On Thursday, September 20, 2012 10:23:15 PM UTC-5, calvin wrote: > After giving so much thought to the existence of a real sqrt of 2, it suddenly occurred to me that one can trivially prove the existence of any root of any positive real number by exactly the same argument and, for example, The 27th root of 113 is the least upper bound of the set of all positive real numbers whose 27th power is less than 113. Big deal. It's really not saying anything. Once you assume the completeness property, then all of this stuff follows, trivially. What's the sqrt of 4? Why it's the least upper bound of the set of positive real numbers whose square is less than 4. Whoopie do. That still doesn't tell us that 2 times 2 is 4.
Why does 113 need so many roots? I mean, like, most numbers do well on just 7 or 8 roots in order to anchor the number and bring it moisure and nutrients.