i understand generators of finite groups but....?

Question

....how do we generate the real numbers as a group under addition?

Answer 1

well the set R or real numbers doesn't have a finite number of generators under addition, the same for Q, the set of rational numbers.

Well any group can be generated by itself, but this is kind of trivial
or R can be generated by the set N of natural numbers and the interval (0,1), for example
So it has an infinite number of generators.

It would interesting to see what is the cardinal of a minimal set of generators for R. I think it's the cardinality of R.

Answer 2

Well, the real numbers do not have a finite set of generators.

This is more along the lines of Fields. A Field is a group under + and where F - {0} is a group under *. It is normally assumed that 1 is an element of the Field and (it has been awhile) I believe the groups are abelian.

So you can take the Field Z and look at polynomials with coefficients in Z. The roots of all these polynomials forms a field -- this is the field of algebraic numbers.

Now algebraic numbers are countable so clearly not all the real numbers. But the point here is that there are other ways to generate infinite groups then generators.

Actually, the concept of algebraic extensions is quite general.

Now, another way to generate the real numbers is to take the closure of the algebraic numbers. You can define + and * on these limits in the obvious way.

Answer 3

You can't do it with a finite number of generators. If you COULD do it with a finite list generators, you'd immediately be able to prove that the reals were countable.