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This reminds me of the married couple handshake problem-

"My wife and I recently attended a party at which there were four other married couples. Various handshakes took place. No one shook hands with oneself, nor with one's spouse, and no one shook hands with the same person more than once. After all the handshakes were over, I asked each person, including my wife, how many hands he (or she) had shaken. To my surprise each gave a different answer. How many hands did my wife shake?"

There is a nice elegant solution to this one but it SEEMS like it shouldn't be possible to answer./P

Have you ever seen the proof of a^2 + b^2 = c^2? No? Of course not.

Not arguing with the rest of your post, but if you've never seen the Pythagorean Theorem proved, your mathematical education missed a really basic and important opportunity. Every geometry course should include at least one version of that proof, and it should be outlined in first-year Algebra.

For your own edification, I suggest reading and understanding a few proofs of the theorem. A very complete collection can be found here: http://www.cut-the-knot.org/pythagoras/index.shtml

Not sure about large numbers, but they certainly had math geniuses
http://www.cut-the-knot.org/proofs/jap.shtml

There are several ways to prove that the angles in a triangle sum to 180.

Axioms in mathematics vary, but here are the most common. It's a while since I studied this, but the first one means (I think), "starting from zero, and counting upwards, we will never again get zero". The second one means something roughly like "for all x and all y, if the successor of x (i.e. x+1) is equal to the successor of y (i.e. y+1) then x is equal to y".

Axioms don't require proofs, but they do require a strict (and preferably very narrow) definition.

I was someone who was once considered to be exceptional in math. Unfortunately, I made the mistake of stopping at calculus.

To regain my mastery of mathematics, I decided to take a single math problem very seriously. I figured that I would try to
understand the solution by grounding all ideas down to postulates.

I figured that this was a great way to learn mathematics anew and really get advanced. I soon learned that there were wonderful
math resources on the web. Wikipedia is really great. There's also MathWorld.com.,
PlanetMath, MathForum.org, and
Cut-The-Knot.org.

Being pretty ambitious, I chose Fermat's Last Theorem and Andrew Wiles's solution as my jump off point. I started this adventure
in 2004. Since then, because the problem is so tough, I started blogging through the different threads of the problem and I find
myself recreating the history of mathematics from the perspective of number theory.

I am not sure that this approach would work for everyone but if you are a solid problem solver, it can really make advanced
mathematics more fun. If you are interested to see what I came up with, you can check out my blog a My math blog.
I also started a general math blog.

Best of luck in learning mathematics.

-Larry

Anyone notice how the company logo looks a lot like those rearranging paper puzzles? Hmm.

I also believe that the square of the length of the hypotenuse of any right triangle is equal to the sum of the squares of the lengths of the other two sides, but I can't prove that either.

You're right, I don't argue with the facts. That would be stupid.

Yet women, of their own free will, choose not to. You can argue that pop culture is brain-washing women into believing they cannot succeed in these fields, and shouldn't even try, or that their fathers are telling them that they'll shame their family if they become engineers, but what exactly does that prove?

Right, that's exactly what I was arguing. You said "why aren't they entering the hard fields" and answered it that "they aren't interested". I asked "why aren't they interested".

Every time a talking Barbie says into her cell-phone "Math is tough!" (and a thousand other examples) we just spoiled mathematics for another young girl. That's not pop-culture? That's not influencing the future? That's not perpetuating the cycle?

I see a lot of sound and motion, but I don't see any progress (either in my post, or your response). We both agree it's nurture -- and yet, I think pop culture influences children, and you don't. Well great. We've come to a real understanding. We've really pushed the debate forward.

I don't think it's a red-herring to apply your argument to another minority. It points to the obvious deficiency in your case that women are held back by themselves despite all the best intentions of whites, I mean men, and not by the institutions of society (and the culture that molds it). This would be telling to your stance, does "affirmative action" work or not? If yes, how, by affecting culture? If not then all the "bending over backward" of university administrators and the opportunities facilitated by others (i.e., men) mean nothing.

Get real, culture has everything to do with it. If you want to mock the argument (let's call it "brainwashing"), that's fine. The question still remains, why are "women not interested"?

I hope all fathers will one day be as enlightened as your grand-dad. Not only allowing their girls to pursue higher education, specifically in science, but encouraging them to do so even in the face of the 'brain-washing' (or should I say, the marketting of gender roles and preferable consumer behaviors).

Are we observing the same pop culture? I don't see anything that would predispose girls against science and math any more than it does boys. Pop culture, as far as I can se, predisposes boys and girls fairly equally against science and math. Yet, despite pop culture, boys make it into the hard sciences, and girls don't.

I guess we aren't observing the same pop-culture. But to an extent I'll further agree with you (and reiterate my belief that our culture is becoming increasingly anti-intellectual) that boys are also -- to a much lesser extent -- being disenchanted with mathematics, pretty much all students are. Anyhow, I didn't think it was that subtle that our culture will begrudingly accept a male nerd (hence the wide popularity of this site, and its over-whelmingly male readership and TV and movies that depict [often hunky] nerds saving the day, e.g., MacGuyver, Farscape [and much sci-fi a sample biased by its authors], etc.) but pretty much universally reviles a female nerd. Notable exceptions... possibly Penny from the 80s Inspector Gadget cartoon (she did have a computer book) and the chacter of Zoe on 24 (although she is portrayed as being socially dysfunctional, hardly a role model, but at least she helps save the day). I don't count Trinity from The Matrix because she is never depicted as a nerd, only as a hot kung-fu babe in skin tight clothing, while an allusion is made in the first movie that she was a super-hacker and thus believed to be a man. Our culture has pretty much accepted the role of woman as doctor so that medical school becomes the haven for "smart girls". I would say that pop-culture drives this, but you would argue pop-culture reflects this.

Well, I'm sad we've gotten no where. Thanks for your time though. Respectfully.

Just referencing the Buffon's noodle article.

Buffon's needle is a way to calculate pi: you throw a needle of length l on a grid with lines spaced D>=l apart. The probability of the needle crossing the grid is related to pi, and Buffon used integral calculus for deriving it.

"Buffon's noodle" is a generalization where you can throw an arbitrary geometric line shape of length l on such a grid, and derive the expected number of grid crossings without even knowing the shape. This leads to a solution of Buffon's original problem which works without integral calculus.

Personally, I prefer Proofs #3, #4 and #9 from here.

Jigsaw puzzles
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Computer Stupidities (warning: may provoke laughbursts)
Math articles
Quicktime panoramas
The world's most famous debunker

Variously educational, baffling, entertaining, or just pretty.

The best hands-on mathematics experience, hands down, is at

http://www.cut-the-knot.org/

The topics are accessible, and often accompanied with applets.

I've used this material to give math talks to high school kids - they love it.

Here is a real favourite:

Make a polygon by picking a bunch of points on graph paper (just the grid intersection points) and connecting these points by straight lines. The spiky looking thing is technically called a lattice polygon. A really cool way to calculate the area is to (A) count the grid points strictly inside the polygon (B) count the grid points lying exactly on the edges and vertices, then do (A)+(B)/2-1 Voila!

The applet and explanation is here:
http://www.cut-the-knot.org/ctk/Pick.shtml

(However, the so-inclined may prefer to fool around with this by themselves, first!)

There are many^(many) phenomena out there like pick's theorem. Call them math paradoxes, or theorems, or whatever, but there's lots of mathematics that is easy to perceive and is mysterious as anything. Mathematics awareness can begin by first learning about and experiencing these brain bending phenomena, and then SEEKING an explanation.

The best hands-on mathematics experience, hands down, is at

http://www.cut-the-knot.org/

The topics are accessible, and often accompanied with applets.

I've used this material to give math talks to high school kids - they love it.

Here is a real favourite:

Make a polygon by picking a bunch of points on graph paper (just the grid intersection points) and connecting these points by straight lines. The spiky looking thing is technically called a lattice polygon. A really cool way to calculate the area is to (A) count the grid points strictly inside the polygon (B) count the grid points lying exactly on the edges and vertices, then do (A)+(B)/2-1 Voila!

The applet and explanation is here:
http://www.cut-the-knot.org/ctk/Pick.shtml

(However, the so-inclined may prefer to fool around with this by themselves, first!)

There are many^(many) phenomena out there like pick's theorem. Call them math paradoxes, or theorems, or whatever, but there's lots of mathematics that is easy to perceive and is mysterious as anything. Mathematics awareness can begin by first learning about and experiencing these brain bending phenomena, and then SEEKING an explanation.

he average adult has a vocabulary of about 20k words, and actually uses much less than that on a routine basis. Let's be really generous, though, and assume we are dealing with highly literate people with a vocabulary of, oh say, 65536 words. ;)

Still overly ptimistic, as a look at Zipf's Law (interesting also if you think RISC) might infer.

Back twenty years I was researching language skills of Turkish children and by then the basic German vocabulary was estimated to be at around 300. Chances are more like that there has not been a big boost since then.

CC.

Even more precisely, pi is transcendental.

AC:You can't prove an axiom.

Who said that 1+1=2 is an axiom?

"Principia Mathematica is the book Russell wrote with Alfred North Whitehead where they gave a logical foundation of Mathematics by developing the Theory of Types that obviated the Russell's paradox. This assertion may become more convincing after a look at the page 362 of Principia Mathematica where Russell and Whitehead finally proved that 1 + 1 = 2."

Russell's Paradox

"In 1913, Russell and Whitehead published "Principia Mathematica," a three-volume set considered one of the intellectual landmarks of the century that began from first principles and developed the laws of arithmetic (proving on Page 362 of Volume 1 that 1+1=2), but failing in the end to prove the internal consistency of mathematical logic and its ability to determine the truth or falsity of a given statement. The project drove Russell to the outer bounds of sanity."

Honolulu Star-Bulletin: No certainty exists in search for truth

AC: What Russell would have "proved" would look something more like...

Pooua: It would have looked like this!:

before we initiate we will admit 3 primitive concepts (this is you understand it without we need a explanation):
ZERO (denoted by 0);
NATURAL NUMBERS (N);
and IS SUCESSOR OF (this concept indicate a number that is the sucessor of another, and we will denote by "a" is the sucessor of "a");

finaly we define the operation sum by i) a+0 = a (the sum of any number with zero is equals to the number)
ii) a+(b)=(a+b) (the sum of any number with a sucessor of other number is equals to the sucessor of the the sum of the numbers);

by notation you have:
0=1; 1=2; 2=3; etc....

now we can calculate 1+1 that is
1+(0)=> by definition (ii) of sum =>(1+0) =>(1+0) => by definition (i) of sum =>(1) =>(1) => by notation equals to 2.

and by this you can know all others sums. note: you cannt say 0+a=a until you proof this is real. so by now, if you want calculate ], for example, 0+2 you must do:
0+2 => by notation => 0+(1)=> by definition (ii) of sum =>(0+1) => by definition (i) of sum => (1) => by notation equals to 2.

1-1/2+1/3-1/4+1/5-...= ln(2)
1-1/3+1/5-1/7+1/9-...=Pi/4

These can be gotten from plugging x=1 in the taylor series for ln(1+x) or arctan(x), but require a fair bit of proof that the series actually converge to what you think they do. It's amazing that series that look so alike can get you pi on one hand and logs on the other!

On the other hand, for sheer drop-your-jaw-in-awe-ness it's hard to beat some of the formulas from Ramanujan's notebooks (see example #5 at http://www.cut-the-knot.org/do_you_know/fraction.s html/ for a couple examples)

Amen to that :)

What 0.9999.. = 1 is really saying is that the more repeated 9s you have, the smaller the amount by which the number is less than 1. With an infinite number of 9s, the difference is infinitely small. This infinitely small quantity is accepted as being equal to zero.

This page has a few more interesting ideas on the subject.

Other neat tricks on Google:

1. Do a search on "1" and then "2" ... "9" and for each number plot the number of hits returned versus the number, and you end up with something like Benford's Law.
2. On Babelfish, enter a phrase in one language, and keep translating it through the set of languages until you hit a limit cycle.

I disagree. I would rather show a student a formal proof than to show them that this formula works for this particular triangle. This method of showing an example is not an acceptable proof in Math, and students should know what make acceptable proofs and what do NOT constitute acceptable proofs.

Of the many proofs of the Pythagorean Theorem, I believe Proof #4 on this page would be the easiest proof.

I also think students need to learn proper proof writing, which includes proof by contradiction, proof my mathematical induction, proof by contrapositive, as well as the regular If P, then Q proof.

However in this case your comment is only slightly wrong and therefore I have some hope that my reply might be a useful contribution.

You are correct that mathematical proofs are based on axioms. However there is still a crucial difference between a mathematical proof and a scientific theory. A mathematical proof is an absolute certainty. Note that I am not claiming that the underlying axioms are certain. I am only claiming that the proof itself is certain.

To put it another way, mathematicians are never certain about their underlying axioms but they are absolutely certain that if those axioms hold then the result stated in the proof also holds. It's kind of like a building with indestructible walls but no foundation.

Scientific theory is a whole different kettle of fish. You cannot prove a scientific theory with absolute certainty. In fact it is not even clear to me how one can define certainty within the framework of the scientific method. You never have any guarantee in science that future observations will be consistent with past observations.

In science you can prove a theory in the sense of preponderance of the evidence. You can even sometimes prove a theory beyond all reasonable doubt. But there is no way to eliminate the unreasonable doubts. Any endeavour based on empirical observation suffers from the fundamental limitation that you can never be sure of the next observation.

Finally, regarding 1+1=2, the foundational proof of this fact using the standard propositional axioms of mathematics really does require 362 pages. You can see the 362nd page on the bottom half of this Russell's paradox site.

good article explaining (George K.) Zipf's/Benfords law here.

good article explaining (George K.) Zipf's/Benfords law here.

Easy as can be. The average of those two numbers falls between them.

So what is that number?

From this web page, I got the following text....

Here's another enlightening argument from Burger . I never met anybody who thought 0.999... greater than 1. So, if it's not equal to 1, it is less than 1. Let's think of the average of 0.999... and 1. As an average of any two numbers, it's greater than 0.999... but is less than 1. Can we determine its decimal expansion? Say, what is its integer part. Since it's less than 1 but greater than 0 < 0.999..., its integer part is bound to be 0. What about its first decimal digit. Since 0.9 < 0.999..., that digit must be 9. And the second one? Since 0.99 < 0.999..., the second digit must also be 9. And so on. It appears like the average of 0.999... and 1 is 0.999... If the latter is denoted as X, (X + 1)/2 = X. X + 1 = 2X. X = 1. The conclusion can't be helped.

Easy as can be. The average of those two numbers falls between them.

So what is that number?

From this web page, I got the following text....

Here's another enlightening argument from Burger . I never met anybody who thought 0.999... greater than 1. So, if it's not equal to 1, it is less than 1. Let's think of the average of 0.999... and 1. As an average of any two numbers, it's greater than 0.999... but is less than 1. Can we determine its decimal expansion? Say, what is its integer part. Since it's less than 1 but greater than 0 < 0.999..., its integer part is bound to be 0. What about its first decimal digit. Since 0.9 < 0.999..., that digit must be 9. And the second one? Since 0.99 < 0.999..., the second digit must also be 9. And so on. It appears like the average of 0.999... and 1 is 0.999... If the latter is denoted as X, (X + 1)/2 = X. X + 1 = 2X. X = 1. The conclusion can't be helped.

Here is another good proof after some Googling . It goes like this....

Do we agree that 0.99999..... is equal to 0.9 + 0.099999...?
If so, then do we also agree that 0.99999.... is equal to 0.99 + 0.00999999....?

Then we find that no matter small a fraction we add to 0.99999... that we get a number greater than 1.

0.9 + 0.1 + 0.099999... > 1
0.99 + 0.01 + 0.009999... > 1
0.999 + 0.001 + 0.0009999.... > 1


No matter how small a fraction 0.0000.......001 you add, you get a number greater than 1.

As I have already stated here. If 0.99999... is less than 1, then please state a number in between the two. For any number A < B, then there exists a number Z such that A < Z < B. Please state Z for me.

This previous school year I took the first year of a two year computer science class (I am part of the first year where the AP exam will be in Java instead of C++ ). I will not be able to take the second year next year because of scheduling conflict but will be able to take it my senior year. Our class left off at advanced math problems being solved by recursion (the Towers of Hanoi problem sticks out in my head.) What would be a good book to review what we learned this year and be prepared for my second year of the class?"

This previous school year I took the first year of a two year computer science class (I am part of the first year where the AP exam will be in Java instead of C++ ). I will not be able to take the second year next year because of scheduling conflict but will be able to take it my senior year. Our class left off at advanced math problems being solved by recursion (the Towers of hanoi problem sticks out in my head.) What would be a good book to review what we learned this year and be prepared for my second year of the class?

If you/they have the right predilections then try cut-the-knot.org.

more recreational math

some lego