We know that a rational number is a number that can be written in the form of [katex]\frac{p}{q}[/katex] where p, q [katex]\in[/katex] Z and q ≠ 0. The numbers 25, 3.765, 4.5, etc., are rational numbers. can be reduced to the form [katex]\frac{p}{q}[/katex] where p, q Z, and q ≠ 0 because [katex]\sqrt{25}=5[/katex]
Examples of rational numbers:
i) 0.26= is a rational number.
ii) 3.6666… is a recurring decimal, it is a rational number.
iii) 5.333…is a rational number.
iv) 0.142857142857… is a rational number.
What is Irrational Numbers:-
Irrational numbers are those numbers that cannot be put into the form of [katex]\frac{p}{q}[/katex] where p, q [katex]\in[/katex] Z and q ≠ 0. The numbers [katex]\sqrt{7},\sqrt{11},\sqrt{\frac{5}{16}}[/katex] are irrational numbers.
Examples of Irrational numbers:
v) 0.0100100010100000001 … is a non-terminating, non-periodic decimal, so it is an irrational number.
vi) 287.12116283873112373573 … is also an irrational number.
vii) 1.4142135 … is an irrational number.
viii) 7.3205080 … is an irrational number.
ix) 1.709975947 … is an irrational number
. x) 3.141592654… is an important irrational number called it (Pi) which denotes the constant ratio of the circumference of any circle to the length of its diameter i.e.,
[katex]\pi = \frac{circumference \: of \: circle}{ diameter \: of\: circle}[/katex]
An approximate value of [katex]\pi[/katex] is [katex]\frac{22}{7}[/katex] ,a better approximation is [katex]\frac{355}{113}[/katex]and a still better approximation is 3.14159. The value of [katex]\pi[/katex] is correct to up to 5 lac decimal places and has been determined with the help of the computer.