What are irrationals?

Irrational numbers

 A number that can not be written in the form p/q, where p and q are integers and q is non zero. So we can say a number which is not rational is irrational. Thus, √2, √5, π, etc. are such examples of irrational numbers.

In the view of decimal expansion

As we  know the decimal expansion of √2 =.  1.4142135623730950488016887242096..., π = 3.14159265358979323846264338327950...., e = 2.71828183, etc., which are irrational numbers.

In above examples we have seen that decimal expansion is non terminating as well as non repeating. Now, let us summarize our results in the following form:

The decimal expansion of a irrational number is neither terminating nor non terminating recurring (non repeating). Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.

         The collection of irrational numbers is denoted by Q' (Compliment of Q). 

The Greek genius Archimedes(287 BC - 212 BC) was the first to compute digits in the decimal expansion of π. He showed 3.140845 < π < 3.142857. Aryabhatta (476 AD - 550 AD), the great Indian mathematician and astronomer, found the value of π correct up to four decimal places. Using high speed computers and advanced algorithms, π has been computed to over 1.24 trillion decimal places.

Properties of irrational numbers:

• No fraction is irrational number.
• No natural number is irrational number.
• The integer 0 is not irrational number.
• All integers are not irrational numbers.
• Here we can't say what is next irrational number, i.e. we can't say successor or predecessor of a irrational number as there are infinitely many irrational numbers, even uncountable.
• Irrationals have a denseness property, i.e. the closure of a collection of irrational numbers is closed.
• Between any two given irrational number, there are infinitely many rationals as well as irrationals.
• Square-root of every every positive integer which is not perfect square is irrational number.
• There is nothing common between rationals and irrationals.