Does the set of irrational numbers form a group?
No - Irrational numbers are not closed under addition or multiplication.
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#
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
If you try the same with cube roots, you find that you need to consider numbers like:
More generally, if