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IMNSHO, but that was the worst proof of infinite number of primes. Why introduce unique factorizability when you don't need to? Why introduce something foreign that you are not going to prove when there is absolutely no need for it?

The most elegant proof I've seen so far (but I don't know any website showing it, so I can't link to it) is this: For any given N, an integer, consider N!+1, which is greater than N (where N! is defined by N! = 1 * 2 * 3 * ... * N). If this number is divisible by no other number than 1 and N!+1, then we are done (i.e. we have proven that given any arbitrary integer, there is a prime greater than tat integer). If this number is divisible by a prime, than that prime can't be less than or equal to N, since any integer (not equal to 1) less than or equal to N divides N! (see the definition of N!) but does not divide 1. Therefore, the prime that divides N! is greater than N. QED.

This proof involves no assumption (additional to assumptions (i.e. axioms) of the set of integers) other than this (which also happens to be much easier to prove than factorizability into primes): if n divides a + b and n divides a, then n divides b as well.

The theory is that there is an infinite number of these numbers

The problem that I have with it is that it's based on scholarly work. Since he's the president of a college, that probably means that he is studying grades, homework, or teacher reports on male vs. female students. This does not take into account factors that effect the outcome before the students even reach his school.

Some studies have shown that teachers tend to call on girls less and expect them to get lower grades in math and science classes, and that parents also can have a negative effect on girls' attitude toward math. Other studies have shown that girls simply don't like math. here are some examples.

So, there are dissenting opinions on the subject. This is something that I don't think we can get a really good control case for. Without a control case, how do we know that we aren't starting out with girls that have not been negatively influenced by parents, teachers, or other peers?

I am female, and have always had a natural ability to do math. I don't particularly like it, but I am good at it. (We are talking about a level below calculus, I did not have any kind of natural ability with that!) I used to tutor people that were in the same class as me.

While I only remember having a couple of teachers pick me out or completely ignore me, I'm not sure if that was because of me as a person, or because of gender. I tend to think that it didn't have anything to do with gender. I may not be getting a general idea of what is going on with this, though, because I went to a small college where a decent percentage (about 1/4) of the math and science graduates were female. I also went to elementary/middle/high school in an extremely small town, there were only about 20 people in my graduating class.

That said about teachers specifically, I think that a lot of what happens has to do with what girls pick up from society and school. Parents are sometimes sexist, they have an idea of what their children should be like and treat them as if they were that way. My dad refused to teach me chess after teaching my younger brother, and my mom is always saying how she can't do this or that because she's a girl. This may be unusual with parents specifically, but this is the kind of influence that girls get from the people around them all the time. I went against parental influence on this particular point, but it would appear that I am unusual in this.

You graph complex numbers in the complex plane, silly =) One axis is the real axis and the other is the imaginary axis. Here's a brief intro from the math forum (a written intro, without graphics), and here's some more examples from a graphic intensive site that shows how you can perform operations on complex numbers like vectors. You can also do neat stuff like find the complex roots of numbers by manuiplating the graph, which is also mentioned in the 2nd site.

What was it that drew you to a life of programming? How old were you when you first used a computer?

I remember seeing this board on tv. Maybe a motherboard for some crappy computer. And I was enchanted. I couldn't have been more than a few years old. And I saw Mr. Wizard's world on Nickelodeon and I liked when they had robots on there. And my brother kept setting himself ablaze with batteries and things you're not supposed to use D-cell batteries with. I was loving it all and couldn't wait to get my hands on a computer.

I finally got my chance when my mother got me a Pocket PC (radio shack, not windows). It had a 1-line display, could be programmed in BASIC, and had an assembler. Then I went to a computer themed middle school, computer-themed high school, and got my degree in Computer Science at University. It always came naturally to me and I didn't need anybody to turn me on to computers.

What pieces of modern software do you think would be a good way to introduce today's kids to the world of computing?

I think Squeak would be good because it's just fun-looking. You get to play with the race car and the mouse's eyes follow your cursor around.

Even better would be a Lego Mindstorms set. Lego has got to be the coolest toy ever and it's programmable. And I don't care how old it is... LogoWriter is big fun. It was compiled, had methods and variables, and we could draw with it. I wish I could find a copy of it.

It's a good thing Slashdot doesn't have this problem.

Are you serious? Teachers saying arithmetic is not a large part of math, and strategy is? And then using that as a reason to discourage kids to advance their math skills?

Sure, there are many 'parts' of math (with varying levels of being pure math, or applied math): calculus, algebra, linear analysis, logic, probability, numerical analysis, discrete math, geometry, etc.

But strategy? No.

Math is exact. Math can be fully proven to be correct without the need to actually make observations. Math is about truth, not about perception. A strategy is "an elaborate and systematic plan of action" (wordnet). To prove that a strategy is correct, you need to observe the result after implementing the plan of action. Ergo, strategy can be science, but it is not math.

What are those teachers smoking?

But, umm, it looks like you'll have to do it yourself if you want to stimulate your kid's obvious interest in math. Browsing around just a bit, I found places like mathforum.org. Maybe those can help?

Incidentally, I found that my brother, who is a freshman in high school, learned multiplication several years ago in one messed-up way (I'm 13 years older than him). While we would simply write this:

137
x23 ...he was taught to break it down into:

137 x 20 + 137 x 3

According to this Fahrenheit chose 96 degrees as the human body temperature. The article gives a brief explanation.

I just found a good little example of some of the concepts in the trachtenberg system here

http://mathforum.org/dr.math/faq/faq.calendar.html /

>> If that is the definition of an uncountable set then all infinite sets are uncountable

That's not true... countable and uncountable sets both refer to infinite set.. but that's another topic.

>>If you (or anyone else) can give links to information on this topic I would be grateful.

http://en.wikipedia.org/wiki/Infinity - I love wikipedia.. read the part about infinity in set theory, it talks about countable and uncountable sets.
http://mathforum.org/library/drmath/view/53352.htm l - this page talks about different sizes of infinity

I hope the links help a bit... if you know anyone who knows set theory well you can talk to them... and maybe number theory, I'm not too sure about that (I've only taken one number theory course so far).

It is not erroneous. While it may not be intuitive, it is correct. I have learned this in both my stats and combinatorics course during my second year at University of Waterloo.

For more info, I found this page using Google: http://mathforum.org/library/drmath/view/59138.htm l

A true geek would never be such a phony. A nerd might. But geeks have skills. You should at least be able to get rid of all the factors below 10 in your head or, failing that, on paper and pencil. There are easy rules here and here. Once you've gotten rid of those, maybe it'd be acceptable to use factor. Better to write your own program; it's not hard.

A true geek would never be such a phony. A nerd might. But geeks have skills. You should at least be able to get rid of all the factors below 10 in your head or, failing that, on paper and pencil. There are easy rules here and here. Once you've gotten rid of those, maybe it'd be acceptable to use factor. Better to write your own program; it's not hard.

As you know, notation has helped the progress of mathematics. Consider, for example, the limitations of the Roman number system, the importance of the invention of a symbol for zero, etc.

Which were, in your opinion, the notations that have permitted the greatest advances in mathematics?

Apropos to the current discussion was this response:

the interest of the question:

> Which were, in your opinion, the notations that have permitted the
> greatest advances in mathematics?

(which is very different from any question concerning the history of math. notations) is very close to the interest of the question: who has been the greatest mathematician in the history, e.g. near zero.

As you know, notation has helped the progress of mathematics. Consider, for example, the limitations of the Roman number system, the importance of the invention of a symbol for zero, etc.

Which were, in your opinion, the notations that have permitted the greatest advances in mathematics?

Apropos to the current discussion was this response:

the interest of the question:

> Which were, in your opinion, the notations that have permitted the
> greatest advances in mathematics?

(which is very different from any question concerning the history of math. notations) is very close to the interest of the question: who has been the greatest mathematician in the history, e.g. near zero.

As you know, notation has helped the progress of mathematics. Consider, for example, the limitations of the Roman number system, the importance of the invention of a symbol for zero, etc.

Which were, in your opinion, the notations that have permitted the greatest advances in mathematics?

Apropos to the current discussion was this response:

the interest of the question:

> Which were, in your opinion, the notations that have permitted the
> greatest advances in mathematics?

(which is very different from any question concerning the history of math. notations) is very close to the interest of the question: who has been the greatest mathematician in the history, e.g. near zero.

Working in the realm of web design and art, one of my favorite 'mathematical concepts' has always been the golden ratio. Now, I realize that this is really more a number than it is an equation, but at any rate it is truly amazing to experiment in the world of design with how shapes adhering to the golden ration often bring about more pleasing and content looking designs as well as playing with the inexpressible startling feel of a design with say, one aspect out of touch with the golden ratio whereas everything else adheres.

Newton's phrase "standing on the shoulders of giants" was a veiled insult to Robert Hooke...

This allegation is made almost every time Newton is mentioned on Slashdot but it has no historical basis.

As this analysis points out, when Newton uses the phrase he is refering to both Descarte and Hook. The most obvious interpretation is that he is complementing Hook by comparing him to Descarte and referring to them both as giants.

Furthermore, Hook was not especially short and in other cases where Newton engaged in scientific debate he specifically avoided what he called "oblique and glancing expressions".

There is thus every reason to suppose that when Newton said he stood on the shoulders of giants, he was acknowledging his debt to Copernicus, Kepler, Galileo, Descartes and Hook, who was at the time England's most eminent scientist.

So skip Apostol.

My special disdain goes out to Dr. Roach and other purists of the University of Texas' School of Mathematics, where a band of rigorists held sway for decades. Their influence and too-early focus on unnecessary rigor [e.g., teaching advanced group theory to freshmen instead of introductory calculus] and upon the (delta, epsilon) interpretation of calculus rippled through numerous universities as their disciples spread out into professorial positions. These bigots ruined the minds of scientists and mathematicians and drove thousands from those fields. May those rigorists stoke the ovens of Hell forever.

A sphere you walk on top of would probably be easier to construct, but unfortunately, either way has the same problem, because you're wrong about one thing.

It would have to be quite large to seem flat.

Thanks to a helpful page on chords at http://mathforum.org/library/drmath/view/57832.htm l, here's what I came up:

Assume a 30 inch step.
That makes the short side of the triangle 15 inches.
Start off with a sphere 10 feet in radius (20 feet in diameter).
15/120 = .0125, which is the sine of the triangle. Cosine(arcsine(short/hypotenuse))=0.94 inches.

A 1 inch height difference would certainly be noticed by me.

Assuming a 0.1 inch difference as small enough to be ignored, your sphere would have to be about 94 feet in radius. (And remember, that's radius. It's almost 200 feet in diameter.

Considering that's what would be required for each person in the game, I think what they've got is definite improvement.

I'm not impressed by the photo, though. It doesn't look like you could (safely) take a step forward, unless those blocks are really fast.

To anyone who complains that I should have done that in metric:
A) I'm a Merkin. (See alt.fan.pratchett on Usenet) We're allowed.
B) I'm at work and trying to be reasonably honest with my employer's time...

World War II might have gone a different way if not for "operational research," which sought decision-making rules for the precise allocation of resources. I hope that anyone with an MBA has heard at least of the Simplex Algorithm from 1947, and thinks of Game Theory as something absolutely precise about best strategies given well-defined input. Even dumbass Excel comes with a suite of tools, both linear and nonlinear, for performing optimizations, and today's desktops are capable of running what-if scenarios that would have required supercomputers just 10 years ago.

This 2x2 matrix idea seems awfully damned fluffy, considering how much is known about optimizing complex systems. Definitely an "airport book," as another Slashdotter described it.

Mean, median, and mode are all types of averages, although the mean is the most common type of average and usually refers to the _arithmetic mean_ (There are other kinds of means that are more difficult).

Interesting. Well, I suppose if the great gurus bless it as 1, it has some merit. However, my understanding of the concept of a limiting value is that the sum never reaches the value. Dr Math seems to have a bet each way. From the page:
http://mathforum.org/library/drmath/view/55748.htm l

.9 is not 1; neither is .999, nor .9999999999. In fact if you stop the expansion of 9s at any finite point, the fraction you have (like .9999 = 9999/10000) is never equal to 1. But each time you add a 9, the error is less. In fact, with each 9, the error is ten times smaller.

You can show (using calculus or other methods) that with a large enough number of 9s in the expansion, you can get arbitrarily close to 1, and here's the key:

THERE IS NO OTHER NUMBER THAT THE SEQUENCE GETS ARBITRARILY CLOSE TO.

Thus, if you are going to assign a value to .9999... (going on forever), the only sensible value is 1.

The key phrases above are 'arbitrarily close to' and 'assign a value'. To me this says that if you're going to treat the series as a constant (e.g. for use in engineering), the best fit / nearest constant is its limit, 1. I totally agree with this assertion -- it's the whole reason we bother to calculate limits, convergence etc. However, it doesn't make the series equal to 1, only arbitrarily close to 1 (it's like casting it to a new type, to use a computing analogy). My textbooks make a big deal out of denoting the limit or using the approximate equality sign in such cases.

But anyhoo, I still have half my degree to go and haven't gotten to the hardcore analysis stuff yet, so I may well be mistaken, especially if the various Dr Math PhDs seem to believe it. Thanks for your polite / non-snide reply.

(1) is the limit of the partial sums! That is, (1) equals

lim_{m --> infinity} SUM {from n = 1 to m} 9/10^n.

The limit of this indeed equals 1 (since you're in a "serious" maths degree, I'll leave the proof to you).

Again, this is not the limit of the sequence a_n = 9/10^n, it is the limit of the partial sums. Now, how is it that 0.9999... is NOT equal to 1? It HAS to be, because 0.9999... is EXACTLY the infinite sum of the above. That is what 0.999999... means.

It's kind of funny that blizzard posted this as an april fools joke thinking it wasn't true, but it is in fact true.

There was (is) a rather large discussion of this on sci.math. Here's a sample link: http://mathforum.org/dr.math/faq/faq.0.9999.html

Notice it cites:
R.V. Churchill and J.W. Brown. Complex Variables and Applications. 0.9999... = 1 ed., McGraw-Hill, 1990.

and

W. Rudin. Principles of Mathematical Analysis. McGraw-Hill, 1976.

Rudin is a serious mathematician and he knows what he's talking about-- not that an argument from authority means anything :) Hence, in conclusion, 1 = 0.99999....

They call it "Mebius" (in reference to Moebius) and their logo is a twisted loop. However, if you look carefully, the strip has two twists in it, and therefore is NOT a moebius strip.
Ok maybe I'm being too geeky here.

FYI, the continuum hypothesis is neither true nor false (or BOTH true and false, depending on how you think about it :).

It is independent of the rest of set theory... much like Euclid's parallel postulate is to geometry. You can assume it's true, or assume it's false, and you get different versions of set theory in the end. Similar to the existence of both euclidean and non-euclidean geometries.

Many people don't realize that there are multiple versions of something as fundamental to mathematics as set theory! Check out the Axiom of Choice for another example of something that's neither true nor false in set theory.

My favorite proof involving cardinality and set theory is the proof that there are the same number of integers as fractions... so simple that a school kid can understand every step, yet so profound a conclusion!

from math import pow

def zeta(s):
t = 0;
# does no. samples affect results?
for x in range(1, 25):
f = 1.0/x
n = pow(f,s)
t += n
return t

print str(zeta(-2))

(ugh stupid 'ecode' tag doesn't indent my python correctly)

For zeta(n<=-1) I get a large positive integer (which I believe translates to infinity). For zeta(n>=1) I get a number approaching 1 for larger values of n.

Am I missing something here? How do (-2, -4, -6...) etc produce a "trivial zero"? Does this function only work with complex numbers?

Thanks for the great post!

OK, I'm too lazy to RTFA, but, if there are infinite numbers, why would there not be infinite prime numbers, and infinite prime twins? Are there also not infinite perfect numbers, or ...?

That's a good question.

The fact that there are an infinite number of numbers doesn't immediately imply that there are an infinite number of primes, but Euclid figured out how to prove this is true in about 300 BC. It only takes a few sentences to explain it; here's one example.

It is definitely not obvious that there are an infinite number of twin primes. It has been an open question for more than a century, and some of the greatest minds in mathematics have worked on it. If this proof is correct, it will be a major result.

I was trying to think of a good example of something that there is not an infinite number of. Browsing through MathWorld, I found Truncatable Primes - there are only 83 of these.

Can anyone think of any other examples of a type of number that only has a finite number of them, even though at first glance it seems like there might be an infinite number of them?

Sorry to nitpick, but zero is even unless you pick some seriously wacky definition for "even."

"So you end up with a trivial result, too - a finite volume can only hold a finite amount of information."
I'm not so sure about that. There's something called Torricelli's Trumpet/Gabriel's Horn that has an infinite area in a finite volume. That implies to me that since there's an infinite area, there would be an infinite capacity for storing information, but I don't think it could really be constructed, and as such may live only in the interesting, but not so useful realm of peculiar math things.

1. The fact that his drawings weren't commented is what tipped us off to his genius in the first place. Everyone knows smart people don't comment.

Ha! Yes they do, here's a guy who is famous for making a helpful comment -- he even put it in the margin!

... he'd have called it RFC 0!

But 0! (zero factorial) is equal to 1, so what's your problem?

If you meant RFC0, I'm working on that right now, and it will be published in 1967 as soon as I can get this flux capacitor to work...

Run a google-search on "trachtenberg math".

You're looking for sites like Trachtenberg Speed System or Trachtenberg Math (Multiplication).

Professor Jakow Trachtenberg was a brilliant mathematician. Imprisoned by the nazis during WWII, he kept his mind busy to survive by applying advanced mathematical techniques to numeric computation. Eventually developing a number of techniques that provide for rapid mental computation without massive rote memorization.

For example:

0 Zero times any number at all is zero.

1 Copy down the multiplicand unchanged.

2 Double each digit of the multiplicand.

3 First step: subtract from 10 and double, and add 5 if the number is odd.
. Middle steps: subtract from 9 and double, and add half the neighbor, plus 5 if the number is odd.
. Last step: take half the lefthand digit of the multiplicand and reduce by 2.

4 First step: subtract from 10, and add 5 if the number is odd.
. Middle steps: subtract from 9 and add half the neighbor, plus 5 if the number is odd.
. Last step: take half the lefthand digit of the multiplicand and reduce by 1.

5 Use half the neighbor, plus 5 if the number is odd.

6 Use the number plus half the neighbor, plus five if the number is odd.

7 Use double the number plus half the neighbor, plus five if the number is odd.

8 First step: subtract from 10 and double.
. Middle steps: subtract from 9, double, and add the neighbor.
. Last step: Reduce the lefthand digit of the multiplicand by 2.

9 First step: subtract from 10.
. Middle steps: subtract from 9 and add the neighbor.
. Last step: reduce the lefthand digit of the multiplicand by 1.

10 Use the neighbor.

11 Add the neighbor to the number.

12 Double the number and add the neighbor.

Handy Kelvin links for all your learning needs..

Explanation as to why we don't say "Degrees Kelvin"
mathforum.org

For all your other Kelvin needs including history about the great man himself
http://zapatopi.net/lordkelvin.html

Let's not make the same mistakes again!!

I'm not sure it lends itself to a "gist" type of proof. In 1993, Andrew Wiles presesnted an indirect proof at some sort of mathematical conference, but it was based in graduate-level maths.

(I should point out that the original statement of the problem required x, y, and z to be positive integers. From there, it's trivial to show if true, it must also hold for all non-zero rationals, but most people stick with the original formulation because it's easier to work with.)

This page claims to have a brief proof that they say should have been discovered long ago, if only mathematicians had considered it. I'm having trouble wrapping my head around their peculiar wording, but it seems to go something like this:

• Because of the effect of exponents on the last digit of a number, one must only prove the theorem holds for n from 3 to 6.
• These cases were already shown to be true several hundred years ago.
• QED

I'm not entirely sure I buy into that first point, though. Their "proof" seems to involve a bunch of hand-waving.

This page seems a little more promising. It's an outline of Dr. Wiles's proof, but it, too, shoves in my face the fact that I'm losing all the higher math I learned at college.

I hope that's helpful.

I always thought that the average of independent guesses produces a surprisingly precise guess (assuming n is high enough). Sorry I couldn't find a better link

There are real world examples of this - the idea that prime numbers are an infinite series, for example. Since nobody has ever been able to find an end to the series, it still stands, but you can't *prove* it.

What an absolute nonsense. That primes are an infinite series was proven centuries ago.

This page, down in the glossary, uses the definition you suggest. That is, it defines "times larger" as purely multiplicative. "Three times larger" means "tripled."

In contrast, Dr. Math agrees with me that "times larger" implies adding to the baseline. Dr. Math says '"Three times larger than N" means "4 * N" - but only if you stop to think about it, as many people do not.'

I think it's amply clear that "X times larger" is ambiguous without the data to disambiguate it. You have one group of people who thinks it clearly means one thing, and another that thinks it clearly means something else. It is therefore (maybe not-so-clearly) ambiguous. Dr. Math sums it up nicely:

So here's my answer: "N times more than X" technically should mean (N+1)X, but is so commonly used to mean NX that it would be dangerous to follow the former interpretation without asking questions. I haven't yet found a dictionary or other authoritative source to support one view or the other (or both, most likely).

Real number can be mapped to a line. You might argue that this line is itself embedded in a plane representing all complex numbers. But there;s nothing in which that plane is embedded; there are no numbers that can't be expressed as a sum of real and imaginary part.

Mr. Pies, meet Quaternary numbers.

Funny things about them:
1. Multiplication is not comutative unless you limit yourself to a sub-body in which the third and four components are zero (that would be the good old complex numbers).
2. Nobody has been able to come up with a THREE-dimensional numeric system that makes sense, although no one has proved such a system is impossible.

Isn't Math just mind-blowing?

"Can you please point to me where the federal government has cut spending on federal aid or services because I want to have a party to celebrate such a momentous occassion."

It is believed (but not quite proven) that there is no highest prime.

Actually it is well proven that there are an infinite number of primes. Here is a really straightforward, simple proof.

Binary math has many special properties in group and number theory. We'd lose those in higher base math, and we wouldn't gain new properties to make up for that loss. Two, the low bound, is special.

Not sure what you mean here. Yes, groups of order 2 have some special properties, but so do groups of various prime and square orders etc. '2' is indeed a special prime, being the smallest absolutely, but that doesn't give it a monopoly on having special properties. See Here for some more resources on prime numbers (OK, now I really sounds like a geek.)

A non-complete can never be assigned

What is a non-complete? Like pi, sqrt(2), or 1/3. So you're saying that 1/3 cannot be assigned.

The whole point of the elipsis is to have a way to express a number such as 0.9999... which most definitely can be assigned.

Also see this.

Now we've been decelerating...then accelerating?

This is the thing that has been driving me absolutely crazy vis-a-vis the Big Bang theory, is that the practitioners seem to operate under the maxim:

"Keep adding terms until the data fits"

That's not the way science is supposed to work.

We've had a fair share of juggling of terms, including:

• "Big Crunch" - gravity will let the universe collapse again
• "Flat Universe" - universe will expand forever, but keep slowing down
• "Inflationary Universe" - universe expanded faster than the speed of light for a tiny moment (addressing the age and isotropy problems)
• Not sure what to call this... "Second wind universe" - universe slows its acceleration before dark energy becomes the reigning cause of repulsion

The Hubble telescope observations are getting awfully close to the predicted age of the universe. I wonder what age-of-the-universe estimate this new theory will predict; something more than 13.7 billion years?

The missing mass in the form of dark matter is, by all accounts, supposed to be mass that attracts; the inflationary universe theory depends on it for flatness. This might be another move 'around' the problem.

The Big Bang theory fell from grace for me over a period of fifteen years. While I don't subscribe to the notions of Velan, I'm curious, yet ambivalent about Alfven's plasma cosmology, there are a number of viable cosmological theories that don't have age, mass or exotic physics problems. It seems we closed the book on alternatives too soon, and are constantly interpreting data so it fits with theory, instead of breaking the back of theory on data.

Proving mathematically that you can never hit a wall must be tempered with observations of a hole in the wall and drunk in front of said wall on his back at a frat party :)

The thing is, you might "solve" Zeno's paradox as much as you want by referring to examples, but most attempts at attacking Zeno's paradox via "logical" examples doesn't do anything to explain it, but merely points at motions and declares the matter solved.

Look at your answer again - you just restated the paradox

If you keep taking increasingly smaller steps, you will never reach your goal.

That is the core of the paradox: During the race, you will always have an infinite number of "half-distances" left.

Yet, the paradox as stated is correct in stating that to move from point A to B (provided they are not the same :), you have to cover every "half-distance" in between - an infinite number of them.

So how do you prove that covering an infinite number of half distance is possible to do in finite time?

That's where the aforementioned limits of infinite series comes in.

Today, this is pretty basic maths, but it had people stumped for a proof for more than two thousand years.

A few hundred years before the invention of the electronic gadgets slasdotters take for granted people were navigating the world in sailing ships and calculating thier longditude and latitude with a sextant to measure the angle from the ground to the sun or a star, a clock and a book of log tables. Napier produced log tables in the 1600's but an accurate shipboard clock was only invented in 1764.

A book of log tables can be used to multiply integers quickly using A*B=antilog(log A + log B) or to calculate triginometic funcitions like sine, cosine and tan.

Original production of a book of log table took a lot of mathematical work. Publishers reputedly seeded the books with errors in the last digit to catch copiers. Link

Quoting the sentence once again:
At 15,000 degrees Celsius (27,032 degrees Fahrenheit), the plasma valve is about 50 times hotter than room temperature when measured in degrees Kelvin.

Now, my "too far from science and math class" comment was just a piece of rhetoric... a slight attempt at humor. I actually do remember how the Kelvin and Celcius scales work, and doing a direct switch(not a conversion) of one to the other is not only bad accounting, but tremendously bad science. The difference between 15000 K and 15273 K is a lot. Switching units in practice can be catastrophic(see that crashed Mars probe of a few years ago), and switching units in explanation is extremely irresponsible.

As for Kelvins... look at the way a temperature on the Kelvin scale is written: the freezing point of water is written as "273K" . Its boiling point is "373K". Notice the lack of a degree sign? It isn't just a character set issue. While "degrees Kelvin" may be pervasive, according to the International Bureau of Weights and Measures it is incorrect. Dr Math explains it, and has a couple supporting links.

This piece of nomeclature really isn't that important to me, but but in a sentence as boneheaded as that one is, why not pick on every little flaw?

(credit also to Bill Nye the Science Guy's Big Blast of Science for confirmation on the Kelvin wording thing)


*honk*

Not at all. The number of things a monkey can do (at least, with respect to Shakespeare replication) is not only countable but finite. The product of a finite set and a countably infinite set is still countable. In particular, assuming totally random key banging at a fixed rate, it straightforward to calculate the probably of Othello being emitted by the Monkey Bard in any particular span of time. No - that probability will never reach 1, but then it isn't zero for very long (relatively speaking) either.

Also, note that the product of two (indeed, N!) countably infinite sets is still countable: see Cantor or the nice little discussion here

"How many gas stations are there in the US?" is actually a Fermi Question, not a Microsoft question.

Enrico Fermi was a physicist at U. Chicago and participated in the bomb making at Los Alamos. I believe U. Chicagos supercollider is named after him. Fermi definitely predates Microsoft.

My high school math teacher introduced our class to Fermi questions. I'm not sure how good they are at interviews - you have a big advantage answering them if you know what they are, if you've answered one in the past and if you know what is expected of you.

More info about Fermi Questions can be found at:
http://mathforum.org/workshops/sum96/interdisc/she ila1.html