(1)  Any real number that cannot be expressed as a ratio of a/b where a , b are integers and b is not equal to zero is an irrational number.

(2) The decimals of irrational numbers cannot be expressed in terms of repeating or terminating numbers.

(3) The story goes that Hippasus, a student of famous Greek mathematician Pythagoras, discovered irrational numbers when he tried to

      express square root of 2 as a fraction. Pythagoras did not agree but could not disprove Hippasus. However  other  pupils   became

      jealous and threw Hippasus overboard while in a boat and he drowned.

(4) Difference with Transcendental numbers : Transcendental numbers can never be expressed as the roots of polynomial equations

     where the co- efficient are   integers, rational or algebraic.

     (Equations   involving addition, multiplication,  subtraction  or  division. No infinite series, logarithms, or trig. functions.). PI , Phi ,  the

     golden angle are transcendental numbers whereas square root of 2 is an irrational number.

e (Euler's number)
√ 2
Log 2
Log 10
Log e 2
Loge 10

(5) There is a beautiful relation among 2 most conspicuous irrational numbers and the imaginary number, thanks to Euler.

     e = -1

(6) Among the irrational numbers,some are more irrational than others. For example, golden ratio i.e  φ is more irrational than e or π .The

     decimal places of PI is 0.141592654.... that is close to 1/7 which is 0.142857 and e which has decimal 0.71828 is close to 5/7   which   is

     0.7142885. Phi does not have any proximity to any such numbers.

(7) Some of the irrational numbers are expressed in terms of series by Ramanujan-

      π/2 =(2/1)*(2/3)*(4/3)*(4/5)*(6/5)*(6/7)........

   √2 = (2/1)*(2/3)*(6/5)*(6/7)*(10/9)*(10/11)........