Slime wrote:
> 
> So, it's kind of interesting that when we can't come up with a way to 
> describe a number besides "one of the roots of this degree-5 
> polynomial," we typically consider that less descriptive than the value 
> of an expression with radicals.
> 

I suspect that the reason here might be largely historical.  IIRC long 
ago, the concept of what number could be defined "exactly" was 
essentially derived from the Greek notion of which number could be 
constructed using only a straightedge and a compass.  It turns out that 
everything you can write with integers, addition, subtraction, 
multiplication, division and square roots can be written in this form. 
Thus if you had a number written in this form, you knew that it could be 
created with a compass-straightedge construction, but other things like 
pi, cube roots, etc. were iffier -- particularly so with roots of 
general polynomials.  This probably set the stage for looking at 
"allowed numbers" as being build by the combination of a set of basic 
functions, even when the set of allowed elementary functions was 
extended to allow things like nth-roots, exp and log which don't 
correspond to circle-straightedge constructions.  Nowadays computers 
allow us to compute roots to arbitrary polynomials relatively 
efficiently, so this way of thinking about things isn't as useful as it 
used to be.

Caveat: The above is pure conjecture.

Post a reply to this message