Does the set of irrational numbers form a group?

The set of irrational numbers does not form a group under addition or multiplication, since the sum or product of two irrational numbers can be a rational number and therefore not part of the set of irrational numbers.

About the simplest examples might be:

`sqrt(2) + (-sqrt(2)) = 0`

`sqrt(2)*sqrt(2) = 2`

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Footnote

Some interesting sets of numbers that include irrational numbers are closed under addition, subtraction, multiplication and division by non-zero numbers.

For example, the set of numbers of the form `a+bsqrt(2)` where `a, b` are rational is closed under these arithmetical operations.

If you try the same with cube roots, you find that you need to consider numbers like: `a+broot(3)(2)+croot(3)(4)`, with `a, b, c` rational.

More generally, if `alpha` is a zero of a polynomial of degree `n` with rational coefficients, then the set of numbers of the form `a_0+a_1 alpha+a_2 alpha^2+...+a_(n-1) alpha^(n-1)` with `a_i` rational is closed under these arithmetic operations. That is, the set of such numbers forms a field.