[reikonomusha]

You can calculate, measure, draw, construct, write a power series for, express as hypergeometric function, etc. the Bring radical too.

All of these concepts, from sine to real numbers, Bring radicals to complex exponentials, can all be defined in different, equivalent ways. What is interesting are the properties invariant to these definitions.

It still doesn't seem to me that a square root should be any more or less contrived than a Bring radical. Maybe we should call it a ultraradical instead?

[xigoi]

For me, what makes the square root more “natural” is that, although it’s usually introduced as an “answer by definition”, it can also be arrived at by wondering what happens if you take something to the halfth power.