(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.
(5) There is a beautiful relation among 2 most conspicuous irrational numbers and the imaginary number, thanks to Euler.
eiπ = -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-
√2 = (2/1)*(2/3)*(6/5)*(6/7)*(10/9)*(10/11)........