Decimal versus binary arithmetic was Re: J4 - presentation/discussion on "Future of the COBOL Standard"
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From: Richard on 8 Apr 2008 01:16 Robert wrote: > The ideal numeration base is an irrational called phi or golden ratio, which is 1.61803... > The only digits are 0 and 1, which makes it ideal for computers. One would expect 'our' > integers to produce infinately repeating fractions while irrational numbers produce > terminated fractions. Not so. In phinary, EVERY number can be represented exactly in a > finite string of 0s and 1s, including e, pi and square root of 2. That's a very valuable > attribute. For more on this, see: In a finite string of 0s and 1s of length n 2^n different values can be be represented. Regardless of what those representable values may be there will always be another number that is not in that set of values, in fact there will still be an infinite number of unrepresented values. Just because pi, e and root2 can be represented exactly does not mean that "EVERY number can be represented exactly".
From: Richard on 8 Apr 2008 01:29 On Apr 8, 2:11 pm, Robert <n...(a)e.mail> wrote: > On Mon, 7 Apr 2008 11:08:23 -0500, "tlmfru" <la...(a)mts.net> wrote: > >Robert <n...(a)e.mail> wrote in message > >news:521gv394rgj59j3d8u86clvp0a8am6857n(a)4ax.com... > >> On Sat, 5 Apr 2008 10:54:00 -0600, "tlmfru" <la...(a)mts.net> wrote: > > >> >One very good reason that pure binary representation was not used is that > >> >decimal fractions cannot (generally) be expressed except as a repeating > >> >number in binary. Examples: 0.2 (decimal) = .00110011 ... (binary). .12 > >> >(decimal) = .000111 etc. (binary). > > >> To pay for something costing .50, you don't tear a dollar bill in half (a > >fraction), you > >> tender 50 pennies (an integer). > > >Nevertheless, if you're calculating what happened, you've spent half a > >dollar - unless you convert ALL the figures to pennies. > > No conversion is necessary. Figures are already in pennies. > > > > >> >To make correct arithmetic using only binary possible, it seems to me > >that > >> >ALL numbers must be implicitly scaled to remove the decimal point. If > >that > >> >is your meaning, Robert, then you haven't made it clear. Binary > >fractions > >> >cannot be represented accurately - period. > > >> I did make it clear. Here's the thread: > > >> >> On Mon, 31 Mar 2008 21:55:10 -0300, Clark F Morris > >> >> <cfmpub...(a)ns.sympatico.ca> wrote: > > >> >>>Try doing a simple divide like calculate the value of 1 / 5 in binary. > > >> >> OK. > > >> >> 01 numerator value 1 binary pic 9(9). > >> >> 01 denominator value 5 binary pic 9(9). > >> >> 01 quotient binary pic 9(9)v9(4). > > >> >> compute quotient = numerator / denominator > >> >> display quotient > > >> >> 0000000002000 > > >> >>>You get a never ending fraction. > > >> >> Looks pretty diadic to me. > > >> >That's because you did not do it in binary, you did it in decimal. > >> >One-fifth is rational in decimal, but irrational in binary. > > >> >Nope, all three numbers are binary INTEGERS. The fallacy is in thinking > >of quotient as a fraction. It is not a fraction, it is an integer. > >> > The compiler SCALED two integers by multiplying by powers of 10, then > >divided two binary integers to get a quotient with no fraction. > >> > Binary 10 divided by binary 5 gives binary 2. > > >> >101111101011110000100000000 / 1100001101010000 = 11111010000 > > >This isn't the point. The point is that with very few exceptions a decimal > >number with figures on both sides of the decimal point can't be represented > >as a pure binary number, as the fractional portion is almost always > >non-terminating. Binary ONE divided by binary 5 gives binary .00110011...... > >A scaled number is not the number itself: it's a transformation of the > >original. > > By definition, integers are whole numbers. There is no fractional part to an integer. > > >Incidentally, how do you decode that string of digits? > > I paste it into a calculator and hit Dec. Hardware does it by doubling the answer and > adding each bit, left to right. > > >> >Incidentally - everybody knows about the simple algoritm to convert base > >10 > >> >numbers to base 2 - i.e. repeated division by 2 - > >> >I had some fun working out the mirror algorithm to convert decimal > >FRACTIONS > >> >to binary fractions. (Very easy, actually). (Both algorithms work for > >any > >> >base). But what puzzles me is I've never seen this written up anywhere. > >> >I'm certain that the early developers of computers (or perhaps IBM > >research > >> >fellows) must have discovered the algorithm. Has anyone seen it written > >up? > > It's of no use when dealing with integers, because they have no fractional part. But they do have fractional parts. If I have my stock valuation total in cents and I have the quantity in units then when I divide to find the average cost per unit I have a fractional part to deal with. This may be rounded using various methods but accounting practice requires this be done in decimal, and binary rounding does not give accurate decimal rounding. > > >Very interesting! But doesn't answer my question. I'd really like to make > >an original contribution to math or computer science but I can't believe > >that "my algorithm" is it. > > Now you're in the domain of abstraction and generalization, a faux pas in these parts.
From: Richard on 8 Apr 2008 01:41 On Apr 8, 2:12 am, Howard Brazee <how...(a)brazee.net> wrote: > On Fri, 04 Apr 2008 21:19:36 -0600, Robert <n...(a)e.mail> wrote: > >That sort of ad hominem is what government and big corporations do when caught in a lie to > >which they have no rational defense. Members of this august forum have more artful > >responses. > > >Robert: The sun rises in the east. > > Most of the following was amusing - but the basis is that your > statement is one that nobody but a fool could argue with. To be > more accurate, you should have started off with: > > Robert: Dogs have spots. I think that clearly indicates the real problem. Robert really does believe that his assertions and generalisations actually are 'universal truths'. He really thinks that whatever he says _is_ equivalent to 'the sun rises in the east'.
From: Richard on 8 Apr 2008 01:54 On Apr 8, 1:24 pm, Robert <n...(a)e.mail> wrote: > On Mon, 07 Apr 2008 08:12:37 -0600, Howard Brazee <how...(a)brazee.net> wrote: > >On Fri, 04 Apr 2008 21:19:36 -0600, Robert <n...(a)e.mail> wrote: > > >>That sort of ad hominem is what government and big corporations do when caught in a lie to > >>which they have no rational defense. Members of this august forum have more artful > >>responses. > > >>Robert: The sun rises in the east. > > >Most of the following was amusing - but the basis is that your > >statement is one that nobody but a fool could argue with. To be > >more accurate, you should have started off with: > > >Robert: Dogs have spots. > > If two out of two dogs had spots, concluding that all dogs have spots would probably, but > not necessaryily, be a hasty generalization. If a thousand out of a thousand dogs had > spots, the generalization would probably be correct. There is a line somewhere between two > and a thousand. Determining where it is requires independent knowledge of the standard > deviation between dogs, and a stated confidence interval. If the deviation where known to > be zero (dogs are clones), a sample size of one would be adequate. If the goal were > certainty, abstraction and generalization would be impossible because we would have to > sample every dog in the past, present and future. > Yeah, but all that rationalisation and argument is completely irrelevant when I have a dog here that has no spots. And that is the real problem with most of your arguments. You have even claimed that the deviation was zero and thus a sample size of one or two was sufficient to claim 'all' when _my_ sample was not like yours at all. > As a rule of thumb, moderate precision (90%) measurements of human opinion require 50-200 > samples; high precision (98%) requires 1,000-2,000. Haven't you seen code written by 100 > programmers? But taking sample opinions of 1000 southern baptists merely shows them _all_ to be wrong.
From: Howard Brazee on 8 Apr 2008 10:47
On Mon, 07 Apr 2008 22:39:38 -0500, Robert <no(a)e.mail> wrote: >In my experience, there are two kinds of shops: those run by managers and those run by >workers. They're split about 50-50. In the ones run by managers, refusing to do something >is grounds for dismissal (as it should be). In the shops run by employee workers, the most >common reason is posing a threat by being better than them. Managers want 'the best'; >workers want to protect the status quo. I haven't come across the 2nd type of shop - but I suspect that if I had, I'd find that the conflict is where at least one party KNOWS that he is right. When that happens, nothing else matters. This is the human characteristic that not only is the cause of most wars, but it also gets in the way of progress of all kinds. |