This is mathematically equivalent to ℝ\ℚ, the set difference between the reals and the rationals
(as a side note, if you are looking for those characters ( the R and the Q) or wonder how to make them with your keyboard, they are called blackboard bold. Have a look at the link.)
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There is no special symbol for Irrational Numbers in general. There are symbols for some special numbers, such as pi and e.
No. Radical(1.21) = 1.1, for example, is rational.
No. The sum of an irrational number and any other [real] number is irrational.
A negative irrational number can be thought of as an irrational number multiplied by -1, or an irrational number with a minus sign in front of it.
If an irrational number is added to, (or multiplied by) a rational number, the result will always be an irrational number.
Any irrational number will, eg √2, π, e.