sez [EMAIL PROTECTED] >> Reason numero uno: A rational number *is* a real number -- the former >> is a subset of the latter > >Is 1/3 the same as 0.333. (0.333 recurring)? Yes, it is.
>Squaring 1/3 = 1/9. What does squaring 0.3333. give? If you square the infinitely recurring decimal .3333..... you get the infinitely recurring decimal .11111... So yes, squaring 1/3 gives you 1/9 regardless of whether you do it as real or rational. Just for grins, try squaring .3, .33, .333, and so on, with an ever-increasing number of 3s after the decimal point. Are there any discernable patterns in the results? >It is a long while since I did my mathematical studies, so I could be >wrong. However, my memory is that the definition of the number 1 as a >real is the limit of 0.99999 recurring. It is -- but .99999... is *not* what you get when you square .3333... You appear to be confusing two different quantities here. >All results, when using real numbers, are limits. Nope. All results, when using *limited-precision approximations of* real numbers, are limits. >However, as to squaring being the inverse of taking a square root try >this: > >Take any calculator, computer, abacus, pen and paper - whatever >calculating device you like - and take the square root of 2. Repeat a >large number of times. Eventually you end up with the answer 1. (The OS >X calculator, when it first shows 1, still holds a decimal part - >continuing long enough makes it disappear). Square 1 as many times as >you like and you will not get back to 2. >This 'inaccuracy' can appear to be an artefact of the limits of the >calculating device. It is not. Sure it is. A calculating device with *infinite* precision *would not* exhibit the behavior you describe above. >The problem is that, once a result >needing an infinite number of decimal places to represent it enters the >system, most operations on it do not have an inverse. Nope. There's the infinite-precision real number you'd *like* to work with... and there's the *approximation to* that real number that you're *forced* to work with, when your "system" only allows for *finite* precision. Two different numbers, even if they are generally very close to one another. >Take pi. This number is known to great accuracy. When taking the square >root of pi do you operate on: > >3.14 >3.142 >3.1416 You use whichever you like; just be aware that each of those numbers is merely an *approximation* *of* the *true* value of pi. How close of an apporximation is good enough? You tell me... >Clearly, however many decimal places you choose, squaring the result >will not yield pi. Sure -- because any *finite* number of decimal places in the expansion of pi *is* *not* *pi*. Why would you expect to get *pi*, if you square the root of some number which *isn't* pi? >Anyway, I suggest any mathematical castigation I deserve be sent off >list as I don't won't to clog up the list, Actually, I thought that your misconception touched on a point that's well worth reminding people of: The limits of precision in our computing machinery. Just as "the map is not the territory", so it is that an N-digit approximation of a real number is not *the number*! _______________________________________________ use-revolution mailing list [EMAIL PROTECTED] http://lists.runrev.com/mailman/listinfo/use-revolution
