There is no single mathematician. It has been known since ancient times that the length of the diagonal of a unit square is sqrt(2) which is irrational.
yes * * * * * No. Rational and irrational numbers are two DISJOINT subsets of the real numbers. That is, no rational number is irrational and no irrational is rational.
Not necessarily. The sum of two irrational numbers can be rational or irrational.
There are an infinite number of irrational numbers.
All irrational numbers are not rational.
All rational and irrational numbers are real numbers.