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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