Is Pi(π) an Irrational number?

Is Pi(π) an Irrational number?

Pi(π) is an interesting personality in the world of mathematics. Actually pi is  a mathematical constant. Pi is approximately equals to 3.14159265.  In the short form , for many(almost all) mathematical calculations we use π=3.14.

      Originally, we define pi(π) as the ratio of a circle's circumference to its diameter.
       

      It is true that, π=22/7 is nothing more than an easy approximation to pi. In reality π is never equals to 22/7 exactly.
 Now we will discuss about the reason why pi is an irrational number.
 In this case, firstly  we should clear about the definitions of rational and irrational numbers .A number which can be expressed as the ratio of two integers  "a" and "b" such that "b" is not equals to zero; is called a rational number. And a number which is not a rational number, is called an irrational number . But, Pi can  not be expressed as the ratio of two integers 'a' and 'b'  such that 'b' is not equals to zero.So, pi  is not a rational number.Thus  pi  is an  irrational number.

Lambert's proof:

In 18-th century(in 1761)  Lambert proved that pi is an irrational number.

 Lambert first proved that the following  continuous fraction expansion holds.

Then he proved that , this expression  is irrational for rational x(not equals to zero).

Now, tan(π/4)=1.

So, π/4 is irrational and consequently, π is irrational.

This is the Lambert's proof.

Many other mathematians like Hermite, Niven, Cartwright also proved that pi is irrational in different ways.

     Actually π is endless. It means that pi has infinite number of digits in its decimal representation. Till now 2.7 trillion digits of pi are known as per the record.
  π=3.141592653589793...
    One more important thing about π is:
In trigonometry , π is very useful. in trigonometric calculations we use,
π radians=180 degrees.

  

 If you find out any incorrect information or know anything more about this , please write it in the comment section!