Hippasus, a Pythagorean philosopher is believed to have proven the existence of numbers that are not rational. However, this is not quite the same as "known for". Many (most?) mathematicians would not know who Hippasus was or what he did.
No. Irrational numbers by definition fall into the category of Real Numbers.
Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
Yes, irrational numbers are never rational numbers because irrational numbers can't be expressed, by definition, as a fraction of two integers.
Yes. The definition of a surd is "an irrational number"
Pi is irrational. Irrational numbers, by definition, have no factors.
the collection of rational and irrational numbeers is known as real numbers
It is a non-terminating, non-repeating decimal representation. That is a definition of irrational numbers.
The set of real numbers is defined as the union of all rational and irrational numbers. Thus, the irrational numbers are a subset of the real numbers. Therefore, BY DEFINITION, every irrational number is a real number.
Because that is the definition of irrational numbers: They are the numbers that cannot be written as a ratio of two integers! A fraction is a ratio of two integers.
Pi and the square root of two are irrational numbers.
All irrational numbers are Real numbers - it's part of the definition of an irrational number. Imaginary numbers are neither rational nor irrational. An example of a number that is both Real and irrational is the square root of two. Another example is the number pi.
There are an infinite number of irrational numbers between 2 and 4. See the link below for the definition of irrational numbers. The two most popular irrational numbers between 2 and 4 are pi (3.14159...) and e (2.71828...).