My post is too long for a.cc to process correctly, so I had to leave out some quotes relevant to my responses. Sorry about that.

Of course it is a contradiction. If you can represent more natural numbers with one base than you can with another, that makes the other base less valid as a choice for what you want to use.

Powers of 10 can represent more numbers than powers of 2, so they are a better representation of numbers.

Computers may use binary, but guess what - computers are intended for human use, and humans use decimal. So for mathematical purposes, they are broken.

You're the one spouting nonsense. Expecting perfect string decimal to data conversion and back is a reasonable expectation for my purposes. There's no good reason that 5.1/0.1 should ever equal something other than 51. You act as if I expect the common implementation of floating point numbers to do what I want it to do. I've said more than once I'm looking for a data type that will perfectly convert to data and back to string, so it should be completely obvious by now I'm not intending to use conventional floating point numbers. If you don't get that, too bad.

There are always twice as many total integers as there are even integers. The fact that the count of both approaches infinity is irrelevant.

No, it can't. Subtract infinity from both sides and you have 0 == n? Since the condition was that n is non-zero, it is obvious 0 doesn't equal n. Infinity + 1 is not infinity. Infinity times 2 is not infinity. The entire number system breaks if you treat it that way, so I won't.

Representing exact decimal numbers exactly in data is not analagous to representing irrational numbers.

@ Audric
Here's what I originally asked for :

I may have misspoke in the last sentence when I said "to double and back", but it should have been fairly obvious that I meant "to data and back", however it would be represented. I have also clarified that several times and stated doubles aren't sufficient for my purposes, so for the last time - I want a data type that can move from string to data and back without any loss of precision. I haven't fully tested it yet, but it appears that the MAPM library can do this for me. So unless you have any suggestions for libraries that could do the same thing as well as or better than MAPM, further comments are likely pointless.

Finally! Someone who actually comprehends what I've been asking for all along.

I'm not sure what the underlying representation of the MAPM data type is, but it appears to be smart enough to know that "5.1" means 5.1 and not 5.0999999999999996. I don't believe that any rounding has occurred during conversion to data or back to string, as default precision is 30 digits, and the only rounding done is during division or else explicitly.

It is for a function expression evaluation tool. You can try out the working prototype over at the

batch files, graph plotting and

thread started by William Labbett. If you run the expression 5.1/0.1 inside the program you can see that it fails horribly at extracting an exact answer (which clearly exists). The current implementation still uses doubles as the data type, and that is why it fails. I want to test MAPM a little more before I convert everything over to the MAPM data type.

@ James Lohr
Trying to achieve accurate computations is the wrong thing? I bow to your wisdom great sir. Which part of my understanding is flawed? The part where I understand floats and doubles can't do what I want, or the part where I asked for a data type that could, or the part where negative powers of two can't represent a decimal number exactly 100% of the time?

Did you miss the part where I said I wasn't concerned about the efficient use of memory? Who cares that base 10 can represent decimal numbers exactly? Oh well, just the people who desire accurate calculations. Never mind them though. For the last time - negative powers of base 2 will never do a better job of evaluating decimal expressions than base 10 would. That's beside the point though - see my next answer to you.

What is elegant about inaccurate calculations (from the use of base 2 with negative powers)? Nothing. You're the one who's naive if you think inaccurate calculations are acceptable where accurate calculations are both possible and desired.

Decimal numbers are what the majority of people use - not binary. The only crude approximation of a real number here stems from the fact that negative powers of two are used, and they cannot accurately reflect most decimal numbers less than one.

@ orz
As I said, I'm not concerned with irrational numbers, or with infinite precision. What I am concerned with however is accurate calculations where they are possible.

Use of positive powers is necessary if you want to use base 2 to accurately represent rational decimal numbers.

If you can express more numbers in a certain base than another, then as far as I'm concerned it is more precise. That said, base 210 is unnecessary since I'm primarily concerned with accurate representation of decimal numbers.

What exactly is it that I'm misunderstanding? You all seem to think I don't understand what is going on, but I've repeatedly shown that I do, otherwise I would be satisfied with the endlessly tiresome suggestion that floats and doubles are sufficient for my needs.

You're disregarding basic mathematics theory here

But infinity minus infinity is not 0, it's undefined

That your problem can easily be solved by just using doubles, without the need to use any of those libs. All you have to understand is how floating point numbers work, and when printing a floating point number, disregard the part that doesn't have the precission required, using rounding (IIRC, 64 bit doubles have 15 significant (decimal) digits, so if you use doubles that means if you limit your output to 15 digits you'll get an exact output (the same you'd get if you used a decimal representation and print a max of 15 digits)).

In your example of 5.0999999999999996. If you only print the first 15 digits, rounding the number, you'll get 5.1 printed.

That's how calculators work (Orz explained it in his second post). This is faster (because you're using a native representation for floats), and also takes less memory (to get the same precission, you need more memory if you're using decimal representation than if you use a binary representation).

Showing you that there are the same amount of even integers as integers does nothing to show you why you do or don't need a different data type. You miss my point entirely.

My point is that you are mistaken about fundamental truths and refuse to take correction. You speak as if you understand more about mathematics than somebody who knows much more than you. (I'm not speaking of myself.)

So you insult the mathematicians among us by pretending to know more than they, and then expect them to entertain a thoughtful and meaningful discussion about the topic at hand.

For fun here's a simple proof that there are as many even integers as integers. Give me any integer, and I will provide a unique even integer that maps to it.

Given integer n, I present you an even integer 2 * n.

If there were twice as many integers as even integers then I would run out of unique even integers to pair.

Again, back to my point. If you would argue against proven mathematics (i.e., the equivalent of saying 1 + 1 = 3), then your arguments against using floats or doubles will be perceived as the same sort of stubbornness, regardless if you have valid concerns.

I understand your reasoning, but your proof is one of intuition and is not mathematically valid. I cannot "extrapolate as far as I want" because infinity has no bounds in that context.

My proof relies on no tricks. We have two sets of numbers:

• Set A: all integers

• Set B: all even integers

We want to know if Set A and Set B are the same size. The mathematical way to prove that is to provide a one-to-one function that maps a single element in A to a unique element in B (and B to A.)

I have provide such a function (although admittedly I have not actually proved that it is one-to-one and onto, but I don't think you are disagreeing with that). It is mathematically sound, and covers the range of everything in the two sets we are discussing.

If you wish to argue, then we aren't even speaking the same language. Mathematics has rules, and if you don't wish to follow them, then there's no common ground for discussion.

I won't try to explain it to you any further. I'm not trying to be "insulting," but I cannot explain an entire course on number theory in a handful of posts. (And it's not like I remember much of what I once learned anyway...)

Here's a small article to get you started.

You are dealing with infinite set, so that is not an operation that means that A is not the same size as B.

You are saying something like:

1. I have infinite sets A, B, C.

2. A is proven to be the union of B and C

3. A is proven to be the same size as B

4. A is proven to be the same size as C

   
   

5. Based on my intuition of how I think infinite sets operate, I declare the mathematical proofs to be invalid.

You cannot throw your hands up at the conclusion and say "it cannot be." You must reshape your beliefs on what you have proven to be correct. Or you must show that you have incorrectly proven one of steps 1 to 4.

Your intuition that the result is a paradox because of your knowledge of how finite sets operate is not sufficient proof.

That's really just circular reasoning. That's precisely the statement that we are trying to prove. It's must like saying "1 + 1 does not equal 3 because 1 + 1 does not equal 3".

It can simultaneously be the same size if you can find a function that maps A to B and a function that maps A to B union C.

I present a simple mapping function for the latter:

• n => n

And we already have n => 2 n for set B, so now we have proven what you said is impossible.

You may find this interesting. A relevant piece:

That's not an infinite set. You are bounding it by '2n', so of course A is twice as big. But if Set A is the set of all integers, and Set B is the set of all even integers, then the mapping from B to A looks like. n => n / 2.

It's not about being politically correct, it's about being mathematically correct.

If you want to deal with "impossibly" large sets, then use this notation: {1, 2, ..., N}. However, that is not the same as infinite sets. Any conclusion you draw from such sets does not apply to infinite sets, even if N is really, REALLY big!

You can not entertain a discussion that is intrinsically mathematical and then decide on using your own definitions and rules, and expect any sort of favorable outcome.

Make n infinite then. You have to keep indefinitely extending the set to make it fulfill your mapping.

{2,4,6...2n}
{1,2,3...n} OOPS {n + 1,n + 2...2n} Is missing!

I guess we have to go up to 4n, I mean 8n, I mean 16n, so on, so forth, forever and ever over again...

You're comparing unequal ranges of numbers. Of course you can say that for every number n there also exists a number 2n, but that doesn't prove for every set of numbers from 1 to n there also exists n numbers from 2 to n by 2.

It just doesn't cut it for me. Sorry.

Well then if that's the way mathematicians deal with infinite sets, then I don't want any part of it, because it's just screwed up.

The cardinality of {2,4...2n} will never equal the cardinality of {1,2...2n}. It just doesn't hold up.

If you can find me a proof that 2*infinity == infinity, then maybe I'll shut up about this. Otherwise, it's probably not likely.

Append for orz
I posted after you did, but hadn't read it yet.

That was in response to this :

Sorry if it was confusing or not matched to the topic. But it mostly applies, as I'm not interested in precisely representing rational fractions. If I wanted that, I would be looking for a fraction library, not an arbitrary precision library.

Can you represent one tenth exactly using negative powers of 2? No. That's why I said that.

I guess I meant all real numbers that don't repeat. Sorry for the confusion.

I guess you were referring to this :

I guess I didn't actually know what that entailed, as I wasn't thinking of the number being rounded that way. Sorry if it seemed like I ignored you, I just wasn't perfectly clear on what that actually meant.

In any case, see my response to Oscar Giner, as it applies equally as a response to you :

@orz
See my edit in my last post.

So the value 2n as n approaches infinity doesn't equal infinity then? Neat trick.

Enlighten me then. What does infinity really mean?

Then WTF are we talking about?

Ah... Back to condescension. There's the old Matthew. I wondered where you went. You were actually being quite generous in your explanations there for a while. I knew it couldn't last though.

I will freely admit that the way infinite quantities are dealt with bothers me. If that makes me an imbecile in your eyes so be it. Meh.

The set of all even integers is a subset of the set of all integers. It can never have the same cardinality. It just doesn't make any fucking sense.

Perhaps. I'll just have to take your word for it. Just stop telling me that 2*infinity equals infinity so my brain stops hurting.

"Infinite" breaks down to "in" (meaning not) and "finite", meaning, well, finite. (or countable). If you could point out a number and say "that's infinity" all I have to do is add one to it to disprove that.

Look at it this way, you'd have to travel an infinite distance to sail a ship off the end of the world. Going twice as fast won't help.

BTW, that library I offered earlier uses binary, so I guess you wouldn't want it.

[EDIT]

This is the routine that decides how many decimal digits can be printed with reasonable accuracy in my bignum lib. I limited the digits to less in the trig example above to preclude someone else coming along and saying "I found an incorrect digit at the end of the 3rd result" or something.

void bn_maxdecset(void)
{
  double tmp;
  //log10(256) ~= 2.4082 but practically it leaves junk at the end
  bn_maxdec = (double) bn_fracbytes * 2.4082;
  //printf("\norgmaxdec = %d",bn_maxdec);
  tmp = (log(bn_fracbytes ));
  tmp *= 1.5;
  tmp = floor(tmp);
  bn_maxdec = (double) bn_maxdec - tmp;
  //printf("\nnew maxdec = %d\n",bn_maxdec);
  //fflush(stdout);
}

This is what I can't grasp.

Oops. Be careful here: If you talk about a line of 1cm, you talk about real numbers, not natural numbers. But, real numbers do have a different cardinality than natural numbers, which amounts to a mathematical proof (originally given by Cantor) that there are indeed more real numbers than natural numbers. So the real numbers do have a different, and greater, cardinality than natural numbers!

In contrast, the rational numbers have the same cardinality as the natural numbers. For a proof (which is giving an bijective map between the two sets) again see Cantor.

Want some more?

It was for a long time unknown if there is any cardinality between the natural numbers and the real numbers. This is known as Cantor's continuum problem. The answer is quite unexpected, and again goes very deep into mathematical logic: You can have both answers... in terms of commonly accepted axioms Cantor's continuum hypothesis is not decidable. That is, you can neither prove it nor disprove. This goes back to the work of Kurt Gödel during the 1930s or so.

The infinite sum is, to be a tiny little bit more precise, a notion for an (infinite) series of numbers that converges to the desired real number. Which is again approximation, where you can approximate the real number to any given degree of exactness by the numbers of that series. Which again means that 1) you don't get the exact value, and 2) you need more members of the series to achieve a better approximation...