one of these?:dubious:

In statistical mechanics we have to be concerned about N-dimensional spaces. And N is so large that, being physicists, we just set huge number=infinity and go with infinte-dimensional spaces Any suggested reading about statistical mechanics?

*Okay, so maybe there is sufficient prompting to model thermodynamics with such spaces, but I hope the point about constructs not necessarily being applicable was made.
We encounter infinity often enough. But usually we just plug it into our equations and see if anything useful comes out. Because despite the mathematicians' protests we will do arithmetic with infinity.

Have you ever wanted to scream at a physicist while watching him doing physicist math?
Depends on how it is handled.. I don't think screaming would help though. I would have a problem setting something (like a number) equal to it unless it was understood that "=infinity" is not an equivalence relation, but a statement about {positive} divergence (as in we're no longer dealing with a huge number). It may turn out for a given scenario that the infinite-dimensional case can be well-approximated by a finite-dimensional case if it can be shown that the infinite case is convergent.

Hehe.


And you actually own one of these?:dubious:

Well, not a PHYSICAL copy, no. It'd be kind of hard to create a 4-dimensional object in real life, and if we did, I sure hope we're not relegating that ability to making Rubik's Cubes! But that virtual copy is just as good.

Infinite dimensionality is just like a sequence. It's nothing too special. A vector is just a sequence with a finite length.

EDIT: In a finite dimensional vector space, of course ;)

Well, not a PHYSICAL copy, no. It'd be kind of hard to create a 4-dimensional object in real life, and if we did, I sure hope we're not relegating that ability to making Rubik's Cubes! But that virtual copy is just as good.

Have you ever solved a 4d or 5d one?

Have you ever solved a 4d or 5d one?

Once or twice. It really is not easy.

Any suggested reading about statistical mechanics?

*Okay, so maybe there is sufficient prompting to model thermodynamics with such spaces, but I hope the point about constructs not necessarily being applicable was made.


Sorry, I can't recommend anything. The course material and even the textbook I read on the subject was in German.

Yes, constructs are not necessarily applicable, but it is quite hard to find any mathematical construct that has not yet been used (or abused) for some application outside of pure mathematics.


Depends on how it is handled.. I don't think screaming would help though. I would have a problem setting something (like a number) equal to it unless it was understood that "=infinity" is not an equivalence relation, but a statement about {positive} divergence (as in we're no longer dealing with a huge number). It may turn out for a given scenario that the infinite-dimensional case can be well-approximated by a finite-dimensional case if it can be shown that the infinite case is convergent.

This careful reasoning about convergence is exactly what physicists usually don't bother with. Until proven otherwise we just assume it converges and go right ahead. If we notice along the way that it doesn't, we try another approach, but as long as a technique works we don't care much about proving that it is applicable.

Leoreth was on the ball with this one.
Hehe.
Got the joke only after you pointed it out. Oh my.

New question: do you have a favourite math joke?

What's purple and commutes?

An Abelian grape.

What's purple and commutes?

An Abelian grape.

Are there any non-abelian food groups?

Hot cross product buns?

This careful reasoning about convergence is exactly what physicists usually don't bother with. Until proven otherwise we just assume it converges and go right ahead. If we notice along the way that it doesn't, we try another approach, but as long as a technique works we don't care much about proving that it is applicable.
At the risk of a huge over-generalization, I would say that physics requires more creativity, while mathematics requires more rigor.

RPN or in-fix?

At the risk of a huge over-generalization, I would say that physics requires more creativity, while mathematics requires more rigor.

I wouldn't say that physics requires more creativity. I think you need quite some creativity to think of a new proof. I would rather say that physics requires the ability to approximate, to determine which terms are important and which can be neglected.

Without simplifying a problem by dropping the irrelevant effects, you don't get very far in physics. So you have to leave behind the mathematical rigor and focus on the important part while ignoring the unimportant. To distinguish these is one of the main skills of physicists.

In mathematics the quest for absolute proofs does not allow for approximations, so you have to be rigorous and cannot just drop unimportant terms. Almost proving a theorem does not count for anything.

RPN or in-fix?

I looked up RPN and am honestly surprised anyone speaks of that with a straight face.

http://s14.postimage.org/v1qagr6tt/rudinbook.jpg
u ready?

What's your survival strategy for real analysis? It has a nasty tendency to break people. ;)

My strategy - figure out what book I'm assigned, and read every chapter of it before the class begins.

Twice.

I wouldn't say that physics requires more creativity. I think you need quite some creativity to think of a new proof. I would rather say that physics requires the ability to approximate, to determine which terms are important and which can be neglected.

Without simplifying a problem by dropping the irrelevant effects, you don't get very far in physics. So you have to leave behind the mathematical rigor and focus on the important part while ignoring the unimportant. To distinguish these is one of the main skills of physicists.

In mathematics the quest for absolute proofs does not allow for approximations, so you have to be rigorous and cannot just drop unimportant terms. Almost proving a theorem does not count for anything.
Sure, I agree. But don't you think that being able to "bend the rules" like that is a form of creativity? Not to mention the creativity involved in designing good experiments.

Why are statisticians so much better than mathematicians?

I don't know if is it me that I'm stupid or that the concept is really hard to get. You see, an operator's arity is a concept easy to get for the most common forms of it (nullary, unary and binary), but it's hard to get when you reach ternary level. I assume that ternary operators work pretty much like a transistor, but what about quaternary or other n-ary operators? And what about a multary (that accept many arities)? I cannot imagine operators like these but maybe you can enlighten me.

Sure, I agree. But don't you think that being able to "bend the rules" like that is a form of creativity? Not to mention the creativity involved in designing good experiments.

Of course it is a form of creativity. But I also consider the ability to apply the rules in new ways to get a new proof a form of creativity. So I wouldn't say one group is more creative than the other.

http://s14.postimage.org/v1qagr6tt/rudinbook.jpg
u ready?

What's your survival strategy for real analysis? It has a nasty tendency to break people. ;)

I shivered when I saw that picture.

No lie.

--

My strategy - figure out what book I'm assigned, and read every chapter of it before the class begins.

Twice.
Bad way to read Rudin.

The good way to read Rudin:

Start at page one. Read until you come to the word Theorem. Do not read the proof. Prove it yourself. Or at least try. If you get stuck read a line or two until you see what to do.

Thrust, repeat.

If you make it through the first six or seven chaptors like this then there shall be no power in the verse that can stop you. Enjoy graduate school. You half way there.

Rudin isn't probably the easiest book to read as the first one on the subject. I'd recommend Avner Friedman's Foundations of Modern Analysis, which is a bit simpler.

Of course it's given that you don't read math books. You must try proving things yourself. :)

http://s14.postimage.org/v1qagr6tt/rudinbook.jpg
u ready?

What's your survival strategy for real analysis? It has a nasty tendency to break people. ;)

You've had a calculus course. Some of the material will be familiar, but now you have to deal with the why.

(Actually, if you've had a discrete math course where proofs were done, you already have a good beginning).

Analysis isn't hard, it's just weird applying rigour to topics which previously haven't required it.

You can always post in my maths thread in sci/tech if you want help.

Analysis isn't hard, it's just weird applying rigour to topics which previously haven't required it.

You can always post in my maths thread in sci/tech if you want help.

Hehe, actually, if I ever need help with something, I usually ask the people at Physics Forums. Sometimes I help other people out there too.

Pfft, Physics. Far too useful.

Pfft, Physics. Far too useful.

They have a dedicated math section too! We once had a wonderful and enlightening debate on why the Riemann Hypothesis is or isn't true. I was on the "is" side, of course.

Of course? That's not the mathematical way of thinking. If you can't prove it, doubt it.

Of course? That's not the mathematical way of thinking. If you can't prove it, doubt it.

Ah, but I never said I don't doubt it. It's just my opinion that the circumstantial evidence in favor of RH being true is a lot more extensive than the circumstantial evidence for it not being true. Plus, I'd really like it to be true. But the other side gave off really convincing arguments. Hell, the whole debate was just interesting and fun to take part in.

Circumstantial evidence? Just one counter example would be enough.

By circumstantial evidence, I mean analogs of the Riemann Hypothesis that have been proven. For instance, an analog of the Riemann Hypothesis for finite algebraic function fields was proven by Weil (in the quadratic case) and Deligne (in general). There are other examples, if you need them listed. And until that one counter-example is found, there's no proof that a counter-example does exist either. (And if it is found, I think it's likely that it'll be found by finding a minimum of Re(zeta(1/2 + i t)) above zero or a maximum of said function below zero, both of which have been proven would violate the Riemann Hypothesis, if they happened)

Mmmkay. You're obviously not as drunk as I am ;)

Been sober for almost 20 years now.

I guess I'm your multiplicative inverse then ;)

Thanks for ignoring my question.

About n-ary operators? They are just mappings.

"plus" can be defined as an n-ary operator, like in LISP (programming language), where

(+ 1 2 3 4) = 10.

So you basically provide the operator with the arity needed and that's about it? And what about the operator ?: ? Can you provide it with any arity other than ternary?

Nope, the ternary operator ?: (which is if/then/else for people who don't speak dialects of the C programming language) needs 3 arguments.

a ? b : c

means if a is non zero, return b else return c (in C anyway, in Java I think you can only test a boolean).

EDIT: Syntax for operators is a bit weird in C as well, I've seen the common error

if( a == b == c )

before, it doesn't do what people expect it to do (== is the equality test operator in C). However, a && b && c does work (&& = boolean "and"), with the caveat that it uses lazy evaluation.

EDIT2: Note: 0 == 2 == 3 evaluates to true in C ;)

Does C evaluate from right to left? Damn, I should know these things.

Depends on the operator. Looks like I have it wrong, == evaluates left to right according to this page: http://msdn.microsoft.com/en-us/library/2bxt6kc4.aspx

So 0 == 2 == 3 is false in C, but 2 == 3 == 0 is true ;)

Should I open an ask a C/C++ programmer thread?

Have you ever done any cryptography? I find that stuff fascinating. But I guess, like most useful things, it's more computer science than maths.

I haven't. There's quite a lot of number theory involved though.

Have you ever done any cryptography? I find that stuff fascinating. But I guess, like most useful things, it's more computer science than maths.I did, and i really wanted to write my thesis in Cryptography, but there were no good Professor availible, sadly.

Cryptography is treated as a Pure Math subject at my university, and it's really close to Algebra and Number Theory. The methods used nowadays are based on unresolved problems in Math, namely the Discrete Logarithm and the Integer Factorization.
I even think that if you manage to solve this problems you might become kinda famous. Me and my friends used to joke about it that you could use this power for good (become famous) or bad (sell it to the Chinese for a buttload of cash). You could easily amass all money in world, cause all online accounts are protected this way.
I guess after all, it's like with physics. With all the applications you might oversee that fact that all is based on non-trivial Math, and they are lost without us:mwaha:

I don't know if is it me that I'm stupid or that the concept is really hard to get. You see, an operator's arity is a concept easy to get for the most common forms of it (nullary, unary and binary), but it's hard to get when you reach ternary level. I assume that ternary operators work pretty much like a transistor, but what about quaternary or other n-ary operators? And what about a multary (that accept many arities)? I cannot imagine operators like these but maybe you can enlighten me.

Arity is another name for how many operands are used. It's a function that may operate on more than one variable..

For example, we could perform an AND operation on 4 boolean operands*. If all four are true, then the result of our operation is also true.

*Technically this is equivalent to 3 binary AND operations:

(A*B)*(C*D) ,

((A*B)*C)*D

'''''''''''''''''''''''''''''''''''''''''''''''

Alternative example: we could have an operation that takes a list of integers and return the largest value in the list (n-ary example that's comp sci related).

AE2: List of names, return the median name from alphabetized list.

Cryptography is treated as a Pure Math subject at my university, and it's really close to Algebra and Number Theory. The methods used nowadays are based on unresolved problems in Math, namely the Discrete Logarithm and the Integer Factorization.
I even think that if you manage to solve this problems you might become kinda famous. Me and my friends used to joke about it that you could use this power for good (become famous) or bad (sell it to the Chinese for a buttload of cash). You could easily amass all money in world, cause all online accounts are protected this way.
I guess after all, it's like with physics. With all the applications you might oversee that fact that all is based on non-trivial Math, and they are lost without us:mwaha:

With Shor's algorithm there is already a possible solution. There is just nothing that can calculate it, yet.

It is kind of funny that a lot of modern classical cryptography depends on the hope that we won't succeed anytime soon in building a scalable quantum computer.

Well, WWII cryptography will be deciphered almost immediately by modern electronic computers. So there's no surprise. Computer architecture kind of things.

It is kind of funny that a lot of modern classical cryptography depends on the hope that we won't succeed anytime soon in building a scalable quantum computer.
I think these hopes are justified, at least for the next decades.

But even if quantum computers would obsolete classical cryptography, wouldn't they also immediately make quantum cryptography possible, which is even harder to crack?

Well it would take time to develop quantum computing encryption algorithms, whereas methods to brute force existing cryptographic algorithms already exist. Quantum computers would be able to encrypt stuff really really fast, but not necessarily more securely; you'd need to develop new algorithms in order to do that.

That reminds me about that one xkcd about encryption and how rather than breaking the code it would be just much easier to pipe wrench the guy who knows the password.

Quantum computers do not necessarily obsolete classical cryptography, they are just able to break some of the most widely used systems. There is a whole field of research called "post-quantum cryptography" about classical cryptographic systems that we can move to using once large scale quantum computers become a reality. Here is a recent book on this subject:

http://www.springer.com/mathematics/numbers/book/978-3-540-88701-0

Post-quantum cryptography is about classical crypto systems which are secure against quantum attacks. This is separate from research into quantum cryptography, which aims to build a quantum crypto system that is secure against both classical and quantum attacks.


Well it would take time to develop quantum computing encryption algorithms, whereas methods to brute force existing cryptographic algorithms already exist. Quantum computers would be able to encrypt stuff really really fast, but not necessarily more securely; you'd need to develop new algorithms in order to do that.


People have been working on designing quantum cryptographic systems since the 1980s. The advantage of quantum crypto systems is not just speed, but that they can provide a fundamentally better level of secrecy than could ever be achieved with a classical crypto system.

The reason for this is that the uniquely quantum property of entanglement can be used to detect eavesdroppers. The basic example is that if two quantum bits, b1 and b2, are entangled, then it is impossible to even look at b2 without disturbing b1. Therefore when a quantum message is compromised by an eavesdropper, the sender and receiver can detect this.

The situation I described above leads to a protocol called "quantum key distribution", where party A can send a secret key to party B (the key is a randomly generated number) , and they only use the key if they can verify that there was no eavesdropper. In practice, the eavesdropper may try to use "quantum hacking" to learn information from the quantum bits without disturbing their entanglement, so just as in classical cryptography we have a sort of arms race between people proposing new crypto systems and people finding attacks that break those systems. There is a lot more research to be done, but the basic idea that quantum entanglement guards against eavesdroppers has prompted enormous interest in quantum cryptography. There are already relatively small quantum key distribution networks being built from fiber optic cable:

http://news.bbc.co.uk/2/hi/science/nature/7661311.stm

I see, thanks for the explanation and the link!

By the way, you can order a quantum computer with 128 qbits today.

http://www.dwavesys.com/en/products-services.html


I'm not sure how this improves the time to factorize big numbers. I think the minimum number of qubits that is required for the factorization rises with the size of the number to factorize.

I think these hopes are justified, at least for the next decades.


Certainly for the next decade, but beyond that I wouldn't be so sure. It is a hot field at the moment. And if you want to keep something secret for a long time this might become relevant.


But even if quantum computers would obsolete classical cryptography, wouldn't they also immediately make quantum cryptography possible, which is even harder to crack?

Not necessarily. Quantum computers are not a prerequisite for quantum cryptography and do not necessarily make it possible. A quantum computer has to transmit quantum information only over a short distance, but quantum cryptography requires long distance quantum communication. There are concepts of distributed quantum computing which would require quantum communication, but fundamentally these are different problems. For example, if there was a successful implementation of a solid state quantum computer with superconducting qubits, this would not help quantum cryptography at all (except for invalidating many classical schemes).

But ti seems that the problems of long distance quantum communication are easier to solve than those of a scalable quantum computer, so when the first quantum computer to crack RSA comes along, quantum cryptography might be well established already.

There are already relatively small quantum key distribution networks being built from fiber optic cable:


In fact, you can already commercially buy complete quantum cryptography systems. Those are not without problems (for example, they don't use single photons but weak laser pulses), but are already beyond the proof of concept.

But one has to keep in mind, that although the protocol may be inherently secure, the implementation is not. The best cryptography system does not help you, if the computer itself is compromised (at one point the information has to be decrypted to be useful) or you fall prey to the 5$-wrench security hole.

By the way, you can order a quantum computer with 128 qbits today.

http://www.dwavesys.com/en/products-services.html


I'm not sure how this improves the time to factorize big numbers. I think the minimum number of qubits that is required for the factorization rises with the size of the number to factorize.

D-wave is not building a general purpose quantum computer, and their machine is not able (or intended) to factor integers.

The computer that D-wave is building uses a process called "quantum adiabatic optimization." The key word is "optimization", an important general class of problems in mathematics and computer science where the goal is to minimize (or maximize) some function. Perhaps the most famous optimization problem is the traveling salesman: given N cities marked on a map, find the path of minimum distance that connects them all.

More generally, suppose you have a function f(z_1, z_2, ... , z_n) where each of the z_i are 0 or 1 (so f the domain of f is n-bit strings). For n = 128 and some particular types of functions f, D-wave claims that they can find the minimum of f faster than any classical machine that exists on earth today.

To summarize, D-wave is not building a universal quantum computer, but rather a machine specifically designed for a certain kind of optimization problem. There are many potential applications of fast optimization, such as pattern recognition and computer learning, but not every type of problem can be solved by optimizing. No one knows how to express integer factorization as an optimization problem, and it is probably not possible to do so.

To answer your other question, Shor's algorithm factors a number N in time O((log N)^(1/3)) and uses space O(log N). In other words, to factor 1024-bit RSA encryption we need a few 1000s qubits. However, error correction and fault-tolerance are a very important part of building a realistic quantum computer, and including these processes we would need 10,000 or even 100,000 qubits to really use Shor's algorithm.

Fortunately, a universal quantum computer with only 20-30 qubits would be very useful for research in physics and chemistry, so the time of useful quantum computers might not be as far off as suggested in the above paragraph. Even though D-wave claims to have 128 qubits, many people dispute the claim (maybe the electrical signals they are measuring aren't really quantum effects, but just classical thermal noise), and even if we accept the claim then their optimization machine isn't intended to solve the physics and chemistry problems that need only 20-30 qubits to be useful.

Wow, that was an enlightening bunch of posts. Thanks guys!

Indeed it was. I'll be honest and say I didn't know that much about quantum cryptography, but now I feel like I've understood a lot more.

Is it true that mathematicians who specialize in the mathematics of infinites can only count up to three? (aleph-null, aleph-one and c)

In other words... have any of the higher infinities ever found anything to count?

Is it true that mathematicians who specialize in the mathematics of infinites can only count up to three? (aleph-null, aleph-one and c)

In other words... have any of the higher infinities ever found anything to count?

There are higher levels of infinity than the three you named, yes. In fact, there are infinite infinities! However, almost all of these infinities do not correspond to anything most people deal with (like, for example, the natural numbers, or the continuum).

EDIT: Using the beth numbers, there ARE two others. (Note that Beth-null is equal to Aleph-null, and Beth-one is greater than or equal to Aleph-one. The continuum hypothesis is equivalent to the statement Beth-one = Aleph-one = c) The first is Beth-two, which is the cardinality of the set of functions from R^m to R^n, or the cardinality of the power set of the reals. The second is Beth-omega, which I believe is the smallest uncountable cardinal.

Is it true that mathematicians who specialize in the mathematics of infinites can only count up to three? (aleph-null, aleph-one and c)

In other words... have any of the higher infinities ever found anything to count?

There is a lot more to count if we work infinite ordinals, rather than infinite cardinals.

Two infinite cardinals are equal whenever there is a bijective function between them, and this relatively coarse notion of equality makes the cardinal hierarchy simple. With the ordinals we use a tighter notion of equality: ordinals are totally ordered sets, and two infinite ordinals are equal only if there is an order-preserving bijection between them.

To explain this, first define the finite ordinals:

0 = {}
1 = {0}
2 = {0,1}
.
.
.
n = {0, 1, ... , n -1}

These are called successor ordinals, the rule for successor ordinals is:

n + 1 = n union {n}

Given any ordinal number A, which is a totally-ordered set, we construct it's successor by taking the union of A with {A}. Ordinals which are not successors are called limit ordinals. The first limit ordinal is also the first infinite ordinal:

w = {0, 1, 2, ... }

(read w as lower case greek omega, that is the standard notation). Now is the fun part, define the the successor of omega:

w + 1 = {0, 1, 2, ... , w}

and, as ordinals, omega is not equal to its successor. This is because there is no order preserving bijection between w and w + 1. Let f: w + 1 -> w be a function, then f cannot be order preserving, because f(w) has to be larger than f(1), f(2), ..., f(n) for every n, but since f(w) is a natural number (it's in w) it can't be larger than every natural number.

In fact, the successor of an ordinal A is never equal to A, it's always a new ordinal greater than A. So we get a sequence:

1, 2 , ... , w , w + 1, w + 2, ...

what comes after the ... ? Well that's another limit ordinal, w + w:

1, 2 , ... , w , w + 1, w + 2, ... , w + w, w + w + 1, ....

If we continue this we'll get to w*w = w^2. We can keep it up until w^w. We can even keep going to w^w^w^...*w times*... .

That last ordinal I just described, w^w^w^...*w times*... , is merely countable, it has the cardinality of aleph_0. :eek:

So even though there is only one countably infinite cardinal, there are countably infinitely many countably infinite ordinals. It turns out that the "infinity plus one" strategy in the playground game of "name the largest number you can" is well founded after all. ;)

Man this is a math thread, not a physics thread.

Man this is a math thread, not a physics thread.

You mean the discussion about quantum computation? Here is an excellent textbook published by the American Mathematical Society:

Quantum Computation: A Grand Mathematical Challenge for the Twenty-First Century and the Millennium (http://www.ams.org/bookstore-getitem/item=PSAPM-58)

Man this is a math thread, not a physics thread.
Physics isn't math?

Physics isn't math?

It's a subset of applied math...:mischief:


Other discussion stuff:

I didn't have any real issues with Real Analysis, but I had already taken several other rigorous proof based classes before. I get the feeling it's not that bad, but for many people it's there first "real" math hence the horror stories.

So I'm currently sitting around waiting to see if I got into any grad schools, fun stuff:p Any ideas on what to do if I don't get in anywhere? I was thinking of applying to Wall Street banks, tech companies, and NSA/CIA to do crypto stuff. But I know for most of that a b.S. won't cut it. Bottom line I refuse to go do crappy non-technical math work:p

Have you taken the Putnam? I got a 12 last year and I think I got two solid problems this year so 20ish hopefully? Definitely one of my better math experiences, I'de definitely suggest participating to any other math majors if you can.

How do you plan on coping with all the stupid people who can't do math making more money than you after college, and all the smart people who can do math dropping out of math for better money-making opportunities to make more money than you after college?

Do you actually believe everyone who struggles with Math is stupid? (Addressed both at Zelig and the threadstarter.)

And what do you mean by that? Also, what are your thoughts on the following comic?
http://zs1.smbc-comics.com/comics/20091116.gif

:lol: If my teacher tried the second approach, I'd say "Alright fine, I'm never gonna be good at math anyway." And my teacher basically already does the first approach (I do work to pass but I really don't care if I retain it past the class. More focused on History and English.

Question: Does anyone who wants to be in a historical or legal profession really need anything more than geometry.

Do you actually believe everyone who struggles with Math is stupid? (Addressed both at Zelig and the threadstarter.)
On some level yes I do, it's obvious to me but when I explain it to people and they look at me like I'm speaking French I think to to myself that there is probably something medically wrong with them.
But I know better than to say anything and I can usually look past it, both my girlfriends have struggled with math and it didn't bother me too much.


:lol: If my teacher tried the second approach, I'd say "Alright fine, I'm never gonna be good at math anyway." And my teacher basically already does the first approach (I do work to pass but I really don't care if I retain it past the class. More focused on History and English.

Question: Does anyone who wants to be in a historical or legal profession really need anything more than geometry.

Not the actual math, but the reasoning skills are very valuable in legal for sure maybe less so for history.

Do you actually believe everyone who struggles with Math is stupid? (Addressed both at Zelig and the threadstarter.)

I think they're stupid... at math. You can't tell someone's general intelligence from how they perform on one subject, though.

Question: Does anyone who wants to be in a historical or legal profession really need anything more than geometry.

Logic is always nice, although there are many who would argue that logic isn't math. And algebra is very helpful, in my opinion. I actually don't think you'd need much geometry in a legal profession. Algebra would likely be more helpful.

Working in Law is unlikely to exercise your math thingies.

No love for computer games or does it pay too little and is too full of long haired Jesus lookalike drunkards?

I think they're stupid... at math. You can't tell someone's general intelligence from how they perform on one subject, though.

Fair enough (Though struggling doesn't necessarily indicate total lack of ability or intelligence in the area.)

Logic is always nice, although there are many who would argue that logic isn't math. And algebra is very helpful, in my opinion. I actually don't think you'd need much geometry in a legal profession. Algebra would likely be more helpful

The phrase should have been "Above" instead of "More than" (Not feeling too well and so not focusing very much). The Math sequence in our high school is Integrated Algebra, Geometry, Trigonometry, Pre-Calculus, and, if you choose to go for it, basic calculus. I know I'll need a basic math to get a math credit college, but unless I need it beyond that, I'd really rather not go much harder than I am right now (I'm currently in the FIRST part of trig. I couldn't even pass in a regular paced trig class. I'm much more a good history and English student.)

No love for computer games or does it pay too little and is too full of long haired Jesus lookalike drunkards?

I like computer games, I like the idea of computer game design, but I don't like MATH.

Well, if you need help with a particular trig issue, feel free to contact me. I'm like literally always online, and I've tutored quite a few people.

The phrase should have been "Above" instead of "More than" (Not feeling too well and so not focusing very much). The Math sequence in our high school is Integrated Algebra, Geometry, Trigonometry, Pre-Calculus, and, if you choose to go for it, basic calculus. I know I'll need a basic math to get a math credit college, but unless I need it beyond that, I'd really rather not go much harder than I am right now (I'm currently in the FIRST part of trig. I couldn't even pass in a regular paced trig class. I'm much more a good history and English student.)

See if you can take Formal Logic in college, it's a philosophy class that skips all the application stuff you probably don't care about and just does some of the analytical reasoning you hit in higher math classes. More useful for law etc. and probably a lot more fun for you. Also depending on the college you might be able to use it to replace your math requirement.

I like computer games, I like the idea of computer game design, but I don't like MATH.

Well, we're not going to suddenly turn you on to liking maths.

What are you doing in this thread!!!!!!!! (j/k)

There's a lot of math in games, mainly involving triangles (trig), matrices (linear algebra - 3d graphics) and statistics (AI).

See if you can take Formal Logic in college, it's a philosophy class that skips all the application stuff you probably don't care about and just does some of the analytical reasoning you hit in higher math classes. More useful for law etc. and probably a lot more fun for you. Also depending on the college you might be able to use it to replace your math requirement.

Will do.

Well, we're not going to suddenly turn you on to liking maths.

What are you doing in this thread!!!!!!!! (j/k)

There's a lot of math in games, mainly involving triangles (trig), matrices (linear algebra - 3d graphics) and statistics (AI).

I don't want to be tunred into someone who likes math. I want the quickest route through it.

Why do Britons pluralize "math?"

Because we invented English, and we rule, hard.

I don't want to be tunred into someone who likes math. I want the quickest route through it.

Well, meh. I don't really like women, but want the quickest route through as many as possible. NOT REALLY!

Because we invented English, and we rule, hard.
Sorry, I couldn't hear you over the sound of your faded Empire.

I don't want to be tunred into someone who likes math. I want the quickest route through it.

Too bad there isn't a like button for posts on CFC like there is for comments on youtube.:(

Sorry, I couldn't hear you over the sound of your faded Empire.

Maybe you should get your ears checked then, if it doesn't bankrupt you ;)

In Plato's Republic he argues that advanced training in mathematics is a prerequisite to doing philosophy. Many people have heard of "the allegory of the cave", but what most secondary sources omit is that in Plato's version the process of leaving the cave is about learning mathematics.

"...the knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient...then geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down. Nothing will be more likely to have such an effect."

He is careful to point out that he does mean we should study mathematics because it's "useful", that is a secondary indirect benefit. The primary reason to study it is to improve ourselves.

To become a lawyer and make a lot of money does not require an elevated nature, in fact such a nature could be a detriment to that goal. In general the primary goal of most animals is to satisfy their desire for sensations, and that does not need mathematics at all.

Here is book VII of the Republic, which includes the "allegory of the cave" and discusses these points:

http://www.abu.nb.ca/Courses/GrPhil/RepBookVII.htm

Question: Does anyone who wants to be in a historical or legal profession really need anything more than geometry.
Why do you think you'll need geometry?

Well, meh. I don't really like women, but want the quickest route through as many as possible. NOT REALLY!
*rimshot*

Too bad there isn't a like button for posts on CFC like there is for comments on youtube.:(
... or dislike buttons :rolleyes:

Why do Britons pluralize "math?"

Yes, mathematic should be contracted math.


Question: Does anyone who wants to be in a historical or legal profession really need anything more than geometry.

Maths is at the center of intellectual history of mankind. Natural and social sciences have always been measured against maths, as well as pretty many things in the world.

This is something most people don't understand, because they think maths is about calculating, while it really is about deducting. That's the Maths Plato too talks about, and that's why it read on the door to his Academia that nobody who knows geometry should enter.

If you check which philosophers were mathematicians, the list is pretty long (and surprising too). I doubt anyone can understand thinkers like Descartes, Berkeley, Spinoza, Husserl etc. without known maths pretty well.

If you learn geometry, do yourself a favour and learn Euclid's geometry (although modern expositions of it). Usually femoetry is taught nowadays as calculating, not deducing.

I would imagine that math would be VERY useful for a litigator - it's the basis for building up an argument that can't be refuted, assuming the premises are correct.

You probably don't need advanced maths in order to be lawyer, but if you aren't capable of doing maths then you probably aren't smart enough to be a lawyer. I don't know any lawyers who aren't smart enough to do at least A-level (16-18 yrs old) maths.

Fermat was lawyer btw. :)

And he sure took his intellectual property to the grave.

Why do you think you'll need geometry?

I don't but I had no choice but to take that:p

So I've (well it's my co-worker) got a problem.

She needs to calculate the minimal size for a sample she's about to do.
The formula used for this is:

N'=(N*z^2*p(1-p))/(z^2*p(1-p)+(N-1)*F^2)

in this formula:
N'=amount of respondents needed
N=population
z=standard deviation
p=chance of a certain answer
F=marge of error

The known values are:

N'=33
N=35
z=1,96
p=50%
F=5%

Yes, I'm ashamed to say I'm not kidding. We know every single value that this formula has to offer. The answer was received trough a very nifty piece of software, unfortunately she has to write down how the problem is solved.

Now you say, that shouldn't be to hard. You have all the values. True, an again I'm ashamed to say, I can't seem to get to the right answer...

If someone could give me a hand how to write this down that would be much appreciated.

The worst part is... I followed a class in this and had the highest score of said class and for some reason I can't seem to solve it :wallbash:

Maybe you should post how you did it and what's the expected answer?

Wild guess of the error is that you entered probabilty as 50, while it should be 0.5. And I'd guess margin of error should be 0.05 then, but don't know much about statistics.

Look at the numerator first:
35 * (1.96)^2 * .5 * .5 = 33.614
(p = .5, so 1-p is also .5)

Now look at the denominator, which has two parts:
The first part: 1.96^2 * .5 * .5 = .9604
The second part: (35 - 1) * .05^2 = .085

So our answer is: 33.614 / (.9604 + .085) = 33.614/1.0454 = 32.15. Since we can't interview a sixth of a person, we must interview 33 people.

Hopefully there are no typos here, as the calculator I have with me doesn't show what I've typed, and I'm too lazy to do it myself.

Maybe you should post how you did it and what's the expected answer?

Wild guess of the error is that you entered probabilty as 50, while it should be 0.5. And I'd guess margin of error should be 0.05 then, but don't know much about statistics.

Hmm that might be prudent.
I did it the following way:

(35*1.96^2*0.5(1-0.5))/(1.96^2*0.5(1-0.5)+(35-1)*0.05^2)
=
(35*3.8416*0.5*0.5)/(3.8416*0.5*0.5+34*0.025)
=
33.614/1.8104
=
18,57

since the answer should be 33 (more or less) it is quite obvious I made a mistake... but for the life of me I can't figure out what :(

Look at the numerator first:
35 * (1.96)^2 * .5 * .5 = 33.614
(p = .5, so 1-p is also .5)

Now look at the denominator, which has two parts:
The first part: 1.96^2 * .5 * .5 = .9604
The second part: (35 - 1) * .05^2 = .085

So our answer is: 33.614 / (.9604 + .085) = 33.614/1.0454 = 32.15. Since we can't interview a sixth of a person, we must interview 33 people.

Hopefully there are no typos here, as the calculator I have with me doesn't show what I've typed, and I'm too lazy to do it myself.

GIANT PACEPALM :wallbash: :wallbash:

thanks :D I found my mistake

for some reason I had 0.05^2=0.025 instead 0.05^2=0.0025

Looks to me like I need to get up earlier... the problem's already solved by the time I get here! :lol:

Looks to me like I need to get up earlier... the problem's already solved by the time I get here! :lol:

Here's a puzzle for you...

You have 2 pieces of string of different, unspecified length, and some matches. Each piece of string takes exactly an hour to burn, but the burn rate is not constant. This means that it could take 59 minutes to burn the first 1⁄4, and 1 minute for the rest. The strings have different burn rates, and of course you don't know the rates anyway.

Using only the matches and the strings, measure 45 minutes.

This is one of my favorite puzzles that I have ever solved. The bonus version I came up with is to show that if given an infinite number of such strings you can measure any quantity of time greater than a certain amount to an arbitrary precision.

Here is my method.
Use some matches and one string to make a pendulum. Burn the other string and count swings to determine the period of the pendulum. (This will be somewhat tedious.) Now use the pendulum as a clock to measure out 45 minutes.

Here is my method.
Use some matches and one string to make a pendulum. Burn the other string and count swings to determine the period of the pendulum. (This will be somewhat tedious.) Now use the pendulum as a clock to measure out 45 minutes.

But this would take an hour to calibrate, and you want to calculate 45 minutes now. Also this uses other assumptions like being on a planet that rotates etc. Which we don't make, there is a much more elegant mathematical solution.

Here is my method.
Use some matches and one string to make a pendulum. Burn the other string and count swings to determine the period of the pendulum. (This will be somewhat tedious.) Now use the pendulum as a clock to measure out 45 minutes.
Channeling Richard Feynman? :D

But this would take an hour to calibrate, and you want to calculate 45 minutes now. Also this uses other assumptions like being on a planet that rotates etc. Which we don't make, there is a much more elegant mathematical solution.

Well, it does take an hour to calibrate, but the problem said "measure 45 minutes", not "measure 45 minutes starting now."

It does assume the existence of a gravitational field that does not vary in time, but it doesn't require anything else. It does not assume a rotating planet.

Whether a mathematical solution is more elegant than a physical solution is entirely a matter of preference.

I had not seen that riddle before, nice one.

Light both ends of string #1, and light one end of string #2. When string #1 is completely gone we know that one half hour has passed, then light the other end of string #2.

Note that this makes the physical assumption that we are in an environment with oxygen and so can burn the matches. :lol:

Here's a puzzle for you...



This is one of my favorite puzzles that I have ever solved. The bonus version I came up with is to show that if given an infinite number of such strings you can measure any quantity of time greater than a certain amount to an arbitrary precision.

Ooh! A puzzle! Fun. I'll try to avoid the spoilers, as I'm sure they all have the right answer! I'll get back to you on this one.

Have you ever thought about the possibility of going crazy :p? In math class two years ago, we watched a video that showed us a bunch of famous mathematicians. Pretty much all of them went crazy. They started talking to numbers and stuff like that.

Have you ever thought about the possibility of going crazy :p? In math class two years ago, we watched a video that showed us a bunch of famous mathematicians. Pretty much all of them went crazy. They started talking to numbers and stuff like that.

Insanity is an unfortunate consequence of being too good at thinking about maths.

Similar with chess, when I played extensively I started seeing in grids and began considering how to eat girls on a night out by approaching them in an L-shaped manner.

Solution 1:
Assuming you have a lot of matches, burn as many matches as you can in one hour (measured by the length of the string). Matches should have a constant burn rate, so once you determine how long each match takes (60 minutes / number of matches burnt in an hour), you can light as many matches as necessary to get to 45 mins.

Solution 2:
Light one end of one string, and both ends of the other string. It'll take half an hour to burn through the 2nd string. Once the 2nd string is burnt, immediately light the other end of the fisrt string. It will take 15 mins to burn. Total time = 45 mins.

Solution 3:
Light one end of one string. At some point, you will have measured 45 minutes.

P.S. I don't see why there can't be several solutions to this puzzle. I mean, if you gave 10 people the same tools and the same problem, they'd come up with several different answers. Some may work better than others, some may be faster or easier, but they all get the job done. I don't see what's wrong with CKS's solution - it has the benefit of being able to measure out any length of time, without destroying as much of your equipment as <other solutions>. It's simple and it's reusable, which, to me, is much more valuable than <other solutions>. If you ask me, his is the best overall solution.

My "mathematical" solution is distinctly non-elegant:

Light both ends of string one. It will take 30 minutes to burn. Then light string 2 at both ends and somewhere in the middle. It will be burning in 4 places. When one of the two sections of string 2 burns up, light the other section in the middle. Repeat until all the string is gone. This will have string 2 burning always at four places, so it will take 15 minutes to burn. It will be a major pain to actually do.

This method can be used to measure other amounts of time, too, but it isn't very convenient. (Keep the string burning in 6 places to get 10 minutes, for example.) Clearly, with lots of strings you can measure to any amount of time you like, just burning each string in the appropriate number of places. Your precision can be as good as you like, but your accuracy will clearly suffer if you can't light the strings infinitely quickly.


I'm a physicist, not a mathematician; we tend not to look so much for elegance, and our perspective is distinctly different from that of mathematicians. While I appreciate mathematics, doing it requires a different sort of pickiness that I don't have.

I'm a physicist, not a mathematician

I suppose an engineer would make a working clock from the matches. A computer scientist would join the matches together to make a series of logic gates, eventually turning the contraption into a binary clock. And a 18th Century French physicist would define "1 minute" as the time it takes for a match to burn, and then burn 45 matches in succession.

I suppose an engineer would make a working clock from the matches. A computer scientist would join the matches together to make a series of logic gates, eventually turning the contraption into a binary clock. And a 18th Century French physicist would define "1 minute" as the time it takes for a match to burn, and then burn 45 matches in succession.

To a man with a hammer everything looks like a nail. I particularly like your 18th century French scientist hypothetical.

Bundle some matches with the string. Drop the bundle from such a height that the drop time is equal to the burn time of a match. Using that information, burn enough matches from to reach 45 minutes.

Do you believe numbers exist?

Do you believe numbers exist?

That depends on what you mean by "existing". There are many people who believe "existing" means that it has some sort of physical presence, that is, that there is something that we can point to and say "that is two" or some such. If that's the definition you're using, I'd say no. Numbers are a mathematical reality, not a physical reality. They're a mental construction that happens to coincide well with what we experience around us.

Oh, and as for the string thingy, I did manage to figure it out.

Burn a string at both ends, and the other one at one end. Now, I'm pretty sure that the string burned at both ends, well, it should take a half hour, although I'm not ENTIRELY certain. I see it as you having a string with an unknown density along the string, but you know that the integral of the density is some number, and so there is definitely a point along the string such that the integral of part of the string's density is half said number. Which should prove that if you light the string at both ends, it'll take a half hour. That said, after the first string is completely gone, a half-hour has passed and you can light the other string, which SHOULD take 15 more minutes.

That depends on what you mean by "existing". There are many people who believe "existing" means that it has some sort of physical presence, that is, that there is something that we can point to and say "that is two" or some such. If that's the definition you're using, I'd say no. Numbers are a mathematical reality, not a physical reality. They're a mental construction that happens to coincide well with what we experience around us.

Is it possible that an alien intelligence would have a completely different idea of numbers, and therefore a completely different set of mathematics?

I understand that much of mathematics is more formally expressed as logic - are there other logics out there in the same way that Euclidean Geometry isn't the only Geometry?

I'm always reminded of the ad hoc nature of layman math when other bases come up - base 2, base 8, base 16 etc.

Does e retain its properties across bases? This is something i've never grasped...

Is it possible that an alien intelligence would have a completely different idea of numbers, and therefore a completely different set of mathematics?

Personally, I'd call it unlikely. However, I wouldn't rule it out.

I understand that much of mathematics is more formally expressed as logic - are there other logics out there in the same way that Euclidean Geometry isn't the only Geometry?

Yes, many. I'm not sure if I could provide examples for you, though.

I'm always reminded of the ad hoc nature of layman math when other bases come up - base 2, base 8, base 16 etc.

Does e retain its properties across bases? This is something i've never grasped...

Of course it does - the base of a number doesn't affect the properties of the number itself. A base is just a representation of the number. pi in base 3 has exactly the same properties as pi in base 10.

What does the number 1E99 mean?

(That's the highest my calculator will go, but I have no idea what it means.)

What does the number 1E99 mean?

(That's the highest my calculator will go, but I have no idea what it means.)

The E is a calculator shorthand for "times ten to the". So this literally means "1 times ten to the 99", or 10 to the 99th power.

1 followed by 99 zeros, essentially.

Anyway, question. What do you think of duodecimal advocacy? The main argument seems to be that there are more divisors of 12 than 10 (2,3,4,6 as opposed to 2,5), leading to easier multiplication/division and less cumbersome fractions to deal with. They also argue that the divisors of 12 are more useful than those of 10 (yes, I am aware that 5 is common, but that may be due to the demands of base-10).

Base 12 is silly. Clearly we are meant to use base 8, because repetitive halving (1/2, 1/4, 1/8, 1/16, 1/32, etc.) works out so nicely there. Plus, you have eight non-thumb fingers. ;)

Base 2. Everything else is needless abstraction.

Even some would argue that base 2 is needless. Why do you need anything more than base 1? :mischief:

The E is a calculator shorthand for "times ten to the". So this literally means "1 times ten to the 99", or 10 to the 99th power.

That makes sense. I had wondered why it wasn't "10E99." Now I know why;)

Another question: What's the point of imaginary numbers? If they aren't even real, why do we need to understand them?

That makes sense. I had wondered why it wasn't "10E99." Now I know why;)

Another question: What's the point of imaginary numbers? If they aren't even real, why do we need to understand them?

10E99 isn't normalised (all scientific notation has the mantissa (part before the 'E' between 1 and 9.9999...)

Imaginary numbers enable you to solve x2 = -1

And it turns out you don't need any "more" numbers to solve all polynomials in real (or complex) numbers either, so they are the algebraic closure of the real (and complex) numbers.

They also have nice geometric properties (multiplication by a complex number is a spiral enlargement) which is very useful for describing physical properties (come up in the equation for solving motion of an oscillating spring, A/C current, etc.).

Imaginary numbers are numbers, too. By extension, imaginary answers are answers, too. How else are you supposed to solve sqrt(-1)? They just exist on a different set of axis than real numbers. The real-imaginary plane allows you to plot all kinds of crazy imaginary numbers and allows you to get some pretty crazy relationships, e.g. e^(pi*i) + 1 = 0.

More generally, imaginary numbers enable electricity to get to your house, as a lot of electrical engineering deals with imaginary numbers. A few other fields, such as differential equations, numeric analysis, and some other stuff more mathematically-inclined people can tell you about, also deal heavily with imaginary numbers.

I was about to say that imaginary and complex numbers are mathematical tricks to make physics easier, but then, all of maths is just a trick to make physics easier.

Anyway, question. What do you think of duodecimal advocacy? The main argument seems to be that there are more divisors of 12 than 10 (2,3,4,6 as opposed to 2,5), leading to easier multiplication/division and less cumbersome fractions to deal with. They also argue that the divisors of 12 are more useful than those of 10 (yes, I am aware that 5 is common, but that may be due to the demands of base-10).

Well, what base a number is in doesn't actually affect mathematics that much, especially since a number in base 10 is still just as valid a number as a number in base 12! However, based purely on convenience, it would have been a lot nicer for civilization if we had developed using a base 12 number system, yes. Much more convenient to have that many divisors. (This is actually one of the reasons why the Babylonian's base 60 is so useful: it has divisors 2, 3, 4, 5, AND 6!) Alas, God/Nature/Evolution gave us ten fingers, and humans love to use their fingers for things.

Another question: What's the point of imaginary numbers? If they aren't even real, why do we need to understand them?

Really, it's unfortunate that imaginary numbers were termed "imaginary". This term was given to the number system by a detractor, much like other scientific concepts, including the "big bang" (which wasn't a bang at all). Imaginary numbers are VERY real, and have very real uses. The most common you hear about is in the field of electrical engineering, and they are useful there, but they're also instrumental in simply and easily solving the differential equation, for example, y'' + y = 0. Now, this equation COULD be solved for y without using complex numbers, but it's easier if you use them (along with the general method for solving ordinary differential equations with constant coefficients) to find the solution.

Really, it's unfortunate that imaginary numbers were termed "imaginary". This term was given to the number system by a detractor, much like other scientific concepts, including the "big bang" (which wasn't a bang at all). Imaginary numbers are VERY real, and have very real uses. The most common you hear about is in the field of electrical engineering, and they are useful there, but they're also instrumental in simply and easily solving the differential equation, for example, y'' + y = 0. Now, this equation COULD be solved for y without using complex numbers, but it's easier if you use them (along with the general method for solving ordinary differential equations with constant coefficients) to find the solution.

I have a feeling I won't understand the answer anyway, but how does the concept of square-rooting negative one make sense, when any number squared is positive? In other words, how CAN the square root of negative one exist when it breaks other mathematical rules?

It doesn't break any rules.

You just make up a number, i, and say that i2 = -1. (-i)2 is -1 as well, of course ;)

We did the same thing when we made up negative numbers...

I have a feeling I won't understand the answer anyway, but how does the concept of square-rooting negative one make sense, when any number squared is positive? In other words, how CAN the square root of negative one exist when it breaks other mathematical rules?

You're close there. Any "real" number squared is positive, and that's still true, by the definition of what "real" means in mathematics. (Hint: It's not the same as what it means in real life! This trips people up!) However, i, referred to as the "square root of negative one", which is kind of an arbitrary thing, is not a real number, and so doesn't have to conform to this rule. Then we say, okay, so we got these numbers that aren't "real", but they make real numbers when you perform a certain operation on them. We'll call them "imaginary" numbers then.

I'd best not mention quaternions then, with their 3 distinct (but related) square roots of -1 ;)

Those are used all the time in 3d graphics of course, since they don't commute in exactly the same way that 3d rotations don't ;)

I have a feeling I won't understand the answer anyway, but how does the concept of square-rooting negative one make sense, when any number squared is positive? In other words, how CAN the square root of negative one exist when it breaks other mathematical rules?
The "any number squared is positive" rule is just something high school teachers like to tell their students because they know they'll never reach the point where this statement becomes false in high school, and those that will study math later on will get by anyway.

It's like telling children in elementary schools that you can't divide 3 by 2 without rest because they don't know rational numbers.

Now "sense" is of course a completely different thing. For mathematicians, sense is secondary, if they can define a set of numbers that allow x² = -1 etc. to have a solution and everything's consistent, they're satisfied. If it has a practical purpose is another issue, but as it turned out in physics, in certain fields there's actually a meaning behind imaginary numbers.

Imaginary numbers are a bit like negative integers: When you were kid and first heard of them, you thought that they're stupid, but when you get used to them, they're perfectly ok. You can also use term "complex number", which is prehaps a little better.

The history behind them is that in middle ages or renaissance some dudes used them like variables in their calculations, but discarded them in the answers. So when they were calculating, they just named this nasty thing i, and later on it could vanish and only real numbers remain. Then they checked if their solution was right, and it was. So you could think it as a nice technique which is ill found however.

When people had been doing this for sometime, some of hem started to use it like a regular number, and if it can be consistently used, why not call it a number? Why would be it any worse than irrational numbers. (Which have similarly funny name, and which too were at least problematic until 19th century).

Then there's the geometric representation of complex (imaginary) numbers: You can think real numbers as a real line, but you can think of complex numbers as a plane. The real numbers is one axis of that plane and imaginary numbers other. I won't go into detail, but wikipedia probably has something about it, or someone else might want to.

Many other areas of maths have evolved the same way, calculus for example, or people integrating "delta function" (which is not a function in the usual sense at ll, and thus can not be integrated). The job of mathematicians is to try and make them well found.

What I've learnt about imaginary numbers:

Everybody conceptualises imaginary numbers differently.

Complex and quoternions was found to be useful in a number of applications but what about octonions, sedenions and higher numbers?

I've once read an article which was about utilizing octonions to simplify something with multidimensional string theory. Half of it flew over my head when I read it and I've forgotten the rest by now, though :(

Well, what base a number is in doesn't actually affect mathematics that much, especially since a number in base 10 is still just as valid a number as a number in base 12! However, based purely on convenience, it would have been a lot nicer for civilization if we had developed using a base 12 number system, yes. Much more convenient to have that many divisors. (This is actually one of the reasons why the Babylonian's base 60 is so useful: it has divisors 2, 3, 4, 5, AND 6!) Alas, God/Nature/Evolution gave us ten fingers, and humans love to use their fingers for things.

Do you think it's worth changing now, though? Or would it be too much cost for not enough return?

What's your definition of natural number? Do you call the number 0 a natural number or not?

And another question. What's in you opinion the most comfortable algorithm to multiply manually? I know of pretty comfortable algorithms to add, substract and divide manually but multiplication is a hell of a operation if you want to do it manually.

The "any number squared is positive" rule is just something high school teachers like to tell their students because they know they'll never reach the point where this statement becomes false in high school, and those that will study math later on will get by anyway.

It's like telling children in elementary schools that you can't divide 3 by 2 without rest because they don't know rational numbers.

Now "sense" is of course a completely different thing. For mathematicians, sense is secondary, if they can define a set of numbers that allow x² = -1 etc. to have a solution and everything's consistent, they're satisfied. If it has a practical purpose is another issue, but as it turned out in physics, in certain fields there's actually a meaning behind imaginary numbers.

Fair enough. I guess it doesn't matter much to me then since I don't want to study math!:)

What's your definition of natural number? Do you call the number 0 a natural number or not?

My definition of a natural number is the standard one, i.e. the set of integers greater than zero. Since zero isn't greater than zero, it's not a natural number.

And another question. What's in you opinion the most comfortable algorithm to multiply manually? I know of pretty comfortable algorithms to add, substract and divide manually but multiplication is a hell of a operation if you want to do it manually.

Hm... well, for relatively small numbers, i.e. numbers less than 100, I find it pretty easily to separate them into tens and ones and then use (a+b)(c+d)=ac+bc+ad+bd. For example, if I were to multiply 37 and 73, I would do something like this:

(30+7)(70+3) = 2100+490+90+21

Then just add the four together. Of course, this gets really difficult for larger numbers. :) If you're multiplying two three-digit numbers or bigger, I'd probably pull out a calculator at that point.

Huh? Just do long multiplication... It's not hard.

I'm with you on the natural numbers being 1, 2, ... though. Dutchfire disagrees!

Huh? Just do long multiplication... It's not hard.

I'm with you on the natural numbers being 1, 2, ... though. Dutchfire disagrees!

Just because it's not hard doesn't make it not tedious. :)

Well, do successive doubling and addition then! Much more exciting. Convert one of the numbers into binary and work it through.

EXAMPLE: 1536 * 11

11 is 8 + 2 + 1 in binary.

1536 * 1 = 1536
1536 * 2 = 3072
1536 * 4 = 6144
1536 * 8 = 12288

so 1536 * 11 = 1536 + 3072 + 12288 = 4608 + 12288 = 16896.

And checked with a calc - right first time ;)

EDIT: Although 15360 + 1536 is probably easier ;)

I use my phone.

EDIT: And yeah, timesing by 11 doesn't need any special methods :p

Well it's completely general, I just used 11 as an example. Was going to use 7 but that's 1 + 2 + 4 so doesn't involve any redundant calculation.

Well, do successive doubling and addition then! Much more exciting. Convert one of the numbers into binary and work it through.

EXAMPLE: 1536 * 11

11 is 8 + 2 + 1 in binary.

1536 * 1 = 1536
1536 * 2 = 3072
1536 * 4 = 6144
1536 * 8 = 12288

so 1536 * 11 = 1536 + 3072 + 12288 = 4608 + 12288 = 16896.

And checked with a calc - right first time ;)

EDIT: Although 15360 + 1536 is probably easier ;)

You missed the point. I was asking for a comfortable algorithm to multiply manually. This algorithm is not comfortable when both factors are pretty large numbers. Do you understand? In other words, what algorithm would you use to solve the multiplication such as 27711.84 * 204929.25 manually?

In other words, what algorithm would you use to solve the multiplication such as 27711.84 * 204929.25 manually?

Write one line under the other like this:
027711.84
204929.25
========
And then start multiplying the upper row with the leftmost digit in the lower row: 4*5=20, write 0 down, and add 2 to the next multiplication, 5*8+2. Write 2 again down and add 4 to the next multiplication 5*1 etc.

The first digit you write down this way goes under the leftmost 5, and you fill from right to left.

Then you start multiplying the upper row with the penultimate digit of the lower row, 4, and start filling from right to left.

At the end you add all the rows you write down, and move the decimal point 2+2=4 left from the end of the sum, and there's the result.

They taught us this when we were 7 or 8 or something like that, and I don't suppose there is any method more comfortable that would be worth learning.

Yeah, long multiplication, I already said that was easy. Apparently it's tedious as well ;)

So no way to get rid of the long multiplicaiton? :(

PS: You suck at algorithmics CFC.

I don't think it's tedious. Just some little calculating. If it's too bothersome, then use a calculator.

Using a calculator is the easy way to get rid of long multiplication.

Most short cut algorithms depend on nice properties of one of the numbers.

EDIT: If there was an easier way, don't you think they'd teach you it in school?

I don't know why you would ever want or need to do that kind of calculation without a calculator.

I do almost all calculations in head or on paper. It's more fun that way. :)

I don't know why you would ever want or need to do that kind of calculation without a calculator.

Have you ever heard the sentence "you can never learn too much"?

Learning when it is best to use a calculator is a valid skill as well ;)

EDIT: I do most calculations manually as well ;) Keeps the Alzheimer's away...

I do almost all calculations in head or on paper. It's more fun that way. :)

Wow! And what algorithm do you use to divide? The same as in the school?

EDIT:

EDIT: I do most calculations manually as well ;) Keeps the Alzheimer's away...

Really? The same happens to languages and linguistics. It's good to have all other branches of knowledge depending on us ;)

Long division. There's a new way to do it apparently, but Carol Vorderman (PBUH) doesn't like it since it doesn't work with fractions ;)

Long division is something you need to relearn when you start dividing polynomials.

Yes.

But it's not like I'd calculate numbers all day long.

Furthermore, mathematicians rarely have much to do with actual numbers. Usually physicists are better calculators for example. Like one dude said: "There are only three numbers for mathematicians: 0, finite and infinite".

Come on, we have 1 (multiplicative inverse), e (base of natural logarithms), pi (you know what that is!) and i (principal square root of -1) as well! -1 is quite handy too, but that's just a unit in the integers multiplied by another member of the set, so it doesn't count ;)

Well yes... It was more of a joke of course, but one that had hint of reality inside it. This guy was proving that some operator is bounded, and was therefore only interested of whether there is a constant c such that |Lx| <=c|x|, not the actual value of c. In a way that applies to e and pi too: you just notice that wow, there is really such number. How nice. And then you go on.

How could I forget the square root of 2 in my list :( Both of them!