You know, you just did. At the very least, you cleared up a misconception I created… When I spoke of numbers who may contain all finite series of numbers, I said these were irrational numbers. Fact is, I spoke of transcendental numbers.
I realise now, also, that there is no such law stating that all finite series of numbers are contained in a transcendental number. Is the following number transcendental?
(In this example, I write Pi_n to say, ‘the nth digit of Pi’, and c_n to say, ‘the nth digit of our supposedly transcendental number’.)
| Pi_n if Pi_n <> 9
c_n = |
| 0 if Pi_n = 9
What do people think? This number seems like it’s transcendental to me, even though there is a rule deducing its digits; it’s the fact that these digits are deduced from a transcendental number.
Clearly, if the above example is transcendental (and I really don’t know if it is), then it does not include all finite series, because any series of digits with a 9 in it is not included.
I use a quite practical definition of “random”, which may be a failing: by random, I mean that there is no human means currently available to deduce one digit based on the preceding ones.
In that sense, a coin’s toss is random, even though it depends largely on the initial position of the coin, the force with which it is tossed in the air, the viscosity of the tosser’s skin, his altitude, atmospheric pressure, etc.
If all these factors were known with enough precision, it can be argued that we could predict the outcome of the coin’s toss. Then it would no longer be random. It is our incertitude that makes things appear random, and generating a true random number is more difficult than it seems. Like I mentioned in a previous post, even numbers generated by most computers are not random, but pseudo-random, i.e., it’s actually impossible on a practical level to make a difference. (Alright, so in numerical simulations in grad school Chaos theory it was’t, but hey. Who said grad school was practical? )
Is the following number random?
837640916027365
It may seem like it is, and establishing a correlation between two digits is gonna be a tough job… I typed it at random on my numpad, and you’d have to know the position of my fingers on the keyboard, the type of keyboard I have, my current mood, whether there was noise distracting and influencing my typing, etc. to determine the actual series of events that took place and be able to extrapolate one digit from the preceding ones.
In my book, that’s random enough… Alright, pseudo-random.
Cervaise, 2/3 is a rational number. Rational numbers are those that can be expressed as the quotient of two integers (including the integers themselves). They happen to be just those whose mantissas (the part right of the decimal point) are either finite or periodic. All this is obviously true of 2/3 = 0.66666…
Irrational number are… well, the others, provided they can (in principle) be written in decimal form. They have infinite (thus “in principle”), non-periodic mantissas and cannot be expressed as a quotient of integers. The square root of 2 is a classic example.
Rationals and irrationals together are called real numbers. Fancy numbers like the complex ones, which can’t be expressed in decimal form at all, are not real and thus neither rational nor irrational.
To be exact, complex numbers also include the real ones. In the previous paragraph, I meant the “really complex” ones, which are called irreal (I think; not sure in English) whose imaginary component is non-zero. (Complex numbers have a real component and an “imaginary” one, the latter being a (real) multiple of the “imaginary unit” i. i is defined as the square root of -1, a number that is beyond most people’s comprehension, including mine. But they work, and they are useful.)
Your definition of transcendental numbers seems to be okay.
Confused enough? Summary:
All complex numbers are either real or irreal.
If real, they can be transcendental or not.
If non-transcendental, they can be rational or irrational.
If rational, they can be integer or not.
If integer, they can be natural (i.e. positive) or not.
Exactly right. The reason they are called
rational numbers is that they can be
expressed as the RATIO of two intergers,
i.e. as a fraction. So 2/3 is quite definitely a rational number.
“. For
instance, knowing a string of 100,000 consecutive
digits of pi won’t give you any edge in guessing the
100,001st. Since that’s generally the sort of thing we
mean by ‘random’.”
Sure it would.
There is a one in ten chance that it’s a digit from one to ten.
A few years ago, an expression for pi was discovered that lets you calculate any digit of it without having to calculate the intervening ones (so long you’re using base 16). (Thus, ditto if your base is any power of 2).
This property gives circumstantial evidence aginst pi’s digits being random.
I’m sorry, but I’ll have to see it before I believe it. People are still flaunting how they manage to calculate Pi to the 1,000,000,000th digit.
Cause otherwise, you could just calculate all digits of Pi separately in base 16 (why base 16???) and convert the whole thing to base 10, which is much easier than calculating 1,000,000 digits to get the 1,000,001st one.
I’m sorry to sound so doubtful, but that’s because I am.
To Holger et al: I stand corrected. Serves me right for poking my head into something that interests me, but to which I can’t really rationally contribute.
I do have an explanation, though. I was reading this thread, fascinated but with wrinkled brow; I went off to an online encyclopedia to check some terminology, and found myself with the classic “Oooohhhhh!” reaction as the light bulb went on. I figured I’d share what I found, but as it turns out, what really went on was Christmas lights, and although they blink attractively, they don’t illuminate the room.
So shame on me for trying to take a concise-to-the-point-of-meaningless definition and translate and expand it for other readers, in the process mangling it beyond recognition. :o
It works for base 16 because they found a formula that works for base 16. At the time I read about it, it wasn’t known whether there’s a similar base-10 algorithm.
If I’ve read this thread correctly (and I may not be), there’s some confusion about transcendental numbers. Forgive me for quoting some definitions:
An algebraic number is a complex number that is a root of an algebraic equation
f(x) = a_ox^m + a_1x^(m-1) + … + a_m
where the a_i’s are all rational numbers and m is a finite positive integer. All integers, rational numbers, and a subset of irrational numbers are algebraic:
for instance, the square root of 2 is algebraic.
Transcendental numbers are all the numbers “left out” of the algebraics. For instance, pi cannot be the root of an algebraic equation as defined above.
It can be shown that “most” (real or complex) numbers are algebraic, but proving that involves a branch of math called measure theory.
Right, a transcendental number with that property is called “normal” – see URL above.
Only a few contrived numbers, not including pi, are provably normal.
One way to get the next digit is to search for the known string. Take this random-looking number: 8831311722359377438741462183. I bet you can guess the next digit. (That string could occur twice and cause ambiguity, but practically speaking it can’t.)
For password/cryptographic purposes, even a sequence of coin flips is not random once it’s been made public. Size of size space is IMO the least tricky way to look at matters.
This definition, by the way, does not imply “normal” in the above sense. Say I generate successive digits to be 1 or 7 by flipping a coin: that’s unpredictable, but will never contain “123”.
A deep and rather satisfying definition of randomness (for infinite sequences – I think there’s no good definition for finite ones) is Martin-Lof randomness: basically, any program to generate the sequence has to get about as long as the sequence itself. Pi is obviously not this sort of random.
Most are not algebraic, that is. Just because each algebraic corresponds to a finite series of integers, and those are countably infinite (because pairs are countable and by induction); whereas the reals are uncountably infinite, so infinitely more numerous.
Then of course if you want to say anything about algebraics being “of measure zero” you’re going to have to drag in measure theory after all.
I know this is a stupid question, and you’ll hate me for asking, but why would anyone care to find out the 10 millionth digit of pi? Is there any practical use of that digit? If not, why not stop at 200 digits? Isn’t that small enough?
A common mistake people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools. -Douglas Adams
Regarding pi— I like apple-with vanilla ice cream if you’ve got it. The finer things in life are not found on mathematical tables, but on kitchen tables. :}
I’ve always used a practical definition of “random” - it is the quality we attribute to events whose outcome we cannot, due to lack of information or time, acccurately predict before they occur.
A coin flip is often portrayed as “random”. Certainly such an event is subject to the laws of physics and is therefore not “truly random”. If we knew enough about the wind, force applied, distance to the landing surface, construction of the coin, etc., and had enough resources we could predict coin flips with 100% accuracy. Do we therefore call a a coin flip “pseudo-random”? Certainly not!
Many things we once considered random can now be predicted with reliable accuracy due to increased knowledge. Birth defects/genetics comes to mind.
Conclusion: Randomness is a matter of perception that is inversely proportional to knowledge. We will never be omniscient, and as a result there will always be the appearance of randomness.
Attention, everyone who enjoys looking for patterns within a long series of apparently random digits — I strongly recommend reading Carl Sagan’s novel “Contact”. I think you’ll love it!
[The movie was okay, but Patterns Within Random Digits was a major aspect of the novel, whereas the movie version put the focus on the more general-interest action-type stuff.]
It may be true of classical reality, but if you consider quantum mechanics, true randomness exists. The wave of probability of an electron, for instance, is a true random function, and only by considering things globally, that is, at a large scale, do they become predictable again.
I’m speechless. They’ve calculated the 5 trillionth hexadecimal digit of Pi with this algorithm, and the only reason this algorithm is used instead of the decimal formula is for speed, given afterwards, converting a string of digits from hex to decimal is trivial.
This is a major advance, and the actual algorithm is relatively simple…
You’re going to have extreme trouble finding more than 85 3’s in a row in pi. There are only about 10^85 electrons in the universe, so it will be hard to represent 10^85 digits in a computer, even if it is very economical in storage (one bit/electron).
Hang on, George. I don’t need to save 10^85 digits to find 85 3’s in a row. All I have to save is enough values to save the state of my calculation (typically this doesn’t take too much storage) and the previous 84 digits. In the 1 - 1/3 + 1/5 … series that Cecil mentioned, I’d only need to save enough digits that I could adjust the previous digits in case I needed to round up or down (for example, …26599 + …00001 would result in …26600), and the last integer that I inverted and added to the series. The sequence of digits you need to save to do the addition without error might be long, but it’s unlikely to be as long as the sequence itself.
Having said all that, if it is indeed possible to get a sequence of 10^85 9’s (or 0’s) in a row, then you would have to save them all to be able to continue the calculation. So far, though, that hasn’t been a limiting factor for anyone.
At the very least, you cleared up a misconception I created… When I spoke of numbers who may contain all finite series of numbers, I said these were irrational numbers. Fact is, I spoke of transcendental numbers.