**IRRATIONAL NUMBERS
**

(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.

**e ^{iπ}
=
-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)........}