precision and reals

Cubist at aol.com Cubist at aol.com
Sun Apr 11 22:03:33 EDT 2004


sez norman at mrsystems.co.uk
>>  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*!


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