I'm a math newbie, so an intuitive explanation is the most helpful for me, but don't pull your punches with the formulas, if you feel like it.
We can construct the rational numbers using the division operation. Take integers a and non-zero b and we can have any rational with the formula a/b.
Similarly, we can define a set of numbers using the root operation. Take positive rationals a and b, and we can have any - let's call them "root numbers", since I don't know any better term for them - a√b, (ath root of b)
Now, most of the root numbers are certainly irrational, but they are countable and algebraic. Also, I think that root numbers are a superset of rationals, since you can write any rational r using 1√r. My questions are:
- Is the root operation with all the four basic operations enough to express any algebraic number? Or do you need something else too, like the exponentiation operation? Is a^b with any rationals b and positive a definable without using the exponentiation?
- Any rational number can be expressed using only one division. If we define root numbers to be expressed using only one root operation, am I right if I say that even if the question 1. is true, using this limited format, we can't express all the algebraic numbers? (Since you could have something like (a√b)√c, and since we originally defined the root numbers with rationals and that is using the root operation with root numbers itself so the result isn't maybe contained in the root numbers?)
The reason which got me into this, is that you can express a limited set of numbers, like rationals or the root numbers using a fixed formula, and that's convenient if you'd need to define some number exactly when programming with a computer using a limited "structure" of storing the number - but I started wondering if there were any such a "fixed" formula for the whole set of algebraic numbers.