An irrational number is any real number which can not be expressed in the form of p/q , where p and q are integers.The irrational are many and uncountable.
Some examples: e, pi and sqrt19.
We know e = Lt (1+1/x)^x as n--> infinity. The value of e is like 2.71828...
We know pi = Length of circumference of a circle/length of diameter of that circle. the value of pi is like 3.14159...
Square root 19. The value we know is between 4 and 5.
The knowledge of pi is essential in geometry. It is used to know the length of circumference given the radius or diameter.
Rather than use , the invention of irrational number has contributed to the completeness of the idea of numbers. It gave a sense of enlightement that between any two distinct perfect square roots , there are still more square roots which need not be in p/q form alone.