tag:blogger.com,1999:blog-11573761.post8399944726133976270..comments2023-09-23T07:38:46.925-07:00Comments on Chris Quirke's Blog: Understanding IntegersChris Quirkehttp://www.blogger.com/profile/05538828571660803875noreply@blogger.comBlogger2125tag:blogger.com,1999:blog-11573761.post-61675304794673400532008-01-03T02:09:00.000-08:002008-01-03T02:09:00.000-08:00On "What does it mean for a number to be irrationa...On "What does it mean for a number to be irrational and how do we represent those values?", I'd say the concept of rational/irrational numbers, as distinct entities, is false. <BR/><BR/>Instead, it's a matter of convenience as to how deeply you want to specify rational numbers. If to only 2 decimal places, then 3.457 is as "irrational" as Pi. If to zero decimal places, then you're talking integers.<BR/><BR/>The Halting Problem implies it is impossible to predict what "irrational" numbers are truly irrational, vs. rational numbers that happen to have an inconveniently high computational overhead to specify exactly what they are.<BR/><BR/>Some of that overhead may be skewed by the type of calculations you use, e.g. it may be "easy" enough to describe 22/7 as rational, but "too difficult" to pin down the same value to x decimal places if using x.xx notation rather than x/x notation.<BR/><BR/>Whereas there's no deep conceptual difference between rationals and irrationals, there is between integers and rationals, in that some contexts only "make sense" for integers. <BR/><BR/>For example, it's not meaningful to speak of half a person, given that sawing a person in half destroys the property of personhood.<BR/><BR/>On division by zero, I agree this gives the (non-)value of infinity, and that leads one to consider the nation of infinity. <BR/><BR/>Cues from approaching values suggest "very large", but some contexts are ambiguous, especially where sign is concerned. <BR/><BR/>Rather than "very large", it may be "all possible values", e.g. where the negative side of the X axis tends to negative infinity on the Y axis, while the positive side tends to positive infinity, as both sides approach zero on the X axis.<BR/><BR/>Sometimes these wobblies can be dispelled by redefining the axes, e.g. from linear to log.<BR/><BR/>Our problem with infinity is that we tend to think of it as "very large", whereas it is a qualitatively different concept. Rudy Rucker groks this, when he describes the four "mind tools" as number/measurement, space/geometry, infinity/whole and information.<BR/><BR/>IKWYM about this opening a large and general can of worms, reminding me of something I'll blog next...Chris Quirkehttps://www.blogger.com/profile/05538828571660803875noreply@blogger.comtag:blogger.com,1999:blog-11573761.post-70283459765999667012008-01-02T04:16:00.000-08:002008-01-02T04:16:00.000-08:00Incredible... I've been kicking around this same t...Incredible... I've been kicking around this same thought for the better part of a 7 years now and specifically "rationalized" my thinking about integers the other day, much as you have laid out here.<BR/><BR/>What does it mean for a number to be irrational and how do we represent those values? For that matter, how do we represent whole numbers? Whole numbers are really just as arbitrary as rational and irrational numbers. Our precision, or perhaps more precisely our resolution limits our ability to find an exact representation. Some numbers are irrational simply because we do not have a vocabulary that allows us to describe them.<BR/><BR/>It is the limitations in the framework of current knowlage that make us blind to other possibilities. I think it is hard for most people to accept the idea of more than 4 dimensions (3 spatial + time) because it is an intangable experience for them.<BR/><BR/>One of the first concepts that I've been challenging is the idea that division by 0 is undefined. What I believe works is that division by 0 is actually infinity. In fact, it wouldn't be too difficult to argue that lim x -> 0 of 1/x is +infinity and that lim x -> 0 of -1/x is -infinity. Left at that, we create a massive discontinuity. But what if we define infinity as the discontinuity between -infinity and +infinity. Now we have bridged both ends of the spectrum just like we do at -0 and +0.<BR/><BR/>This is distasteful because it is a departure from Euclidian geometry and doesn't allow for the Cartisian coordinate system that we use today. But it doesn't have to. Curvature k is defined as d(tangental angle)/d(arc length). As we have defined division by 0 now we can have a lim arc length -> 0 and the tangental angle would continue to decrease. At some value +0 or -0, there would effectively be no curvature. This is the realm of the Cartisian planes that we are taught in elementry school.<BR/><BR/>Any number that we can think of will exist in that coordinate system, and will be so much closer to 0 than infinity, that for all intents it will fall along a straight line. Like you, this is more of an RFC than fact... I'd certainly like to explore this further and maybe someone else already has.<BR/><BR/>Now, this is where I came in the other day, and where I see your examination of rational numbers syncing up closely with my logic. Each number, 7.45 to use your example, is defined as sum of discrete parts. Today we define each units place as posessing a unit value of 0-9; an offset 7 from outside the system if you will. In fact we are taught that this is 7/1 + 4/10 + 5/100. The reality is that it is just as appropriate to construct that rational number from 37/5 + 1/20. The sum of these pieces are just that, pieces. What previously took me 3 informational quanta to describe was reduced to 3 informational quanta. This made me quesiton the atomic nature of integers.<BR/><BR/>Look at http://books.google.com/books?id=-vPtcriflH0C&pg=PA35&lpg=PA35&dq=distribution+of+floating+point+numbers&source=web&ots=TYDpMYHYUB&sig=StBi2X9J_MaSC80vPOvfrGgjtaY<BR/><BR/>On that page, a little down the page, you will see the output from floatgui.m, a MatLab program to graph the distrubution of floating point numbers. If we have these holes in our system of floating point numbers, might we have similar gaps in our number system that we can't describe; similarly, might this be where we find our irrational numbers?<BR/><BR/>While I know why we have our limitations in floating point, I can't help but wonder if we have similar deficiancies in rational numbers. Floating point works well with numbers closer to 0, perhaps rational numbers have a similar hidden defect. For numbers close to 0, i.e. anything that you can actually compose, rational numbers serve us very well, but as you transition to numbers closer to infinity you begin to run into more irregularities.<BR/><BR/>The uncertainty falls out of this problem. The more precisely you define something, the more likely you are to find one of these areas that you cannot define. A particle that is at rest somewhere between x and x+1 will need to be defined as being either x or x+1 when it really falls somewhere in between.<BR/><BR/>This is where I get stumped. Doesn't this suggest that there is a finite resolution to everything? We are told that there are just as many numbers between 0 and infinity as there are between 0 and 1. To me this seems to suggest otherwise.<BR/><BR/>What is surprising me more is that I can seem to apply this "bifurcation" analysis to everything from math and science to philosophy and religion. I'm going to have to kick this around for a few more years, but for certain it is something that I'm going to write more about at a later date.Anonymousnoreply@blogger.com