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Well, HE was beheaded in the French Revolution because a jealous colleague found that there was nothing to invent anymore in chemistry.

Antoine Lavoisier, 1743-1794

This was resolved when analysis (calculus) was formalized. Read more about it at MathWorld.

Interestingly enough, the Stade Paradox, mentioned on the same page, seems relevant to the whole 'grainy' picture of quantum mechanics that the original poster suggests would solve Zeno's dichotomy motion paradox.

I'm still a bit confused about how calculus resolves the dichtomoy motion paradox, though. Yes, we can calculate that certain infinite geometric series will converge, but don't we do so by looking just at the series and not at the infinite number of items in that series? That is, don't we solve the problem precisely by not performing the infinite number of calculations that would be required without calculus? If so, I don't see how this actually resolves the paradox, which is a paradox precisely because, according to our notions of being and motion, we must actually go through the infinite number of steps involved: we are not allowed to just look at the series as a whole, but must go through it.

Put another way, Zeno's motion paradox was not that we can't calculate such a series: he was not so foolish as to say that a body travelling one mile per hour will take anything other than one hour to travel one mile; it was not a mathematical paradox. It rather dealt with whether mathematical concepts, or any human concept, such as space or being or motion, are applicable to the world: can there actually be a body travelling at one mile per hour? We might think things move, but our definition of movement is paradoxical, and so what we call movement can't actually be what's going on.

But like I said, I simply haven't seen an explanation for why this is otherwise.

it seems like you could tap the signal with some strategically placed mirrors

Nah, with a mirror you'd be caught since their signal would go down. What you want is a beamsplitter.

Wow, I apologize for my "condescending" reply. For some reason I can't fathom I thought you were sending me to Mathworld here. Which would've been a bit smart assed. Still no excuse really. Sorry. Also I should have inserted "no calculus required" from the start.

This damn nicotine patch is way too small.

Not to mention it tastes terrible and is hard to light.

You say that in jest, but it's partially true, and not because of some silly misinterpretation of the way numbers are represented. One structure of interest to mathematicians (algebraists and number theorests in particular) is the cyclic groups of integers under modulo addition. In one of these groups, (Z_3, +), two plus two equals one. For more details see MathWorld's article on the subject. (Note: I use the underscore to denote subscripts when the subscript tag is unavailable as it is on /., so Z_3 is Z with a subscripted 3)

In 2000, Clay Mathematics Institute offered a $1 million prize for proof of the Riemann hypothesis. Interestingly, disproof of the Riemann hypothesis (e.g., by using a computer to actually find a zero off the critical line), does not earn the $1 million award.

An example that operates on the exact opposite principle of awarding prizes is the recent battle between Kasprov and Deep Jr: He gets 500k regardless, and 300k extra if he wins, 200k extra if he loses, or 250k if draw (i think the last case took place).

Talk about being stingy! I'd think that disproving the Riemann Hypothesis would be equally interesting as proving it - There are soooo many theorms out there that basically begins with "We assume that the Riemann Hypothesis to be true, and so forth so forth."

Yeah, a formula for calculating the nth digit of Pi. Yup - nobod'ys ever done that. Ever.

Interesting question, but I'm afraid not. Pi is a mathematical abstraction, defined as the ratio of a circle's circumference to its diameter. These are idealized, mathematical circles and lines. If you were looking at real circles and lines in practical applications, you would truncate after a few dozen decimal places at most. Given that we have computed billions of decimal places, the distinction between the abstract and practical is important to remember.

Will Zeno's Paradox [mathacademy.com] no longer be a paradox since it would no longer be about traveling an infinite series of infinitely small distances but rather traveling a large finite number of miniscule 'space grains'?

This was resolved when analysis (calculus) was formalized. Read more about it at MathWorld.

Could the relativity of time be more about different sizes of 'time grains' and a little less about where an observer might be standing? The rate of passage of 'time grains' being universally constant but the size of the grains dependant on local conditions?

There are some good books out there that are accessible to anyone with a bit of knowledge about relativity and quantum mechanics. See my tangentially related post for some reading references. I particularly enjoyed Smolin's book.

Interesting question, but I'm afraid not. Pi is a mathematical abstraction, defined as the ratio of a circle's circumference to its diameter. These are idealized, mathematical circles and lines. If you were looking at real circles and lines in practical applications, you would truncate after a few dozen decimal places at most. Given that we have computed billions of decimal places, the distinction between the abstract and practical is important to remember.

Will Zeno's Paradox [mathacademy.com] no longer be a paradox since it would no longer be about traveling an infinite series of infinitely small distances but rather traveling a large finite number of miniscule 'space grains'?

This was resolved when analysis (calculus) was formalized. Read more about it at MathWorld.

Could the relativity of time be more about different sizes of 'time grains' and a little less about where an observer might be standing? The rate of passage of 'time grains' being universally constant but the size of the grains dependant on local conditions?

There are some good books out there that are accessible to anyone with a bit of knowledge about relativity and quantum mechanics. See my tangentially related post for some reading references. I particularly enjoyed Smolin's book.

Mathematical proofs follow from logic giving them a certain sense of certainty. If I were to follow the proof, or you, or anyone else, we would all come to the same conclusion.

The skeptecism in using a computer comes up when we let a chip 'think' for us (or rather, just follow the steps). A mathematician may argue that while a human's logical argument is always sound in a formal system (let's just ignore Godel shall we) there is no guarantee that the same will be true in a computer simulation. There is no guarantee that a couple of extra electrons won't pass through some transistor giving me a 2 instead of a 1. There are a lot of things that can go wrong. We can run the program a million times and be reasonably sure we got the right answer, but never a 100% sure.


In reply to sql*kitten:
Computers can be used to prove infinite cases, if the problem is approached in the right way. Look up the proof of the Four-Color Theorem to see what I mean.

---I don't understand. They're comparing a bunch of X Servers versus a bunch of Dell PC's?

That's what it seems. Yeah, I know. The compairison sucks.

---What about the guy who's playing MP3's at his desk?

What? Lopster and XMMS arent good enough for him?

---What about the guy who wants to sync to his Palm Pilot?

There's already good sync software for Linux. Just un-endorsed. Hell, They might actually make a "legit" tool if stuff like this happens.

---What about the guy who's using Messenger?

There's buttloads of tools for IM on Linux.

---What about the guy who *NEEDS* a specific piece of software to communicate with his peers?

In limited cases, WIndows is the only answer for now. But as sysad, you could put heavy pressure on a company who does such.

---What about the guy who's burning DVD's of classroom presentations?

Get him a Mac. Most unix dudes could get one working.

---What about the guy who wants to run mid-priced shrink wrapped applications like Mathematica or MATLAB or IDL (all probably less than $10,000 for a single user license, but could get expensive for a big machine).

OK... Your point ?

---What about the guy who runs small simulations -- the kind of thing a reasonable desktop could do in an evening or a weekend? People who run computer centers often complain about 40 hours of computer time on the big boxes.

Help his department build a small cluster for job crunching. COuld even be a "beowulf" cluster if his apps support it. Then he could 'job' out time to other departments. That'll avoid cpu munchers on the main system.

---In short, what about all the flexibility that the Personal Computer gives the user? Why ins't that included in their "TCO" at all?

How about the flexibility of "use the tool that works for the job"? Trust me, you really dont NEED windows anymore.Look at all 3 links at your Math program question.

HOWEVER, it is worth noting that packages like Mathematica and Maple can do *some* exact calculation solutions, for example some algebraic problems (see Wolfram's discussion) and a few integral problems that happen to have known "named" transcendantal solutions (you know, Pi, Euler's number, and so forth). Beyond that, you're in the realm of floating point and all the hassles that entails -- a subject worthy of a quarter's or semester's study in understanding how these errors propagate and how to minimize the impact of such inaccuracies.

Here's an interesting exercise that one of my professors pointed out:

int n = 4;
while (n is the sum of two primes) n = n + 2;

The question "is n the sum of some two primes?" is of course always computable in finite time; just try all the prime numbers less than n/2 until you find one that is different from n by another prime number.

If you can show whether this program halts or not, then congratulations, you've solved the Goldbach conjecture, one of the most famous open problems in mathematics.

As others have pointed out "rows" are probably "series."

It is said that mathematics is the universal language. It is still damn hard to have a conversation about it if both parties don't speak the same language.

As others have pointed out "rows" are probably "series."

It is said that mathematics is the universal language. It is still damn hard to have a conversation about it if both parties don't speak the same language.

Its not the number of sides thats an issue but the area to perimeter ratio(this is known as the isoperimetric problem)

(Disclaimer: I didn't write these)

Q: What's purple and commutes?
A: An abelian grape .

Q: What's yellow, and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

(with links for the math-impaired)

(Disclaimer: I didn't write these)

Q: What's purple and commutes?
A: An abelian grape .

Q: What's yellow, and equivalent to the Axiom of Choice?
A: Zorn's Lemon.

(with links for the math-impaired)

Formula for computing the n-th Fibonacci number directly

So it is parallelizable.

Get yer Mathematica for Linux right here..

or if you prefer Matlab for Linux...

or perhaps Maple for Linux?

(I haven't used any of these on Linux; I've used Matlab on a Sun back in the days of fvwm2 so I'm *guessing* it will run under more modern window managers without having to mess with anything..)

Water is a very bad conductor of heat without good convection - you can boil water in the top half of a test tube while ice is happily sitting at the bottom - wrapped in gauze to make it sink, obviously.

Just because my desk calculator performs multiplications faster than me, doesn't mean that it is better at mathematics than I am.

I seem to recall a story about another young Irish student who had developed a "revolutionary" encryption engine a while back. That was largely all claim and no solid documentation as well, and what has become of her efforts since then? Not much, not even a single update.

Bullshit. Get your facts straight before you malign someone. Sarah Flannery

• won the Ireland's Young Scientist of the Year, and
• the European Young Scientist of the Year awards,
• was awarded a third-place Karl Menger Memorial Award from the American Mathematical Society and a fourth-place Grand Award in Mathematics,
• won Intel Fellows Achievement Award,
• wrote a paper on her algorithm, with a postscript exposing a successful attack,
• wrote a book, In Code: A Mathematical Journey, on her experiences (5 stars, 13 reviews, sales rank=35K).

She used Mathematica, so the Wolfram website has review of the book.

Here's a quote from Bruce Schneier in his 15 Dec 99 newsletter .

To me, this makes Flannery even more impressive as a young cryptographer. As I have said many times before, anyone can invent a new cryptosystem. Very few people are smart enough to be able to break them. By breaking her own system, Flannery has shown even more promise as a cryptographer. I look forward to more work from her.

All of this was easily found with a Google search that garned 24,000 hits.

The encryption story wasn't snake oil, and had very solid documentation. Sarah Flannery won Irish young scientist of the year, and subsequently the EU-wide prize, for her work. Her paper is here.

The Cayley-Purser algorithm she developed was subsequently shown to have security flaws; I don't recall if this was before or after the EU prize, but thats immaterial, the work was original and interesting, and worth a prize for a 16 year old!

She has subsequently written a book , which is a pop science introduction to crypto, and I understand from the blurb she's now studying maths at Cambridge.

-Baz

We all remember the Flannery episiode, right. She was awarded the first prize at the Irish Young Scientist compition in 2000 for work on speeding up the processing time of the RSA algorithm. I remember slashdot covering this (although I can't find the story) but I also remember reading that it made breaking the encryption almost trival. Still the IYS award is a compition thats been running for 30-40 years now and is a credit to our small corner of the world.

Think about the space required to store that many primes...your method of "trial division" is also known as "brute force".
Here is a very fast program that generates primes using the concept of Wheel factorization
Wheel factorization sounds like a neat way to factor composites, but I tried it and it cannot compare to the quadratic sieve or the number field sieve

A page on factoring algorithms

burris

The fact of the matter is that the dominant style of programming (procedural) is pretty difficult to understand. The basic problem is that there is a lot of stuff going on behind the scenes that is not explicitly visible to the programmer. And so, if you want to investigate the behavior of a little piece of code you end up needing to run the context of the entire program, because that little piece depends on a the initialization of a pretty complex state.

The very proccess of compilation and testing is pretty time consuming and so people tend to test only at "milestone" stages - but adding a significat chunk of code can greatly increase the possible number of states the program can have, and typically only a tiny tiny fraction of those can be explored by the tester.

I know a number of programming languages, and for a few years I've been vaguely considering getting involved in open source - like most others, usually to improve an existing program to suit my purposes better. But invevitably it becomes a major effort to just figure out the structure of the program - what source files do what kind of thing, how the various modules are related and what sort of information they pass to each other in what order. Pretty obvious questions that are really hard to answer without painstaking manual labor.

Procedural programing is partly to blame, but also very important is the fact that most programming languages are generally lacking an overall organizing principle - so that for example there is a distinction between data and functions (like in c), or between functions and the code defining them (like most oo languages). BTW, the consequence of the latter case is that its hard to perform operations on the source code that try to discover structure, or automation tasks that could be used in something like for example a debugger.

At a meta-level, consider this: What does it say about the languages we use that it seems impossible to develop a systemitized way of discovering and teaching ourselves how a given piece of code works?

For an example of how things can be different, see the Mathematica programming language. Basically its like regular expressions on 'roids - you can transform absolutely anything according to patterns, which is made possible by the fact that everything is an "expression" which has a certain simple structure. Furthermore, every expression is self-contained, which makes it easy to pull out little sections of a program and test them with various input (also easily generated).

Of course the 1.8k pricetag is a bit steep - but they've been talking about releasing the language component for years and I predict its going to happen soon. So check it out.

Energy, as far as we know, does not generate gravity

I beg to differ: If we believe in GR and Einsteins field equations, then space-time curvature (i.e. gravity) is determined by the energy-momentum tensor (i.e. rest-mass AND energy, esp. energy of fields). Therefore, both rest-mass and fields generate gravity.
Einsteins field equations are, e.g. here. The left side is the space-time (Ricci-tensor and Ricci-scalar) and the right side is the generating source term: T the stress-energy tensor, it includes rest-mass of particles and electric and magnetic field energy. Unfortunately T also includes the metric tensor, i.e. gravity acts back on itself, making the equation non-linear and solution highly complicated.

Actually, such an effect is predicted by general relativity. It's called gravitomagnetism. If NASA ever launches Gravity Probe B, it should be able to measure it.

Incidentally, the Doppler effect for gravity is more complicated than just compression, because the gravitational field is a tensor, not a scalar. Not only does it "compress", it also "twists", which makes it act differently on fast particles than it does for slow ones. A very similar effect holds for the electric force - when an electron moves very fast (i.e. generates a current), the electric force begins to twist in such a way that it affects moving charges (especially charges moving parallel to the original charge) in a different way from static charges. This is of course just the familiar phenomenon of magnetism, and what I just described is a simple consequence of Ampere's law.

And yes, magnetism travels at the speed of light too. Have you heard of "electromagnetic radiation"? What do you think light is, anyway?

Terry

The Fields Medal is basically the Nobel Prize for Mathematics (since there is no Nobel Prize in that category). It's awarded every four years. Mathworld has some more info.

Dr. Yorke has found the universal mechanism underlying such nonlinear phenomena.

Can someone clarify what part Mitchell Feigenbaum played compared to Yorke and a likely reason why Feigenbaum wasn't included in this prize?

See also The Feigenbaum Discovery and of course James Gleick's book Chaos.

• How can a range nearly equal to that of one of the factors itself be considered scientific?

• What is "95" percent level of confidence" based on?

• How do we know we're looking at "old" star clusters?

• Couldn't they have been reformed once or twice in the expanding and collapsing process?

• How will we ever guarantee that we can see enough of the picture to know we have a statistically representative sample?

• How can a range nearly equal to that of one of the factors itself be considered scientific?

• What is "95" percent level of confidence" based on?

• How do we know we're looking at "old" star clusters?

• Couldn't they have been reformed once or twice in the expanding and collapsing process?

• How will we ever guarantee that we can see enough of the picture to know we have a statistically representative sample?

The "classic" way (due to Hubble) to guess at the "age of the Universe" was as follows:

1. If we observe galaxies outside the Local Group, we see their light as being red-shifted. This indicates that they're moving away from us with some speed.
2. There is a simple relation (called Hubble's law) between the recession speed v, and the distance r between us and the galaxy. This is v = Hr where H is a constant.
3. Nothing can travel faster than the speed of light. So stick v = c into the above equation, and see what r is. Call this "the radius of the Universe".
4. The "age of the Universe" is the time that a photon would take to travel a distance r

Stick all that together, and you get t = 1/H. The problem being that finding H is fairly difficult - we can't accurately find distances to far-away galaxies. Estimates range from 50 km/s/Megaparsec to 100 km/s/Mpc

So how else could we measure the "age of the Universe"? Well, we could work out the age of the oldest stars we can see, make some guesses at how long they would take to form from hot matter, and take that as our "age". After quickly RTFA-ing, I think this is what they've done, with a revised method to obtain the age of a star.

Ofcourse, even numbers are skipped immediately. But as for the rest.. Computers use binary instead of decimals. So for a computer, numbers only end in 1 or 0.

But fortunately, in august last year, scientists from India found a way to test if a number is prime in a really (mathematically speaking) fast way (**). Still, before applying the algorithm they found, it is probably still even faster to first try a couple of frequently occuring cases (dividebility by 2, 3, 5, etc; the computer won't count digits, it will just skip every 2nd, 3rd and 5th number it tries).

Making lookup tables, like other posters already noted, is useless since they would be astronomically huge (between 1 and 2^2048, there are approximately 10^613 primes!(*) if you had every atom in the universe to store them on, you need to store 10^533 primes on each one!).

Besides, a lookup table of previous primes isn't needed. Once you decide the number x is a product of 2 (previous) numbers, you've already found x isn't prime, no matter of the 2 numbers it consists of were prime.

(*) according to Gauss, the number of primes less than x is appoximately x/ln(x).
(**) in O(ln(n)^12). for math/cs people, here is an article with a reference to the paper. (***)
(***) beware, prime testing isn't factorizing. so you can't use this algorithm on the key and get which factors it's made out of; you'll only get the knowledge that it ain't prime (which we already knew).

I would suggest doing some more research about Matlab and Mathematica (as well as Maple).

Matlab is mostly used for creation of and use of complex algorithms, DSP simulations, and other "heavy math" tasks. It's a great swiss army knife and integrates easily with most C compilers for compiled-performance (rather than interpreted). One of the many "modules" included with Matlab is a symbolic math package based on the Maple engine (see below).

Mathematica and Maple are little more than symbolic math packages. (Don't get me wrong, they can do A LOT, but neither comes close to the full Matlab package). Each has its pros and cons, but either will do quite well for any math undergrad university student and most grad students. The merits of Mathematica vs Maple are often heavily debated on the usenet and in other forums.

Matlab, Mathematica, and Maple are all very powerful packages... they can do **WAY** more than any of the lame "MathCAD" type apps you probably used in high school.

All three are available for Windows, Mac OS X, Linux, and most flavors of Unix (Solaris, AIX, IRIX, HP-UX, Tru64). Each has a rather simple interface and "looks" like a native application with the exception of the Linux/Unix version of Matlab -- it's a quick port from Windows with some lame crossplatform toolkit. Its GUI widgets look as though they're straight out of Windows. This cannot be changed without a lot of hackery. Despite the ugly interface, I would recommend Matlab for students... the student price is about the same as that of Mathematica or Maple, yet includes so much more (plus all of the symbolic math features straight from Maple 8).

If you don't need (or don't want) all that Matlab offers, Maple may do the trick for you. I used Maple 6 for years and only recently moved to Matlab (for compatibility reasons). Maple, even the current Maple 8, is a clean lightweight application. It's easy on the disk and ram, and even easier on the CPU. And, (IMHO), it does just as much as Mathematica would for me.

Also, all three have a full-featured command line interface alternative to their GUIs. Learning how to key in equations without the mouse and tool palettes will help you in the long run -- you'll be able to enter data much faster. Brushing up on TeX and/or MathML will also prove helpful.

These days, my workstation runs little more than Matlab, LyX, and sometimes Framemaker.

World of Science and World of Mathematics covers Astronomy, Biography,Chemistry, Mathematics,and Physics. It's a public service resource by Wolfram Research, the Mathematica people.

Throwing karma to the wind... that's not a corollary.

From Mathworld:

A corollary is "an immediate consequence of a result already proved. Corollaries usually state more complicated theorems in a language simpler to use and apply."

You're proposing, I don't know, another axiom or something.

(All right, I apologize. I'm a geek.)

In any case, it's probably correct.

No. But possibly "normal" is.

They will not find simple repetition. That would mean that pi = p/q for two ints p and q and we know for a fact that pi is not only irrational (there is no p and q) but that it's transcendental (not the root of any polynomial with integer coefficients).

However, that doesn't mean they won't find a pattern. It isn't known whether pi is a normal number. A normal number is an irrational number where each digit 0-9 occurs 1/10 of the time, each pair of digits 00-99 occurs 1/100 of the time, etc. Pi is believed to be a normal number because it looks like one, but nobody has proven it.

A normal number may or may not occur in a predictable sequence. For example, this is the Champernowne constant:

0.12345678910111213141516171819202122232425.....

This is irrational and transendental and still there's an obvious pattern.

This is the Copeland-Erdos constant, which is like the Champernowne constant except you only use primes:

0.235711131719232931374143475359616771737983...

The Thue constant is an example of an irrational, transcendental number that is not normal. The nth digit is 1 if n is not divisible by 3 and is the complement of the (n/3)-th bit if n is divisible by 3. This is what it looks like in base 2:

0.110110111110110111110110110110110111110110...
(and in base 10: 0.85909979685470310490357250...)

Try writing that as a fraction.

It's possible that on the way to the trillionth digit of pi they might find that something weird happens, like there's no digits except 0 and 9 after a certain point, but I doubt it.

They will not find simple repetition. That would mean that pi = p/q for two ints p and q and we know for a fact that pi is not only irrational (there is no p and q) but that it's transcendental (not the root of any polynomial with integer coefficients).

However, that doesn't mean they won't find a pattern. It isn't known whether pi is a normal number. A normal number is an irrational number where each digit 0-9 occurs 1/10 of the time, each pair of digits 00-99 occurs 1/100 of the time, etc. Pi is believed to be a normal number because it looks like one, but nobody has proven it.

A normal number may or may not occur in a predictable sequence. For example, this is the Champernowne constant:

0.12345678910111213141516171819202122232425.....

This is irrational and transendental and still there's an obvious pattern.

This is the Copeland-Erdos constant, which is like the Champernowne constant except you only use primes:

0.235711131719232931374143475359616771737983...

The Thue constant is an example of an irrational, transcendental number that is not normal. The nth digit is 1 if n is not divisible by 3 and is the complement of the (n/3)-th bit if n is divisible by 3. This is what it looks like in base 2:

0.110110111110110111110110110110110111110110...
(and in base 10: 0.85909979685470310490357250...)

Try writing that as a fraction.

It's possible that on the way to the trillionth digit of pi they might find that something weird happens, like there's no digits except 0 and 9 after a certain point, but I doubt it.

They will not find simple repetition. That would mean that pi = p/q for two ints p and q and we know for a fact that pi is not only irrational (there is no p and q) but that it's transcendental (not the root of any polynomial with integer coefficients).

However, that doesn't mean they won't find a pattern. It isn't known whether pi is a normal number. A normal number is an irrational number where each digit 0-9 occurs 1/10 of the time, each pair of digits 00-99 occurs 1/100 of the time, etc. Pi is believed to be a normal number because it looks like one, but nobody has proven it.

A normal number may or may not occur in a predictable sequence. For example, this is the Champernowne constant:

0.12345678910111213141516171819202122232425.....

This is irrational and transendental and still there's an obvious pattern.

This is the Copeland-Erdos constant, which is like the Champernowne constant except you only use primes:

0.235711131719232931374143475359616771737983...

The Thue constant is an example of an irrational, transcendental number that is not normal. The nth digit is 1 if n is not divisible by 3 and is the complement of the (n/3)-th bit if n is divisible by 3. This is what it looks like in base 2:

0.110110111110110111110110110110110111110110...
(and in base 10: 0.85909979685470310490357250...)

Try writing that as a fraction.

It's possible that on the way to the trillionth digit of pi they might find that something weird happens, like there's no digits except 0 and 9 after a certain point, but I doubt it.

They will not find simple repetition. That would mean that pi = p/q for two ints p and q and we know for a fact that pi is not only irrational (there is no p and q) but that it's transcendental (not the root of any polynomial with integer coefficients).

However, that doesn't mean they won't find a pattern. It isn't known whether pi is a normal number. A normal number is an irrational number where each digit 0-9 occurs 1/10 of the time, each pair of digits 00-99 occurs 1/100 of the time, etc. Pi is believed to be a normal number because it looks like one, but nobody has proven it.

A normal number may or may not occur in a predictable sequence. For example, this is the Champernowne constant:

0.12345678910111213141516171819202122232425.....

This is irrational and transendental and still there's an obvious pattern.

This is the Copeland-Erdos constant, which is like the Champernowne constant except you only use primes:

0.235711131719232931374143475359616771737983...

The Thue constant is an example of an irrational, transcendental number that is not normal. The nth digit is 1 if n is not divisible by 3 and is the complement of the (n/3)-th bit if n is divisible by 3. This is what it looks like in base 2:

0.110110111110110111110110110110110111110110...
(and in base 10: 0.85909979685470310490357250...)

Try writing that as a fraction.

It's possible that on the way to the trillionth digit of pi they might find that something weird happens, like there's no digits except 0 and 9 after a certain point, but I doubt it.

Also check out Pi at Eric Weisstein's World of Mathematics.

Well, at least pi is a computable number - other interesting ones, like Chaitin's Omega, aren't, at least not ahead of time. In fact, if I recall correctly, it is even (provably) possible to construct Turing machines for which no single digit of Omega can be computed at all, but I'm not really sure about this anymore.

Pi is represented usually by a fraction or relatively simple equation, it's just the division that makes the number go on for ever.

Nope. If pi was rational (a fraction), it wouldn't go on for ever without repeating. (reference)

In fact pi is irrational, i.e. there are no integers p, q such that pi = p / q. (proof)

You can approximate pi as a fraction, which is what projects like this do. (pi is approximately equal to 31/10, or 314/100, or 31416/1000, or ... but these are just approximations; 22/7 is a good enough approximation a lot of the time, but that's just an approximation too)

No, Einstein never denied that Quantum Mechanics fits the known experimental data perfectly or claimed that further experiments would show that QM was wrong. Einstein was himself one of the founding fathers of QM and a master in using the predicting powers of the theory, predicting QM-phenomena like LASERs and Bose-Einstein condensation, decades before they were seen in any lab.

What Einstein never accepted was the interpretation given to the mathematical framework of QM by Bohr, Heisenberg, Born and others. Einstein was not alone in resisting the philosophical/physical interpretation by the "Copenhagen school" , he was joined by people like Planck, Schroedinger, and de Broglie who all knew a bit about QM. (But as always, the old generation dies out and the new generation have gotten used to the new world view.)

Einstein believed in a deterministic universe (just as Newton, Laplace and the other classic mechanics guys before), where when you knew the starting conditions perfectly, you could calculate what happened. This is how to understand the statement "God does not play with dices". "God" knows what is going to happen, He does not only know the odds are for something to happen. This is contrary to Bohr who claim that "God" (or the physicist) can only know the different possible outcomes from some given starting condition and the probability of the different outcomes. According to the uncertainty principle "God" can not even hope to know the starting conditions perfectly.

The answer to QM by Einstein was the so-called "hidden variables" theory, variables that behave in a deterministic way but lead to behaviour that looks random in the experiments that were used to "prove" QM. Einstein also made famous thought experiments to show the inconsistency in the logic of the Copenhagen school, like the EPR paradox.

Today most physicist believe Einsteins objections to QM has been shown to be wrong, and Bohr's interpretation has become the dogma. But who knows? Newton thought light consisted of particles, but was proven wrong. Then Einstein showed that light can be seen as both waves and photon-particles. So, maybe in some hundred years Einstein's objections to QM can be shown to be a "bit" correct :-).

-- MarkusQ

P.S. If you don't get it, don't waste a lot of time trying to figure it out.

I sincerely hope that soon small-to-medium enterprises can own supercomputers. With all the low budget physics stuff going on at Universities around the world, cheap supercomputing can only be a good thing.

Actually they can with software like that from Dauger Research, Project Appleseed and Wolfram Research with gridMathematica

The cool thing here is that this code can be run on all of the desktop computers that already occupy companies and universities world wide allowing for easy access to supercomputer level computational speed (for those problems that can be attacked using parallel computation of course) using the same computers normally used for productivity.

Very cool.