use-revolution Digest, Vol 7, Issue 55

Norman Winn norman at mrsystems.co.uk
Sun Apr 11 19:42:42 EDT 2004


>  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)? Squaring 1/3 = 1/9. What 
does squaring 0.3333. give?

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. Anyway, I am willing to be 
wrong on this. I am not at home so cannot consult the books from which 
my learning came.

All results, when using 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. 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.

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

Clearly, however many decimal places you choose, squaring the result 
will not yield pi. Algebraically, this is no problem.


Anyway, I suggest any mathematical castigation I deserve be sent off 
list as I don't won't to clog up the list,

Norman



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